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Beginning
in the early 1930's and extending through the early 1960's, a number of
researchers have proposed various schemes and hypotheses to explain
population dynamics and commonly observed population interactions in the
field. Leading authorities have been Smith (1935), Nicholson (1933),
Nicholson & Bailey (1935), Solomon (1949), MIlne (1957a,b, 1958),
Thompson (1929a,b, 1930a,b, 1939, 1956), Andrewartha & Birch (1954), Lack
(1954), and Holling (1959), Watt (1959), Chitty (1960), Pimentel (1961), to
mention some of the more vociferous authorities. Toward the end of this
period considerable conflict of opinion developed with the introduction of
ideas by Turnbull (1967), Turnbull & Chant (1961), van den Bosch (1968),
Force (1972), Huffaker (1958), Huffaker et al. (1963, 1971). Presently the
debate continues with publications by Ehler (1976), Ehler & Hall (1982),
Hokkanen (1985), Hokkanen & Pimentel (1984), Goeden (1976), Myers &
Sabbath (1980). The matter
of population interaction is too complex for simple discussion; the subject
must be treated in a mathematical manner, separating the various kinds of
population systems (eg., single host single natural enemy, single host
multiple natural enemies, multiple hosts multiple natural enemies, patchy
distributions versus relatively uniform ones, etc., etc.) In general the
modern theories (since the 1960's), which encompass many different types of
interactions between species, accord well with the observed outcomes of
experimental populations in both laboratory and natural settings. This is
especially true for simpler systems (e.g., laboratory studies, studies of
single species), partly because there is more data available on such systems.
In each of the many systems that have been studied during recent times, from
the simple single-species systems to the multispecies systems, the theories
indicate a complex range of dynamics that can arise from simple regulatory
mechanisms. This range generally includes stable equilibria, stable cyclic
behavior and chaotic dynamics. The presence of these behaviors is standard to
systems, which include time-lags or time-delays, such as the developmental
time between oviposition and adult emergence in insects or the time between
infection and subsequent infectiousness of diseased individuals. The actual dynamics of any particular system
depends on the strength of the interactions among the member species. Thus
increased intensity of competition leads from stable equilibrium to cyclic behavior
in single-species and competing-species systems. Increasing the effects of
delayed density dependence in host-parasitoid systems (by decreasing the
contagion in attacks among hosts) leads from stable equilibrium (when
contagion is significant) to unstable cyclic behavior when search is more
independently random. Inherent overcompensation in some host-pathogen models
leads directly to cyclic behavior without any intervening sphere of stable
equilibria (Bellows & Hassell 1999). For many of the systems theory has been
developed for both homogeneous and patchy environments. In general, patchy
environments permit greater degrees of stability for most interactions, even
permitting global persistence of interactions that are intrinsically unstable
in homogeneous environments. A thorough updated review on the subject of
population regulation was presented by Bellows & Hassell (1999), which
shows the complexity of considerations necessary and offers a clearer
explanation for some of the possible population interactions. These authors
emphasized that natural regulation of populations necessarily involves
interactions among species. By understanding the potential and likely
outcomes of these interactions and the relationships between particular
mechanisms and their consequences, we can better interpret the outcomes of
biological control experiences and better direct future efforts toward
achieving goals of population suppression and regulation. Issues of natural population regulation lie at
the core of biological control. Characteristic of "successful"
biological control are the reduction of pest populations and their
maintenance about some low, non-pest level. Such outcomes are frequently
recorded as being achieved (e.g., DeBach 1964), but documented evidence is less
common (Beddington et al. 1978). The reduction in density of the winter moth,
Operophtera brumata Cockerell in Nova
Scotia following introduction of natural enemies is one such example, while
in the laboratory similar outcomes have been reported. The objective of
biological control programs is to enhance such natural control of
populations, and an understanding of the principals involved in biological
control necessitate an appreciation of mechanisms of population regulation. Biological control has as a principle aim the reduction
of pest species. In this context the objectives are two-fold, first to reduce
or suppress the density of the species and secondly to regulate
the pest species around this new lower level. Thus there are two concepts,
suppression and regulation, which encompass the objectives of biological
control. While mechanisms of population suppression are in many cases as
simple as increasing the level of mortality acting on a population, issues of
regulation, or what will be the dynamical behavior of the population once the
new mortality factor has been added, are more complex and can be affected by
density-dependent responses of both the pest and natural enemy population,
natural enemy search behavior, patchiness of the environment, additional
natural enemies in the system, and other interactions, both behavioral and
stochastic, among the populations. (Please see Legner et al. 1970, 1992, 1973, 1983, 1983, 1975, 1980). These questions of population suppression and
regulation have been the subject of a considerable amount of research, both
theoretical and experimental. It is then well to consider features of
interacting population which can contribute to either suppression or
regulation (or both). The discussion begins with single species systems and
interspecific competition, proceeds to interactions between a host or prey
and a natural enemy, and concludes with considerations of systems with more
than two species (of either prey or natural enemy). (Bellows & Hassell
1999). The topics are developed generally within an analytical framework of
difference equations but, where these are significantly distinct, also
consider the implications of continuous-time systems. The implications of
heterogeneous environments are also addressed, where resources such as food
plants or prey are distributed in patches (rather than homogeneously) over
space. In general theories and mechanisms are considered which are supported
by experimental evidence as having some effect on the dynamical behavior of
populations. Although there is an abundance of information on the effects of
herbivory on the performance of plants, there is little data on the effects
of insect herbivory on plant population dynamics (Crawley 1989). For this
reason most of the discussion on hosts and natural enemies is centered on
interactions of populations of insect predators and parasitoids and their
prey, interactions for which there exists a large body of literature on experimental
investigations (Bellows & Hassell 1999, Hassell 1978). Single age-class
systems Single-species population dynamics has
relished a long history of both theoretical and empirical development, centering
largely around mechanisms of population growth and regulation. The structure
in which the concepts are developed is one of population growth in discrete
time, where the population consists largely of individuals of only a single
generation at any one time. Such populations are characteristic of many
temperate insects and additionally of many tropical insects which occupy
regions with pronounced wet and dry seasons. The algebraic framework is
straightforward: N_{t+1} = Fg(N_{t})N_{t}. (1) Here N is the host population denoted by
generations t and t=1, and Fg(Nt) is the per capita net
rate of increase of the population dependent on the per capita fertility F
and the relation between density and survival g (which is density
dependent for g<1). The fundamental concept represented in
equation (1) regarding population regulation is that some resource, crucial
to population reproduction, occurs at a finite and limiting level (when g=1,
there is no resource limitation and the population grows without limit).
Individuals in the population compete for the limiting resource and, once the
population density has saturated or fully utilized it, the consequences of
this intraspecific competition bring about density dependent mortality and
growth rates reduced from the maximum population potential. Such competition
can be by adults for oviposition sites (e.g., Utida 194_, Bellows 1982a), by
larvae for food (e.g., Park 193_, Bellows 1981), or by adults for food (e.g.,
Park 193_, Nicholson 1954). The dynamics of populations subject single
species competition in discrete generations can span the range of behaviors
from geometric (or unconstrained) growth (when competition does not occur),
monotonically damped growth to a stable equilibrium, damped oscillations approaching
a stable equilibrium, through cyclic behavior. The type of behavior
experienced by any particular population is partly dependent on the
mechanisms and outcome of the competitive process. Species with contest
competition have more stable dynamical behavior, while species with scramble
competition may show more cyclic or oscillatory behavior (May 1975, Hassell
1975). Most insect populations appear to experience monotonic damping to a
stable equilibrium (Hassell et al. 1976, Bellows 1981). The exact form of the function used to
describe g is not particularly critical to these general conclusions
and many forms have been proposed (Bellows 1981), although different forms
may have specific attributes more applicable to certain cases. Perhaps the
most flexible is that proposed by Maynard Smith & Slatkin (1973), where g(N)
takes the form g(N) = [1+(N/a)^{b}]-1. (2) where the relationship between proportionate survival and density
is defined by the two parameters a, the density at which
density-dependent survival is 0.5, and b, which determines the
severity of the competition. As b approaches 0, competition becomes
less severe until it no longer occurs 9b=0), when b=1 density
dependence results in contest competition with the number of survivors
reaching a plateau as density increases, and for b>1 scramble
competition occurs, with the number of survivors declining as the density
exceeds N-a. Multiple age-class
systems.--Most
populations are separable into distinct age or stage classes, and this is
particularly important in competitive systems. In most insects the
preimaginal stages must compete for resources for growth and survival, while
adults must additionally compete for resources for egg maturation and
oviposition sites. In such cases, competition within populations divides
naturally into sequential stages. Equation (1) may be extended to the case of
two age classes (May et al. 1978) and, where competition occurs primarily
within stages (e.g., larvae compete with larvae and adults with adults), A_{t+1} = g1(L_{t})L_{t} (3a) L_{t+1} = Fg_{a}(A_{t})A_{t} (3b) where A and L denote the adult and larval
populations. In such multiple age-class systems, the dynamical behavior of
the population is dominated by the outcome of competition in the stage in which
it is most compensatory. Hence in a population where adults exhibit contest
competition for oviposition sites while larvae exhibit scramble competition
for food, the population will show monotonic damping to a stable equilibrium,
characteristic of a population with contest competition. This result is
extendable to n age classes, so that any population in which
competition in at least one stage is stabilizing or compensatory (i.e.,
contest), the dynamics of the population will be characterized by this stabilizing
effect (Bellows & Hassell 1999). A review of insect populations showing density
dependence in natural and laboratory settings indicates that most such
populations exhibit monotonic damping towards a stable equilibrium (Hassell
et al. 1976, Bellows 1981). This does not preclude the possibility of
scramble competition in insect populations (e.g., Nicholson 1954, Goeden
1984), but does imply that compensatory competition exists in at least one
stage in most studied populations. More complex approaches to constructing models
of single-species insect populations can be taken which involve many
age-classes and great detail in description of biological processes. Many of
these have been designed to consider only the problem of describing
development of the population from one stage to another and do not bear
directly on mechanisms of natural population regulation. Others consider
internal processes which may limit population growth (e.g., Lewis 19__,
Leslie & Gower 1958, Bellows 1982a,b) and consequently do touch on
population regulation. In one comparative study, Bellows (1982a,b) found
little difference in dynamical behavior between simple one and two age-class
models and more complex systems models with several age classes. hence at
least for single-species population models, the distinction between two and
more age classes in the analytical framework may be of little consequence.
This may not be the case for systems with more than one species (Bellows
& Hassell 1999). The preceding unfolding is particularly
applicable to homogeneous environments and uniformly distributed resources.
For many insect populations, however, resources are not distributed either
continuously or uniformly over the environment but rather occur in
disjunctive units or patches. For such cases equations (1) through (3)
generally will not apply, for the distinction between homogeneous and patchy
environments has significant consequences for population dynamics.
Populations competing for resources in patchy environments may be expected to
show the same range of qualitative behaviors-- stable points approached
either monotonically or by damped oscillations, periodic cyclic behavior and
disarray, but the formulations representing them shed new light on the
importance of dispersal, dispersion and competition within patches. Considering an environment divided into j
discrete patches (e.g., leaves on trees) which are utilized by an insect
species, adults (N) disperse among the patches and distribute their compliment
of progeny within a patch. Progeny deposited in a patch remain in the patch
and compete for resources only within the patch and only with other
individuals within the patch. The population dynamics is now dependent partly
on the distribution of adults reproducing in patches OE and partly on
the density dependent relationship that characterizes preimaginal
competition. Population reproduction over the entire environment (i.e., all
patches) can be characterized by the relationship by deJong (1979): N_{t+1} = jFZOE(n_{t})n_{t}g[Fn_{t}] (4) (Z = summation sign) where n is the number of adults in a particular patch and OE(n)
is the proportion of patches colonized by n adults. DeJong (1979) considered four distinct
dispersion distributions of individual adults locating patches. In the case
of uniform dispersion, equation (4) is equivalent to equation (1) for
homogeneous environments. For three random cases, positive binomial,
independent (Poisson), and negative binomial, the outcome depends somewhat on
the form taken for the function g. For most reasonable forms of g,
the general outcomes of dividing the environment into a number of discrete
patches are a lower equilibrium population level and enhanced numerical
stability in comparison to equation (1) with the same parameters for F
and the function g. Two additional features arise: (1) there is an
optimal fecundity for maximum population density and (2) for a fixed amount
of resource, population stability increases as patch size decreases and the
number of patches increases (the more finely divided the resource the more
stable the interaction) to an optimal minimum patch size. The addition of
more patches of resource (increasing the total amount of resource available
but holding patch size constant) does not affect stability per se but
increases the equilibrium population level (Bellows & Hassell 1999). In the same way that competition for resources
among individuals of the same species can lead to r1estrictions on population
growth, competition among individuals of different species can similarly
cause density dependent constraints on growth. Although Strong et al. (1978)
suggested that competition is not commonly a dominant force in shaping many
herbivorous insect communities, it certainly is an important potential factor
in insect communities, especially those which feed on ephemeral resources
(e.g., Drosophila spp.) and
additionally in insect parasitoid communities (e.g., Luck & Podoler
1985). The processes and outcomes of interspecific competition in insects
have been studied widely in the laboratory (e.g., Crombie 1945, Fujii 1968,
Bellows & Hassell 1984) as well as in the field (Atkinson & Shorrocks
1977). Homogeneous Environments Single age-class systems.--Many of the same mechanisms implicated in
intraspecific competition for resources (e.g., competition for food,
oviposition sites, etc.) also occur between species (e.g., Crombie 1945,
Leslie 194_, Park 1948, Fujii 1968, 1970). The dynamics of these interspecific
systems can be considered in a framework very similar to that for single
species populations. Equation (1) can be extended to the case for
two (or more) species by considering the function g to depend on the
density of both competing species (Hassell & Comins 1976), so that the
reproduction of species X depends not only on the density of species X
but also on the density of species Y (and similarly for species Y): X_{t+1} = Fg_{x}(X_{t}+alpha
Y_{t})X_{t} (5a) Y_{t+1} = Fg_{y}(Y_{t}+Beta
X_{t})Y_{t} (5b) Here the parameters alpha and Beta
reflect the severity of interspecific competition with respect to
intraspecific competition. Population interactions characterized by
equation (5) may have one of four possibilities: the two species may coexist,
species X may always exclude species Y, species Y may always exclude species
X, or either species may exclude the other depending on their relative
abundance. Coexistence is only possible when the product of the interspecific
competition parameters alpha Beta<1 (when alpha Beta>1 one
of the species is driven to extinction). For coexisting populations, the
dynamical character of the populations is determined by the severity of the
intraspecific competition and may take the form of stable equilibria
approached monotonically, stable cyclic behavior, or chaos (Hassell &
Comins 1976). It is conventional to summarize the character
of the interspecific interaction by plotting isoclines which define zero
population growth in the space delimited by the densities of the two
populations. In these simple, single age-class models with linear
interspecific competition, these isoclines are linear. When they have an
intersection, the system has an equilibrium (stable for alpha Beta<1);
when they do not intersect the species with the isocline farthest from the
origin will eventually exclude the other (e.g., Crombie 1945). The biological
interpretation applicable to this analysis is that each species must inhibit
its own growth (through intraspecific competition) more than it inhibits the
growth of its competitor (through interspecific competition) for a persistent
coexistence to occur. Multiple age-class systems.--Many insect populations compete in both
preimaginal and adult stages, perhaps by competing as adults for oviposition
sites and subsequently as larvae for food (e.g., Fujii 1968) and in some
cases the superior adult competitor may be inferior in larval competition
(e.g., Fujii 1970). The analytical properties of such multiple age-class
systems may be considered by treating separately the dynamics of the adult
and preimaginal stages (Hassell & Comins 1976): X_{t+1} = x_{t}g_{x}l(x_{t}+alpha
1y_{t}) (6a) Y_{t+1} = y_{t}g_{x}l(y_{t}+Beta
1y_{t}) (6b) x_{t+1} = X_{tF}xg_{x}
alpha(X_{t}+alpha alpha Y_{t}) (6c) y_{t+1} = Y_{tF}y_{gy}
alpha(Y_{t}+beta alpha X_{t}) (6d) where x and y are the preimaginal or larval stages
and X and Y are the adults. Here larval survival of each
species is dependent on the larval density of both species, and adult
reproduction of each species is dependent on the adult densities of both
species. Larval competition is characterized by the larval competition
parameters alpha_{l} and Beta_{l}, while adult
competition is characterized by alpha_{a} and Beta_{a}. The simple addition of competition in more
than one age has important effects on the dynamical behavior of the competitive
system. The isoclines of zero population growth are now no longer linear, but
curvilinear, and multiple points of equilibrium population densities are now
possible. It is even possible to have more than one pair of stable
equilibrium densities (Hassell & Comins 1976). Such curvilinear isoclines
are in accord with those found for competing populations of Drosophila spp. (Ayala et
al. 1973). More complex systems can be visualized with
additional age classes and with competition between age classes (e.g.,
Bellows & Hassell 1984). The general conclusions from studies of these
more complex systems are similar to those for the two age-class systems, vis.
that more enigmatic systems have non-linear isoclines and consequently may
have more complicated dynamical properties. More subtle interactions may also
affect the competitive outcome, such as differences in developmental time
between two competitors. In the case of Callosobruchus
chinensis and Callosobruchus maculatus, the intrinsically
superior competitor (C. maculatus) can be outcompeted
by C. chinensis because the latter develops faster and thereby
gains earlier access to resources in succeeding generations. This earlier
access confers sufficient competitive advantage on C. chinensis
that it eventually excludes C.
maculatus from mixed species
systems (Bellows & Hassell 1984). Patchy Environments Many insect populations are dependent on
resources which occur in patches (e.g., fruit, fungi, dung, flowers, dead wood).
Dividing the resources for which populations compete into discrete patches
can have significant effects on the consequences of interspecific
competition. Two general views of competition in a patchy
environment have been proposed. In the first coexistence is promoted by a
balance between competitive ability and colonizing ability (Skellem 1951,
Cohen 1970, Levins & Culver 1971, Horn & MacArthur 1972, Slatkin
1974, Armstrong 1976). An alternative view proposed by Levin (1974) is that
competition in a patchy environment may result in a persistent coexistence if
both species inhibit their own growth less than their competitors, so that in
any patch the numerically dominant species would exclude the competitor; each
species would have a refuge in those patches where it is numerically
dominant. A idea has been proposed by Shorrocks et al.
(1979) and Atkinson & Shorrocks (1981), where each patch is temporary in
nature but is regularly renewed. Such resources may be typical for many
invertebrates (Shorrocks et al. 1979). In this case the competitively
inferior species is not constantly driven out of patches because the patches
are ephemeral in nature. Because of this, coexistence can occur when
competition between the species can be more severe than in the homogeneous
case because its frequency of occurrence is reduced by the fraction of
patches which contain only one species. This view emphasizes the importance of
aggregated spatial dispersion among patches in the populations of the
competing species. Atkinson & Shorrocks (1981) investigated the
consequences of this by using the negative binomial distribution of
individuals among patches in a two-species competitive model. The conclusions
of this work were primarily that coexistence of competitors on a divided
resource is possible under many more scenarios than in the homogeneous case.
Specifically, coexistence is promoted by dividing a resource into more and
smaller breeding sites, by aggregation of the superior competitor, and
especially by allowing the degree of aggregation to vary with density. Equation (1) may be extended for single
species populations in a homogeneous environment to include the additional
effect of mortality caused by a natural enemy. The particular details of the
algebra espoused would depend to some extent on what biological situation it
is desired to express. Following previous work (Nicholson & Bailey 1935,
Hassell & May 1973, Beddington et al. 1978, May et al. 1981), the insect
protolean parasites or parasitoids are considered. Such systems have
attracted much attention for both theoretical and experimental studies
(Hassell 1978). Pursuing the discrete framework of the preceding sections,
the dynamics of these interactions may be summarized by: N_{t+1} = Fg(fN_{t})N_{t}f(N_{t,P}t) (7a) P_{t+1} = cN_{t}{1-f(N_{t},P_{t})} (7b) Here N and P are the host and
parasitoid populations; Fg(fNt) is the per capita net rate of increase
of the host population, intraspecific competition is defined as before by the
function g with density dependence for g<1; the function f
defines the proportion of hosts which are not attacked and embodies the
functional and numerical responses of the parasitoid, and c is the
average number of adult female parasitoids which emerge from each attacked
host. In such analytical frameworks, different dynamics can result depending
on the sequence of mortalities and reproduction in the hosts life cycle (Wang
& Gutierrez 1980, May et al. 1981, Hassell & May 1986). Equation (7)
reflects the case for parasitism acting first followed by density dependent
competition as defined by g (May et al. 1981 give a discussion of
alternatives). This model then represents an age-structured host population
in which density dependence (if any) occurs in a distinct post-parasitism
stage in the life cycle. This design has a long heritage, and has been
utilized with many versions of the functions f and g. Bellows
& Hassell (1999) stated that early workers incorporated no density
dependence in the host (g=1) and functions for f which implied
independent random search by individual parasitoids (e.g., Thompson
1924, Nicholson 1933, Nicholson & Bailey 1935). A simple reference to
Nicholson (1933) and Nicholson & Bailey (1935) will reveal how emphatic
these authors were to distinguish nonrandom searching by individuals
from random searching by populations (see section on
"Searching". Thus it is difficult to understand the current
statements by Bellows & Hassell, although they have been made before by
Varley et al. (1973) and Milne (1957a,b). In the cases referred to by Bellows
& Hassell (1999), the model design becomes somewhat simpler: N_{t+1} = FN_{t}exp(-aP_{t}) (8a) P_{t+1} = N_{t}{1-exp(-aP_{t})} (8b) where f(N,P) is represented by the zero term of the
Poisson distribution in keeping with the assumptions of independent search by
parasitoid adults. The parameter a is the area of discovery an adult
parasitoid, characterizing the species searching ability. This model
incorporates a somewhat mechanized search behavior for the parasitoid, with
search for hosts being continuous and successful subduing of hosts
instantaneous upon discovery, with no such limits on search as physiological
resources or egg depletion. These works laid useful groundwork but, as
they reflected an interest in biological control and population regulation,
proved inadequate because such simple systems did not include regulatory
population dynamics; quite simply, there is no direct density dependence in
equation (8) and thus no stabilizing feature in the model. In contrast, these
simple systems suggest a destabilizing effect of parasitism on the host
population (delayed density dependence), with such matched populations
exhibiting oscillations of ever-increasing magnitude until extinction
occurred. Experiments conducted in the laboratory (under artificial
conditions) were applied to examine the suitability of such models and
affirmed that such simple systems were characterized by unstable oscillations
(Burnett 1954, DeBach & Smith 1941). Subsequent work considered the dynamics of more
complex forms of equation (8) which have attempted to capture additional
behavioral features of predation and parasitism. Holling (1959a,b, 1965,
1966) introduced the idea of characterizing the act of parasitism or
predation by component behaviors, such as the separate behaviors of attack
and subsequent handling of prey. This view permitted different types of
functional responses to be characterized by different component behaviors
(Holling 1966). Parasitism and predation in insects are largely typified by
Type II functional responses, viewed by Holling as characterized by two
parameters, the per capita search efficiency a and the time taken to
handle a prey T_{h}. These were incorporated into the
structure of equation (8) by Rogers (1972), who added the limitations of
handling time to independently searching parasitoids. Equation (8) becomes N_{t+1} = FN_{tf(}exp(-aP_{t}/(1+aThNt))) (9a) P_{t+1} = N_{t}{1-f(exp(-aP_{t}/(1+aThNt))} (9b) The result of this addition to the earlier
design was increased biological realism, but decreased population or system
stability. The addition of handling time increased the destabilizing effect
of parasitism without contributing any stabilizing density dependence
(Hassell & May 1973). Truly the principles involved in type II functions
responses (as in equation (9b) are inversely density dependent and thereby
destabilizing, contributing to the instability caused by the delayed density
dependence. In more realistic situations, the outcome of
search by a parasitoid population may not be typified by independent random
search. Many processes (spatial, temporal and genetic) will combine to render
some prey individuals more susceptible to predation than others. This unequal
susceptibility between individuals will result in non-independence of attacks.
One approach to capturing this non-independence is to employ the negative
binomial distribution to characterize the distribution of attacks, so that
the function f becomes f(N,P)
= [ aP ]^{-k} [1 +_______] [
k(1+aT_{h}N)] (10) and the simplest
case with no host density dependence becomes N_{t+1}
= FN_{t[{}1+aP_{t/(k(1+aThNt))}}^{-k}] (11a) P_{t+1}
= cN_{t}{1-{1+aP_{t/(k(1+aThNt))}-k}} (11b) Here once again a is the per capita
search efficiency of parasitoid adults and T_{h} is their
handling time. The differential susceptibility of prey host individuals to
attack is characterized by contagion in the distribution of attacks among
individuals, representing the outcome that more susceptible individuals are
more likely to be attacked. This contagion is depicted by the parameter k
of the negative binomial. Contagion increases as k->0, where as in
the opposite limit of k-> attacks become distributed independently
and the Poisson distribution is recovered (equation 8). As May and Hassell (1988)
have discussed, the outcome of a parasitoid's searching behavior cannot
usually be fully characterized so simply as equation (10) (Hassell & May
1974, Chesson & Murdoch 1986, Perry & Taylor 1986, Kareiva &
Odell 1987). Nonetheless, the use of equation (10) with a constant k
permits the dynamical effects of non-random or aggregated parasitoid
searching behavior to be examined without introducing a large list of
behavioral parameters. More complex cases, such as the value of k
varying with host density, can be considered (Hassell 1980), but have little
effect on the dynamical aspects of the host-parasitoid interaction. The simple change from independently random
search foreseen by early workers (equation (8)) to the more general case of
equation (11) can have profound effects on the dynamics of such systems.
Although equation (11) still contains the destabilizing affect of delayed
density dependence inherent in such difference-equation systems, the system
can not be stable when k takes values between 0 and 1, implying some
degree of contagion in the distribution of attacks. This contagion is a
direct density dependence in the parasitoid population which can stabilize
the otherwise intrinsically unstable system. For values of k>1 the
contagion is insufficiently strong to stabilize the system. Hassell (1980) presents an application of this
analytical framework to the case of winter moth, Operophtera brumata
Cockerell, in Nova Scotia parasitized by the tachinid Cyzenis albicans
(Embree 1966). Drawing on quantitative studies from the field, values for the
parameters a and k were obtained and, in this case, T_{h}
approximated by 0. The resulting model outcomes characterized well the known
outcomes in the natural system, vis. the host population declined and remained
at a lower level following the introduction of the parasitoid. The analytical
framework appears sufficiently general that it may have wider application to
other "successful' cases of biological control, and perhaps even to
"unsuccessful" cages where contagion or differential susceptibility
to attacks was insufficiently pronounced to contribute to stability. Future
examination of the roles of natural enemies may benefit from determining the
distribution of attacks in the host population. The preceding discussion has focused on
situations where there has been no implicit host density dependence, with the
function g=1. This may be an appropriate framework for many
situations, particularly where biological control agents are established and
populations are substantially below their environmentally determined maximum
carrying capacity. In other cases, however, the relative roles of regulatory
features of both host and natural enemy populations must be addressed. Such
situations are probably more characteristic of cases where a host populations
is without natural enemies prior to their introduction and has reached an
environmental maximum density. In these cases the function g will no
longer be negligible, and consideration of natural control must include the
relative contribution of both intraspecific competition and the action of the
natural enemies. The design presented in equation (7) can be
used to explore the joint effects of density dependence in the host together
with the action of parasitism. This has been accomplished by Maynard Smith
& Slatkin (1973) for a two-age-class extension of this design with
independent random parasitism (the Nicholson-Bailey model) and by Beddington
et al. (1975) who employed a discrete version of the logistic model together
with random parasitism. To more fully examine the relative contributions of
intraspecific regulatory processes and parasitism a model must be used in
which parasitism can also act as a regulating or stabilizing factor. May et
al. (1981) approached this by using equation (10) for the function f
(the proportion surviving parasitism) with the addition of a discrete form of
the logistic for the host density dependence function g, where g=exp(-cN).
One important feature of these discrete systems
incorporating both host and parasitoid density dependence is that the
outcomes of the interactions will depend on whether the parasitism acts
before or after the density dependence in the host population. May et al.
(1981) envisaged two general cases, the first where host density dependence
acts first and the second where parasitism acts first (their models 2 and 3).
They employed equation (10) with no handling time (T_{h}=0)
for function f, and the two resulting systems are: Host density dependence acts before parasitism: N_{t+1}
= F_{g}(N_{t})N_{t}f(P_{t}), (12a) P_{t+1}
= N_{t}g(N_{t}){1-f(P_{t})}; (12b) parasitism acts
first: N_{t+1}
= F_{g}(fN_{t})N_{t}f(P_{t}), (13a) P_{t+1}
= N_{t}{1-f(P_{t})}.
(13b) Equation (12) is a specific case of equation
(7) with the specified functions for f and g. Beddington et al. (1975) and May et al. (1981)
have explored the outcomes of such interactions by considering the stability
of the equilibrium populations in the host-parasitoid system. This stability
can be defined in relation to two biological features of the system: the
host's intrinsic rate of increase (log F) and the level of the host
equilibrium in the presence of the parasitoid (N*) relative to the
carrying capacity of the environment (K) (the host equilibrium due
only to host density dependence in the absence of parasitism). This ratio
between the parasitoid-induced equilibrium N* and K is termed q,
q=N*/K. The relationship between F and q
varies depending on the degree of contagion in the distribution of attacks
(the parameter k of equation 11), and further depends on whether
parasitism occurs before or after density dependence in the life cycle of the
host. In both cases the degree of host suppression possible increases with
increased contagion of attacks. The new parasitoid-caused equilibrium density
may be stable or unstable, and for unstable equilibrium the populations may
exhibit geometric increase or oscillatory or chaotic behavior. For density
dependence acting after parasitism and for k<1 any population
reduction is stable. Additionally, special combinations of parameter values
in this latter case can lead to hypothetically higher equilibria in the
presence of the parasitoid. This only applies to over compensatory density
dependence, where it is possible to envisage parasitism reducing the number
of competitors to a density more optimal for survival than would occur in its
absence, leading to a greater density of survivors from competition than when
parasitism is not present (May et al. 1981). Also see Bellows & Hassell
(1999) for graphed figures. More generally, much of the parameter space for
both cases implies a stable reduced population whenever k<1. This
reduction would be less for equivalent parasitism acting before density
dependence in the life cycle of the host rather than after. Patchy Environments In the same way that single-species and
competing species population may occur in heterogeneous or patchy
environments, populations which are hosts to insect parasitoids may occur in
discrete patches (Hassell & May 1973, 1974, Hassell & Taylor 198_).
The consequences of such heterogeneous host distributions on the dynamics of
the host-parasitoid system can depend significantly on the numerical
responses of the parasitoid population to prey distributed in patches.
Several mechanisms exist which tend to lead to aggregations of natural
enemies in patches of higher prey densities. Denser patches may be more
easily discovered by natural enemies (Sebalis & Laane 1986), search
behavior may change upon discovery of a host in such a fashion as to lead to
increased encounters with nearby hosts (Murdie & Hassell 1973, Hassell
& May 1974), and the time a predator spends in a patch may depend on the
encounter rate with prey (Waage 1980) or on the prey density (Sebalis &
Laane 1986). The result of each of these mechanisms is an aggregation of
natural enemies in patches of higher prey densities. Consider analytically the consequences of such
aggregations, a simple model of host and parasitoid distributions over space.
If an environment is divided into j patches of areas in the
environment, the fraction of hosts in each area can be specified by alpha_{i}
and the fraction of parasitoids in each area by Beta_{i}, with
the condition that the entire population is represented in the environment,
so that Zalpha_{i} = 1,
ZBeta_{i} = 1. [Z = summation sign] Equation (7) can be modified to express this
distribution over space, N_{t+1} = FN_{t }Zg(falpha_{i}N_{t})alpha_{i}f(alpha_{i}f(alpha_{i}N_{t},Beta_{i}P_{t}), (14a) P_{t+1} = cN_{t} Z alpha_{i}-f(alpha_{i}N_{t}Beta_{i}P12_{t})} (14b) Adopting some of the simplifications employed
in equation (8) (i.e., independent random search by solitary parasitoids, so f(P)=exp(-exp(-aP)
and c=1, and no host density dependence, so g=1, gives the
explicit model: N_{t+1} = FN_{t} Z alpha_{i}exp(-alpha
Beta_{i}P_{t}), (15a) N_{t+1} = N_{t} Z alpha_{i}{1-exp(-aBeta_{i}P_{t}). (15b) The key parameters affecting the dynamical
behavior of this system are host fecundity F and the distribution of
hosts and parasitoids over patches (Hassell & May 1973, 1974). In
equation (15) there is a general model for exploring the effects of any
specific host and parasitoid distributions. In particular the case may be
considered where the natural enemy distribution (Beta_{i}) is
dependent in some way on the host distribution (alpha_{i}), Beta_{i} = c alpha_{i}. (16) In equation (16) the relationship between the
host and parasitoid distributions is determined by the parasitoid
aggregation index (c is a normalizing constant which permits ZBeta_{i}=1).
In this way the distribution of parasitoids in patches can vary from uniform
(= 0) through distributions where parasitoids "avoid" patches of
high host density (<1), parasitoids have the same distribution as the host
population (= 1), to distributions where parasitoids aggregate in patches of
high host density (>1). In each patch parasitoid search is random
according to equation (15). In this system the dynamical behavior is now
largely determined by the host rate of increase F (as before), the
number of patches, and the parameter which determines the degree of aggregation
of the natural enemy population. Generally, conditions for stable population
interactions are enhanced by increasing the number of patches, values of
>1 (aggregation of natural enemies in patches of high host density) and
low values of F. A necessity is an uneven distribution of hosts; if
the host distribution is uniform over patches the system is equivalent to the
intrinsically unstable Nicholson-Bailey formulation of equation (8). This analysis permits some interpretation of
the circumstances under which the distributions of populations over patchy
environments may be significant in regulation of hosts by natural enemies.
First, aggregation of natural enemies is likely only to be an effective
regulatory mechanism if host distributions are non-uniform. Secondly, the
parasitoid distribution must be nonuniform, but not necessarily more so than
the host (i.e., it is not necessary that natural enemies aggregate more
intensely than their hosts). Finally, a host rate of reproduction which is
sufficiently can lead to instability. Inherent in most insect populations is the
concept of age- or stage-structure. Insects grown through distinct
developmental stages, and hence the concepts of age and stage are linked,
although in some systems more closely than others. Many of the analytical
frameworks constructed in the previous sections take such developmental
stages into account. Equation (4) is one such example, where considering
dispersal to occur prior to competition in a patchy resource implies a
dispersing reproductive stage (adults) followed by a non-dispersing stage
which competes for resources (larvae). Other examples are considerations of
the interactions of density-dependence and the action of natural enemies
(equations (12) and (13), e.g.). These implied sequences of events are for
the most part easily handled in the single-step analytical frameworks
presented previously. However, there are a number of implied
assumptions in the previously presented frameworks which limit their applications.
In particular, there are several assumptions about the timing of events
(e.g., that all parasitism occurs simultaneously, that all competition occurs
either before or after parasitism, that all dispersal occurs at once, and
that host and parasitoid populations are so synchronized that all members of
the parasitoid population are able to attack hosts at the same time that all
members of the host population are in the stage susceptible to parasitism).
Systems that are characterized by biologies, which are at significant
variance to these assumptions, may not be well characterized by these
analytical frameworks. The solution to exploring the theoretical
repercussions of more complex biologies frequently has been to construct more
complex models, often called system or simulation models, which
incorporate more biological detail at the expense of analytical tractability.
This approach has been used not only to address issues of population dynamics
but also to address matters relating to population developmental rate,
biomass and nutrient allocation, community structure and management of
ecosystems (Bellows et al. 1983). Here are considered only those features of
such systems which bear on population regulation in ways which are not
directly addressable in the simpler analytical frameworks presented above. Synchrony of Parasitoid
and Host Development.--The implied synchrony of host and
parasitoid development in the discrete-time formulations used above is one of
the simplest assumptions to relax in order to consider the implications of
asynchrony. The degree of synchrony between host and parasitoid development
is a component of each of the evaluations considered in this section. Here
will begin the simplest case followed by building upon it: Insect populations in continuously favorable
environments (e.g., laboratory populations, some tropical environments) may
develop continuously overlapping generations, but in the presence of
parasitism as a major cause of mortality they also may exhibit more or less distinct
generations (Bigger 1976, Taylor 1937, Metcalfe 1971, Notley 1955, Utida
1957, White & Huffaker 1969, Hassell & Huffaker 1969, Banerjee 1979,
Tothill 1930, van der Vecht 1954, Wood 1968, Perera 1987). Godfray &
Hassell (1987) constructed a simple system model in which they considered an
insect host population growing in a continuously favorable environment (with
no intraspecific density-dependence) which passes through both an adult
(reproductive) stage and preimaginal stages. They chose a discrete-time-step
model in which individuals progress through stages (or ages) each time step;
the adult stage reproduces for more than one time step, thus leading
eventually to overlapping generations and continuous reproduction. The model
for the host population is identical in structure to the matrix model of
unconstrained population growth of Lewis (1945) and Leslie (1948), and left
uninterrupted the host population would grow without limit and attain a
stable age-class structure with all age classes present at all times. To this
host population is added a parasitoid which also develops through preimaginal
and adult (reproductive) stages. The length of the preimaginal developmental
period was varied to examine the effect of changes in relative developmental
times in host and parasitoid populations. Attacks by the parasitoid adult
population were distributed using equation (10) with T_{h} =
0 (May 1978). The dynamical behavior of the system was
characterized either by a stable population in which all stages were continuously
present in overlapping generations, populations which were stable but which
occurred in discrete cycles of approximately the generation period of the
host, and unstable populations. These dynamics were dependent principally
upon two parameters, the degree of contagion in parasitoid attacks, k,
and the relative lengths of preimaginal developmental time in the host and
parasitoid population. Very low values of k (strong contagion)
promoted continuous, stable generations. Moderate values of k (less
strong contagion) were accompanied by continuous generations when the
parasitoid had developmental times approximately the same length as the host,
approximately twice as long, or very short. When developmental times of the
parasitoid were approximately half or 1.5 times that of the host, discrete
generations arose. For even larger values of k, unstable behavior was
the result. From these examples it can be seen that
asynchrony between host and parasitoid could be an important factor affecting
the dynamical behavior of continuously breeding populations, particularly for
parasitoids which develop faster than their hosts. In particular, parasitoids
developing in approximately half the host's developmental time could promote
discrete (and stable) generations. Parasitism and Competition
in Asynchronous Systems.--Utida (1953) reported the dynamics of a
host-parasitoid system which had unusual dynamical behavior characterized by
bounded, but aperiodic, cyclic oscillations. These oscillations appear
chaotic in nature but are not typified by the dynamics of any of the discrete
systems considered earlier. The laboratory system consisted of a regularly
renewed food source, a phytophagous weevil, and a hymenopteran parasitoid.
Important characteristics of the system were host-parasitoid asynchrony (the
parasitoid developed in 2/3rds of the weevil developmental time), host
density dependence (the weevil adults competed for oviposition sites and
larvae for food resources), and age-specificity in the parasitoid-host
relationship (parasitoids could attack and kill three larval weevil stages
and pupae, but could only produce female progeny on the last larval stage and
pupae). A system model of this system was constructed
by Bellows & Hassell (1988), which incorporated detailed age-structured
host and parasitoid populations, intraspecific competition among host larvae
and among host adults, and age-specific interactions between host and
parasitoid. The dynamics of the model had characteristics similar to those
exhibited by the experimental population and distinct from those of any
simpler model. Important features contributing to the observed dynamics were
host-parasitoid asynchronous development, the attack by the parasitoid of
young hosts (on which reproduction was limited to male offspring), and
intraspecific competition by the host. The interaction of these three factors
caused continual changes in both host density and age-class structure. In
generations where parasitoid emergence was contemporaneous with the presence
of late larval hosts, there was substantial host mortality and parasitoid
reproduction. This produced a large parasitoid population in the succeeding
generation which, emerging coincident with young host larvae, killed many
host larvae but produced few female parasitoids. The reduced host larval
population suffered little competition (because of reduced density). This
continual change in intensity of competition and parasitism contributed
significantly to the cyclic behavior of the system; simpler models without
this age-class structure would not account for these important aspects of
this host-parasitoid interaction. Invulnerable Age-classes.--The two previous models both incorporated
susceptible and unsusceptible stages, ideas which are inherent to any
stage-specific modelling construction for insects where the parasitoid
attacks a specific stage such as egg, larvae or pupae. The consequences of
the presence of invulnerable stages in a population has been considered
analytically by Murdoch et al (1987) in a consideration of the interaction
between California red scale, Aonidiella
aurantii (Maskell), and its
external parasitoid Aphytis melinus (DeBach). They
constructed a system model which includes invulnerable host stages, a vulnerable
host stage, juvenile parasitoids and adult parasitoids. This model contains
no explicit density dependence in any of the vital rates or attack
parameters, but does contain time-delays in the form of developmental times
from juvenile to adult stages of both populations. Murdoch et al (1987) developed two models, one
in which the adult hosts are invulnerable and one in which the juvenile hosts
are invulnerable. The particular frameworks that were constructed permitted
analytical solutions regarding the dynamical behavior of the systems. In
particular, it was found that the model could portray stable equilibria
(approached either monotonically or via damped oscillations), stable cyclic
behavior or chaotic behavior. The realm of parameter space which permitted
stable populations was substantially larger for the model in which the adult
was invulnerable than for the model when the juvenile was invulnerable. The
overall conclusion is that an invulnerable age class can contribute to the
stability of the system. Whether this contribution is sufficient to overcome
the destabilizing influence of parasitoid developmental delay depends on the
relative values of parameters, but short adult parasitoid lifespan, low host
fecundity and long adult invulnerable age class all promote stability. Many insect parasitoids attack only one or few
stages of a host population (although predators may be more general), and
hence many populations possess potentially unattacked stages. In addition,
however, many insect populations host more than one natural enemy, and
general statements concerning the aggregate effect of a complex of natural
enemies attacking different stages of a continuously developing host
population are not yet possible. Nonetheless, it appears that in at least the
California red scale--A. melinus system the combination
of an invulnerable adult stage and overlapping generations is likely a factor
contributing to the observed stability of the system (Reeve & Murdoch
1985, Murdoch et al. 1987). Spatial Complexity
and Asynchrony.--In predator-prey or parasitoid-host systems
which occur in a patchy heterogeneous environment, there is a distinction
between dynamics which occur between the species within a patch and the
dynamics of the regional or global system. Here there is a distinction
between "local" dynamics (those within a patch) and
"global" dynamics (the characteristics of the system as a whole).
Also, while still interested in such dynamical behavior as stability of the
equilibrium, there is also a desire to understand what features of the system
might lead to global persistence (the maintenance of the interacting
populations) in the face of unstable dynamical behavior at the local level.
One set of theories concerned with the global persistence of predator-prey
systems emphasizes the importance of asynchrony of local predator-prey cycles
(those occurring within patches) (e.g., den Boer 1968, Reddingius & den
Boer 1970, Reddingius 1971, Maynard Smith 1974, Levin 1974, 1976; Crowley
1977, 1978, 1981). In this context, asynchrony among patches implies that, on
a regional basis, unstable predator-prey cycles may be occurring in each
patch at the local scale but they will be occurring out of phase with one
another (prey populations my be increasing in some fraction of the
environment while they are being driven to extinction by predators in
another); such asynchrony may reduce the likelihood of global extinction and
thus promote the persistence of the populations. An example of one such system is the model of
interacting populations of the spider mite Tetranychus urticae
Kock and the predatory mite Phytoseiulus
persimilis Athias-Henriot
constructed by Sebalis & Laane (1986). This is a regional model of a
plant-phytophage-predator system that incorporates patches of plant resource
that may be colonized by dispersing spider mites; colonies of spider mites
may in turn be discovered by dispersing predators. The dynamics of the
populations within the patch are unstable (Sebalis 1981, Sebalis et al. 1983,
Sebalis & van der Meer 1986), with overexploitation of the plant by the
spider mite leading to decline of the spider mite population in the absence
of predators, and when predators are present in a patch they consume prey at
a rate sufficient to cause local (patch) extinction of the prey and
subsequent extinction of the predator. In contrast to the local dynamics of the
system, the regional or global dynamics of the system was characterized by
two stages, one in which the plant and spider mite coexisted but exhibited
stable cycles (driven by the intraspecific depletion of plant resource in
each patch and the time delay of plant regeneration), and one in which all
three species coexisted. This latter case was also characterized by stable
cycles, but these were primarily the result of predator-prey dynamics; the
average number of plant patches occupied by mites in the three-species system
was less than 0.01 times the average number occupied by spider mites in the
absence of predators. Thus in this system consisting of a region of patches
characterized by unstable dynamics, the system persists. Principal among the models features, which
contributed to global persistence, was asynchrony of local cycles. Because of
this it was unlikely that prey could be eliminated in all patches at the same
time, and hence the global persistence. This asynchrony could be disturbed
when the predators became so numerous that the likelihood of all prey patches
being discovered would rise toward unity, a circumstance which could
eventually lead to global extinction of both prey and predator. Other
features of the system were also explored by Sebalis & Laane (1986). If a
small number of prey were able to avoid predation in each patch (a prey
"refuge" effect), the system reached a stable equilibrium, while other
parameter changes led to unstable cycles of increasing amplitude. The results of this exercise accord with
certain experiments reported in the literature. Huffaker (1958) found
self-perpetuating cycles of predator and prey in spatially complex environments,
and Huffaker et al. (1963) found that increasing spatial heterogeneity
enhanced population persistence. Three features of these experiments were in
accord with the behavior of the model of Sebalis & Laane (1986): (1)
overall population numbers in the environment did not converge to an
equilibrium value but oscillated with a more or less constant period and
amplitude; (2) facilitation of prey dispersal relative to predator dispersal
enhanced the persistence of the populations (Huffaker 1958); (3) increase in
the amount of food available per prey patch resulted in the generation of
abundant predators at times of high prey density, and the areas were
subsequently searched sufficiently well that synchronization of the local
cycles resulted, leading to regional extinction (Huffaker et al. 1963). Results reported in larger-scale systems,
particularly glasshouses, include reports of elimination of prey and
subsequently of predator (Chang 1961, Bravenboer & Dosse 1962, Laing
& Huffaker 1969, Takafuji 1977, Takafuji et al. 1981), perpetual
fluctuations of varying amplitude (Hamai & Huffaker 1978), and wide
fluctuations of increasing amplitude (Burnett 1979, Nachman 1981). Specific
interpretation of these results relative to any particular model must be made
with caution because of differences in scale, relation of the experimental
period to the period of the local cycles, and relative differences in ease of
prey and predator redistribution in different systems. Nonetheless, it is
clear that asynchrony among local patches can play an important role in
conferring global stability or persistence to a system composed of locally
unstable population interactions. The preceding has focused on natural enemies
whose population dynamics have been intimately related to that of their
hosts. Such systems might be considered typical of specialist natural
enemies, parasitoids whose reproduction depends primarily on a specific host
species or population. Many species of natural enemies, however, feed or reproduce
on a variety of different hosts, and in such cases their population dynamics
may be more independent of a particular host population. These may be
considered under the term generalist natural
enemies, which are characterized by populations which have densities
independent of and relatively constant over many generations of their hosts,
as distinguished from the specialist whose
dynamics is integrally bound to the dynamics of the host. Equation (11) may be modified to represent a
host population subject to a generalist natural enemy, N_{t+1} = Fn_{t}[{1+aG_{t}/(k(1+aT_{h}N_{t})}^{-k}], (16b) where G_{t} is now the number of generalist
natural enemies attacking the N_{t} hosts, and the other
parameters have the same meaning as before. This equation includes a type II functional
response for a generalist whose interactions with the host population may be
aggregated or independently distributed (depending on the value of k).
One further important feature, the numerical response of the generalist, may
now also be considered. Where such responses have been considered in the
literature, the data to show a tendency for the density of generalists (G_{t}_{)}
to rise with increasing N_{t} to an upper asymptote (Holling
1959a, Mook 1963, Kowalski 1976). This simple relationship may be described
by a formula derived from Southwood & Comins (1976) and Hassell & May
(1986): G_{t} =
m[1-exp(-N_{t}/b)]. (17) Here m is the saturation number of
predators and b determines the prey density at which the number of
predators reaches a maximum. Such a numerical response implies that the
generalist population responds to changes in host density quickly relative to
the generation time of the host, as might occur from rapid reproduction
relative to the time scale of the host or by switching from feeding on other
prey to feeding more prominently on the host in question (Murdoch 1969,
Royama 1979). The complete model for this host-generalist interaction
(incorporating (17) into (16) becomes: am[1-exp(-N_{t/b)]-k} N_{t+1}
= FN_{t}[1 + ________________] (18) [
k(1+aT_{ht) }] This equation represents a reproduction curve
with implicit density dependence. Hassell & May (1986) present an
analysis of this interaction and present the following conclusions: At first
the action of the generalist reduces the growth rate of the host population
(which in the absence of the natural enemy grows without limit in this case).
Whether the growth rate has been reduced sufficiently to produce a new
equilibrium depends upon the attack rate and the maximum number of
generalists being sufficiently large relative to the host fecundity F.
The host equilibrium falls as predation by the generalist becomes less
clumped, as the combined effect of search efficiency and maximum number of
generalists (the overall measure of natural enemy efficiency ah)
increases, and as the host fecundity (F) decreases. A new equilibrium
may be stable or unstable (in which case populations will show limit cycle or
chaotic dynamics). These latter persistent but non-steady state interactions
can arise when the generalists cause sufficiently severe density-dependent
mortality, promoted by low degrees of aggregation (high values for k),
large ah, and intermediate values of host fecundity F. Insect populations can be subject to infection
by viruses, bacteria, Protozoa and fungi, the effects of which may vary from
reduced fertility to death. In many cases these have been intentionally
manipulated against insect populations; reviews of case studies have been
presented by Tinsley & Entwhistle (1974), Tinsley (1979) and Falcon
(1982). Much of this early work was largely empirical,
and a theoretical analysis for interactions among insect populations and
insect pathogens was until recently lacking. An analysis of underlying
dynamical processes in such systems has recently been developed by Anderson
& May 9181) (also see May & Hassell 1988). The principal features of
this framework are as follows: Considering first a host population with
discrete, non-overlapping generations (envisaging perhaps such univoltine
temperate Lepidoptera as the gypsy moth, Lymantria
dispar, and its nuclear
polyhedrosis virus disease) which is affected by a lethal pathogen which is
spread in an epidemic manner via contact between infected and healthy individuals
in the population each generation prior to reproduction. A variant of
equation (5) may be applied to describe the dynamics of such a population
(where g=1 so that there is no other density-dependent mortality): N_{t+1} = FN_{t}f(N_{t}), (19) where f(N_{t}) now represents the fraction
escaping infection. This fraction f which escapes infection as an
epidemic spreads through a population density N_{t} is given
implicitly by the Kermack-McKendrick expression, f=exp{-(1-f)N_{t}N_{T}}
(Kermack & McKendrick 1927), where N_{T} is the threshold
host density (which depends on the virulence and transmissibility of the
pathogen) below which the pathogen cannot maintain itself in the population.
For populations of size N less than N_{T} the epidemic
cannot spread (f=1) and the population consequently grows
geometrically while the infected fraction f decreases to ever smaller
values. As the population continues to grow it eventually exceeds N_{T}
and the epidemic can again spread. This very simple system has very
complicated dynamical behavior; it is completely deterministic yet has
neither a stable equilibrium nor stable cycles, but exhibits completely
chaotic behavior (where the population fluctuates between relatively high and
low densities) in an apparently random sequence. May (1985) has reported in
more detail on this model and its behavior. Many insect host-pathogen systems which have
been studied differ from equation (19) in that transmission is via
free-living stages of the pathogen (rather than direct contact between diseased
and healthy individuals). Additionally, many such populations may have
generations which overlap to a sufficient degree that differential, rather
than difference, equations are a more appropriate framework for their
analysis. Primarily for these reasons the study of many insect host-pathogen
systems have been framed in differential equations. To construct a simple differential framework,
it is first assumed that the host population has constant per capita birth
rates a and death rates (from sources other than the pathogen) b.
The host population N(t) is divided into uninfected (X(t)) and
infected (Y(t)) individuals, N=X+Y. For consideration of insect
systems the model does not require the separate class of individuals which
have recovered from infection and are immune, as may be required in
vertebrate systems, because current evidence does not indicate that insects
are able to acquire immunity to infective agents. This basic model further
assumes that infection is transmitted directly from infected to uninfected
hosts as a rate characterized by the parameter B, so that the rate at
which new infections arise is BXY (Anderson & May 1981). Infected
hosts either recover at rate a or die at rate b. Both infected
and healthy hosts continue to reproduce at rate a and be subject to
other causes of death at rate b. The dynamics of the infected and healthy
portions of the population are now characterized by dX/dt =
a(X+Y)-bX-BXY+Y, (20a) dY/dt = BXY-(alpha+b+)Y. (20b) The healthy host population increases from
both births and recovery of infected individuals. Infected individuals appear
at rate BXY and remain infectious for average time 1/(alpha+b+) before they
die from disease or other causes or recover. The dynamics of the entire
population are characterized by: dN/dt = rN-alphaY, (21) where r=a-b is the per capita growth rate of the
population in the absence of the pathogen. There is no intraspecific density
dependence or self-limiting feature in the host population, so that in the
absence of the pathogen the population will grow exponentially at rate r. Considering now a global feature of the system
what the consequences are of introducing a few infectious individuals into a
population previously free from disease. The disease will spread and
establish itself provided the right-hind side of equation (20b) is positive.
This will occur if the population is sufficiently large relatively to a
threshold density, N>NT, where NT is defined by: N_{T} = (alpha + b + C)/B (22) Because the population in this simple analysis increases exponentially in the absence of the disease, the population will eventually increase beyond the threshold. In a more general situation where other density-dependent factors may regulate the population around some long-term equilibrium level K (in the absence of disease), the pathogen can only establish in the population if K>N_{T} Once established in the host population, the
disease can (in the absence of other density-dependent factors) regulate the population
so long as it is sufficiently pathogenic, with alpha > r. In such
cases, the population of equation (20) will be regulated at a constant
equilibrium level N*=[alpha/{alpha-r)]N_{T}. The proportion of
the host population infected is simply Y*/N*=r/a. Hence the
equilibrium fraction infected is inversely proportional to disease virulence,
and so decreases with increasing virulence of the pathogen. If the disease is
insufficiently pathogenic to regulate the host (A < r), the host
population will increase exponentially at the reduced per capita rate r'=r-A
(until other limiting factors affect the population). The relatively simple system envisaged by
equation (20) permits some additional instructive analysis. First, pathogens
cannot in general drive their hosts to extinction, because the declining host
populations eventually fall below the threshold for maintenance of the
pathogen. Additionally, the features of a pathogen, which might be implicated
in maximal reduction of pest density to an equilibrium regulated by the
disease, should be considered. In particular what degree of pathogenicity
produces optimal host population suppression. Pathogens with low or high
virulence lead to high equilibrium host populations, while pathogens with
intermediate virulence lead to optimal suppression (Anderson & May
1981) This is a vital point because many control programs (and indeed
many genetic engineering programs) often begin with an assumption that high
degrees of virulence are desirable qualities. While this may be true in some
special cases of inundation, it is not true for systems which rely on any
degree of perpetual host-pathogen interaction (May & Hassell 1988). A number of potentially important biological
features are not considered explicitly in the basic representation of
equation (20) (Anderson & May 1981). Several of these have fairly simple
impacts on the general conclusions presented above. Pathogens may reduce the
reproductive output of infected hosts prior to their death (which renders the
conditions for regulation of the host population by the pathogen less
restrictive). Pathogens may be transmitted between generations
("vertically") from parent to unborn offspring (which reduces N_{T}
and thus permits maintenance of the pathogen in a lower density host population).
The pathogen may have a latency period where infected individuals are not yet
infectious (which increases N_{T} and also makes population
regulation by the pathogen less likely). The pathogenicity of the infection
may depend on the nutritional state of the host, and hence indirectly on host
density. Under these conditions the host population may alternate
discontinuously between two stable equilibria. Anderson & May (1981) give
further attention to these cases. A more serious complication arises when the
free-living transmission stage of the pathogen is long-lived relative to the
host species. Such is the case with the spores of many bacteria, protozoa and
fungi and the encapsulated forms of many viruses (Tinsley 1979). Most of the
analytical conclusions for equations (20) still hold, but the regulated state
of the system may not be either a stable point or a stable cycle with period
of greater than two generations. Anderson & May (1981) show that the
cyclic solution is more likely for organisms of high pathogenicity (and many
insect pathogens are highly pathogenic--Anderson & May 1981, Ewald 1987)
and which produce large numbers of long-lived infective stages. The cyclic
behavior results from the time-delay introduced into the system by the pool of
long-lived infectious stages. Such cyclic behavior appears characteristic of
populations of several forest Lepidoptera and their associated diseases
(Anderson & May 1981). In one case where sufficient data were available
to estimate the parameters required by the analytical framework, thee was
substantial agreement between the expected and observed period of population
oscillation (Anderson & May 1981, McNamee et al. 1981). This field of
endeavor will benefit from additional work relating actual populations and
relevant analytical development. The analysis of the simple, two species
interactions considered thus far have focused primarily on single- or
two-factor systems, where the principal features acting on the population
where either intraspecific competition, interspecific competition in the
absence of natural enemies, the action of a natural enemy, or (in some cases)
the action of a natural enemy together with intraspecific competition. In
many populations there may be more than two species interacting, and such
systems would necessarily involve additional interactions, such as herbivores
competing in the presence of a natural enemy or different natural enemies
competing for the same host population. Four such cases are now considered, with
an examination of their dynamical behavior and the relative role the
different interactions may play in population regulation. [
Please also see Cichlid Research ] In many natural systems phytophagous species
are attacked by a entourage of natural enemies, and plants are often attended
by a complex of herbivores. In biological control programs attempts to
reconstruct such multiple-species systems have often met with some debate in
spite of their ubiquitous occurrence. Some researchers have suggested that
interspecific competition among multiple natural enemies will tend to reduce
the overall level of host suppression (Turnbull & Chant 1961, Watt 1965,
Kakehashi et al. 1984). Others view multiple introductions as a potential
means to increase host suppression with no risk of diminished control (van
den Bosch & Messenger 1973, Huffaker et al. 1971, May & Hassell 1981,
Waage & Hassell 1982). The significance of this issue probably varies in
different systems, but the basic principles may be addressed analytically. The dynamics of a system with a single host
and two parasitoids may be addressed by extending the single host-single
parasitoid model of equation (7) to include an additional parasitoid. One
possibility is the case described by May & Hassell (1981): N+1 = FN_{t}h(Q_{t})f(P_{t}) (23a) Q_{t+1} = N_{t}{1-g(Q_{t})}, (23b) P_{t+1} = N_{t}h(Q_{t}){1-f(P_{t})} (23c) Here the host is attacked sequentially by
parasitoids Q and P. the functions h and f
represent the fractions of the host population surviving attack from Q
and P, respectively, and are described by equation (10); the
distribution of attacks by one species is independent of attacks by the
other. Variations on this theme have also been considered, such as when P
and Q attack the same stage simultaneously (May & Hassell 1981);
the general qualitative conclusions are the same. Three general conclusions arise from an examination
of this system. First, the coexistence of the two species of parasitoids is
more likely if both contribute some measure of stability to the interaction
(e.g., the attacks of both species are aggregated: they both have values of k<1
in equation (10)). Secondly, if in the system the host and
parasitoid P already coexist and an attempt is made to introduce
parasitoid Q, then coexistence is more likely if Q has a
searching efficiency higher than P. If Q has too low a
searching efficiency it will fail to become established, precluding
coexistence. If the search efficiency of Q is sufficiently high, it
may suppress the host population below the point at which P can
continue to persist, thus leading to a new single host-single parasitoid
system. Examples of such competitive displacement include the successive
introductions of Opius spp.
against Dacus dorsalis in Hawaii and the
displacement of Aphytis lingnanensis by A. melinus in interior southern California (Luck &
Podoler 1985). Third, and finally, the successful
establishment of a second parasitoid species (Q) will in almost every
case further reduce the equilibrium host population. For certain parameter
values, it can be shown that the equilibrium might have been lower still if only
the host and parasitoid Q were present, but this additional depression
is slight. In general, the analysis points to multiple
introductions as a sound biological strategy. Kakehashi et al. (1984) have considered a case
similar to equation (23) but where the distributions of attacks by the two
parasitoid species are not independent but rather are identical, indicative
of the extreme hypothetical case where two species of parasitoids
respond in the same way to environmental cues, and in locating hosts they
have exactly the same distribution of attacks among the host population. This
alteration does not change appreciably the stability properties of equation
(23), but does change the equilibrium properties. In particular, a single
host-single parasitoid system with the superior parasitoid now has a greater
host population depression than does the three-species system. In natural
systems complete covariance between species of distribution of parasitism may
be less likely than more independent distributions (Hassell & Waage 1984)
and the conclusions regarding this extreme case may be less applicable.
Nevertheless, this is an example where general, tactical predictions can be
affected by changes in detailed model assumptions, emphasizing the importance
of a critical review of the biological implications underlying them. Generalist and
Specialist Natural Enemies The preceding discussion on competing natural
enemies concerns those whose dynamics are inherently related to the dynamics
of their hosts, as is appropriate for such fairly specific natural enemies as
many insect parasitoids. Alternatively, natural enemies with more generalist
prey habits are considered whose dynamics may be more independent of a
particular host species, and turn now to interactions between populations of
specialist and generalist natural enemies. Starting with an analytical
framework, the biological implications are considered with respect to
coexistence of the natural enemies and the effect on the host population
equilibrium and stability. As mentioned earlier in the section of natural
enemies and host density dependence, discrete systems with more than one
mortality factor may have different dynamics depending on the sequence of
mortalities in the hosts life cycle. A situation is presented where the
specialist natural enemy acts first, followed by the generalist, both
preceding reproduction of the host adults. The general framework for this
sequence of events is equation (13), which can now be employed to explore the
particular case of specialist natural enemy followed by generalist (Hassell
& May 1986): N_{t+1} = FN_{t}f(P_{t})g[N_{t}f(P_{t})], (24a) P_{t+1} = N_{t}{1-f(P_{t})}. (24b) Here g(N_{t}) is the effect of
the generalist which, following developments earlier, incorporates a
numerical response together with the negative binomial distribution of
attacks (which allows for independently random to contagious dispersion of
attacks). If it is assumed that handling time is small relative to the total
searching time available, so T_{h}=0 : [ am[1-exp(-N/b)]-k g(N) = [1 + _____________]. (25) [ k ] The function f(P) is the proportion
surviving parasitism and, similarly incorporating the negative binomial
distribution of attacks (and allowing T_{h}=0), is given by: f(P) = [1+a'P/k]^{-k}. (26) Other formulations of these ideas are
possible, in particular structuring equation (24) after (12) to represent the
situation where the specialist natural enemy follows the generalist in the
life history of the host, but the conclusions regarding roles and regulation
are similar. It might now be asked under what circumstances
the generalist and specialist can exist together and what their combined
effect on the host population will be. In particular, a specialist natural
enemy can coexist with the host and generalist most easily if the effect of
the generalist is small (k and am are small, indicating low
levels of highly aggregated attacks) and the efficiency of the specialists is
high and their is low density dependence in the numerical response of the
generalist (Hassell & May 1986). Simply, if the effect of the generalist
is small in terms of the proportions of the population subject to it and in
its regulatory effect, there is greater potential that the host population
can support an additional natural enemy (the specialist). On the other hand
if the host rate of increase F is low or the efficiency of the
generalist population (am) too high, then a specialist is unlikely to
be able to coexist in the host-generalist system. Generally, the parameter
values indicating coexistence of the specialist and generalist are somewhat
more relaxed for the case of the specialist acting before the generalist in
the host life history, because there are more hosts present on which
reproduction of the specialist can take place. In each case the equilibrium
population of the host if further reduced in the three-species system than in
either two-species system. Further details are presented by Hassell & May
(1986). Parasitoid-Pathogen-Host
Systems Another type of system in which there occur
more than one type of natural enemy is that where a host is subject to both a
parasitoid (or predator) and a pathogen (Carpenter 1981, Anderson & May
1986, May & Hassell 1988). These systems may be considered cases of
two-species competition, where the natural enemies compete for the resource
represented by the host population. As in the case for interspecific
competition they are characterized by four possible outcomes: (1) the
parasitoid and pathogen may coexist with the host, (2) either parasitoid or
pathogen may regulate the population at a density below the threshold for
maintenance of the other agent, (3) there may be two alternative stable
stages (one with host and parasitoid and one with host and pathogen), with
the outcome of any particular situation depending on the initial condition of
the system, and (4) the dynamical properties of the component systems may
each be represented in the joint system and additionally may interact and
thereby lead to behavior not present in each individual system. Consequently,
any of the four possible outcomes of the interaction may be characterized by
a steady equilibrium, stable cycles or chaos (May & Hassell 1988). The complex effects of a
host-pathogen-parasitoid system may be illustrated with reference to a simple
model of their combined interactions. The models of equations (7) and (19)
are combined to represent a population which is first attacked by a lethal
pathogen (spread by direct contact) with the survivors then being attacked by
parasitoids: N_{t+1} = FN_{t}S(N_{t})f(P_{t}), (27a) P_{t+1} = cN_{t}S(Nt){1-f(P_{t})}. (27b) Here S(N) is the fraction surviving the
epidemic given earlier (equation (19)) by the implicit relation S=ext[-(1-S)N_{t}/N_{t}],
and f has the Nicholson-Bailey form f(P)=exp(-aP) representing
independent, random search by parasitoids. The dynamical character of this system has
been summarized by May & Hassell (1988). For acN_{T}(lnF)/(F-1)<1
the pathogen excludes the parasitoid by maintaining the host population at
levels too low to sustain the parasitoid. For parasitoids with greater
searching efficiency, or greater degrees of gregariousness, or for systems
with higher thresholds (N_{T)}, so that acN_{T}(lnF)/(F-1)>1,
a linear analysis would suggest that the parasitoid would exclude the
pathogen in a similar manner. However, the diverging oscillations of the
Nicholson-Bailey system eventually lead to densities higher than N_{T}
and the pathogen can repeatedly invade the system as the host population
cycles to high densities. The resulting dynamics can be quite complex, even
from the simple and purely deterministic interactions of equation (27). Here
the basic period of the oscillation is driven by the Nicholson-Bailey model,
with the additional effects of the (chaotic) pathogen-host interaction
leading to stable (rather than diverging) oscillations. As May & Hassell
(1988) discuss, in such complex interactions it can be relatively meaningless
to ask whether the dynamics of the system are determined mainly by the
parasitoid or by the pathogen. Both contribute significantly to the dynamical
behavior, the parasitoid by setting the average host abundance and the period
of the oscillations, and the pathogen providing long term
"stability" in the sense of limiting the amplitude of the
fluctuations and thereby preventing catastrophic overcompensation and
population "crash." Competing Herbivores and Natural
Enemies The presence of polyphagous predators in
communities on interspecific competitors can have profound effects on the number
of species in the community and in the relative roles which predation and
competition play in population dynamics. Classic experiments by Paine (1966,
1974) demonstrated that communities of shellfish contain more species when
subject to predation by the predatory starfish Pisaster ochraceus
than when the starfish is absent, and since that time considerable attention
has been devoted to theoretical considerations of the relative roles of
predation and competition in multispecies communities. Much of this work has
dealt with interactions in homogeneous environments (Parrish & Saila
1970, Cramer & May 1972, Steele 1974, van Valen 1974, Murdoch & Oaten
1975, Roughgarden & Feldman 1975, Comins & Hassell 1976, Fujii 1977,
Hassell 1978, 1979; Hanski 1981). One general conclusion of this work is that
the regulating influence of natural enemies can, under certain conditions,
enable competing species to coexist where they otherwise could not. This
effect is enhanced if the natural enemy shows some preference for the
dominant competitors or switch between prey species as one becomes more
abundant than the other. This work has also been extended to the case
of competing prey and natural enemies existing in a patchy environment
(Comins & Hassell 1987), where the work of Atkinson & Shorrocks
(1981) on two-species competition was used as a foundation. Comins &
Hassell considered the cases for competing preys which are distributed in
patches and either a generalist natural enemy (whose dynamics were unrelated
to the dynamics of the prey community) and for a natural enemy whose
population dynamics was intrinsically related to the prey community (a
"specialist", but polyphagous on the members of the competition
community). For both cases the findings generally supported the earlier
results that the action of natural enemy populations can, in certain cases,
add stability to an otherwise unstable competition community. This is more
readily done by the generalist than the specialist by virtue of the assumed
stability of the generalist population. In all cases aggregation by the
natural enemy in patches of high prey density (which leads to a
"switching" effect) is an important attribute for a natural enemy
to be able to stabilize an otherwise unstable system. Predation which is
independently random across patches is destabilizing for both the generalist
and specialist cases. Coexistence of competing prey species is possible in
this spatially heterogeneous model even when the distributions of the prey species
in the environment are correlated, and when interspecific competition is
extreme. An examination of the problem of searching in
animals shows that it is fundamentally very simple, provided the searching within
a population is random. It is important to realize that we are not
concerned with the searching of individuals, but with that of whole
populations. Many individual animals follow a definite plan when
searching (e.g., a fox follows the scent of a rabbit, or a bee moves systematically
from flower to flower without returning on its course). However, there is
nothing to prevent an area that has been searched by an individual from again
being searched systematically by another, or even the same individual. If
individuals, or groups
of individuals, search independently of one another, the searching within the
population is unorganized and therefore random. Systematic
searching by individuals improves the efficiency of the individuals, but
otherwise the character of the searching within a population remains
unaltered. Therefore, in competition, it may safely be assumed that the
searching is random. The area searched by animals may be measured
in two distinct ways: (1) we may follow the animals through the whole of
their wanderings and measure the area they search, without reference to
whether any portions have already been searched, and so measured, or not:
this is called the area traversed. Or, we may measure only the
previously unsearched area the animals search: this is called the area
covered. Thus, the area traversed represents the total amount of
searching carried out by the animals, while the area covered
represents their successful searching, i.e., the area within which the
objects sought have been found. Competition Curve.--Nicholson (1933) gave
an example of this process. Suppose we take a unit of area, say a square
mile, and consider what happens at each step when animals traverse a further
tenth of that area. When the animals begin to traverse the first tenth of the
area, no part of the area has already been searched, so that in traversing
one-tenth the animals also cover one-tenth of the area. At the
beginning of the next step only 9/10th os the area remains unsearched, so as
the animals search at random (= their populations now), only
9/10ths of the second 10th of the area they search is previously unsearched
area. Consequently, after traversing 2/10th of the area the animals have covered
only 2.71 tenths. At each step of 1/10th of area traversed, the animals cover
a smaller fraction of the area than in the preceding step. Because at each
step the animals cover only 1/10th of the previously unsearched area, the
whole area can never be completely searched. This is true only if the
total area occupied by the animals is very large (not one square mile as
suggested here, necessarily). The results of this progressive calculation
approximates Nicholson's competition curve. Although the competition curve gives the
general character of the effects produced by progressively increasing
competition, it actually only approximates the true form. When the animals
have nearly completed their search of the first 10th of the area, only
slightly more than 9/10ths of the area remains unsearched. This is because
even while traversing the first 10th of the area the animals spend some small
part of the time searching over areas that have already been searched, and
the same type of effort runs through the remainder of the curve. The curve
would become more accurate as its calculations were based on indefinitely
smaller and smaller steps. Bailey (1931, p. 69) gives a formula for this
curve which is the most accurate of all. Examination of the competition curve shows
that as the area traversed increases there is a progressive slowing down in
the rate of increase of the area covered. The searching animals have
progressively increasing difficulty in finding the things they seek. With
random searching, this relation is independent of the properties of the
animals and those of their environments. Because the competition curve represents a
probability, if small numbers of animals and small areas are taken, it is
likely that the relation between the area traversed and that covered will not
be found to be exactly as shown on the curve. This does not mean that there
is anything wrong with the curve, but it does mean that small samples of a
statistical population are not good representatives of the large population
from which they area taken. The Limitation
of Animal Density.--Necessary considerations in the limitation of
animal density determined from the competition curve are the power of increase and the area of discovery. The power of increase
is the number of times a population of animals would be multiplied in each
generation if unchecked. This value is fixed for a given set of conditions
(eg., temperature, RH, host distribution including pattern, etc.). It
determines the fraction of the animals that needs to be destroyed in each
generation in order to prevent increase in density. The area of discovery
is the area effectively traversed by
an average individual during its lifetime. Area of discovery is also a fixed
value for a given set of conditions (e.g., temperature, terrain, etc.). If an
average individual fails to capture, e.g., one-half the objects of the
required kind it meets, then the area of discovery is 1/2 the area traversed.
The value of the area of discovery is determined partly by the properties of
the searching animals, and partly by the properties of the objects sought.
Thus, it is dependent upon the movement, the keenness of the senses and the
efficiency of capture of an average individual when searching. It is also
dependent upon the movement, size, appearance, smell, etc. and the dodging or
resistance of the average object that is being sought. Therefore, under given
conditions, a species has a different area of discovery for each kind of
object it seeks. The value of the area of discovery defines the
efficiency of a species in discovering and utilizing objects of a given kind
under given conditions. It determines the density of animals necessary in
order to cause any given degree of intraspecific competition. The power of
increase and the area of discovery together embrace all those things that
influence the possible rate of increase of the animals and all those that
influence the efficiency of the animals in searching (Nicholson 1933,
Nicholson & Bailey 1935). They are not merely properties of species, but
properties of species when living under given conditions. The same species
may have different properties in different places, or in the same
place at different times. It is also important to notice that climatic
conditions and other environmental factors play their part in determining the
values of these properties, for they influence the vitality and activity of
animals. Therefore, although such environmental factors may not be
specifically mentioned, they appear implicitly in all investigations in which
values are given to the powers of increase and areas of discovery of animals. STEADY
DENSITIES (Steady State) The concept of a steady density has led to
much debate over the years, but in general is misunderstood, for in reality
there is no steady density possible in
animals. It is a mathematical concept, which is useful in showing
population trends. Nicholson (1933) summarized the concept of steady
density. He considered it to be the point where further increase of a
population is prevented when all the surplus animals are destroyed, or when
the animals are prevented from producing any surplus. When this happens, the
animals are in a state of stationary balance with their environments,
and maintain their population densities unchanged from generation to
generation under constant conditions. Because constant conditions are not
possible, the actual steady state is never reached, however. Whenever the
animals' densities reach the mathematical calculation of zero population
growth, this is referred to as the steady state:
the densities of animals when at this position of balance area their steady
densities under the given conditions. The steady densities of animals are determined
from the values of their areas of discovery and powers of increase. An
example was given in Nicholson (1933) as follows: An entomophagous parasitoid attacks a certain
species of host. One host individual provides sufficient food for the full
development of one parasitoid. The area of discovery of the parasitoid is
0.04. The power of increase of the host is 50. There are no factors operating
other than the above. The steady state will be reached when the
parasitoids are sufficiently numerous to destroy 49 out of every 50 hosts,
and when there are sufficient hosts to maintain this density of parasitoids.
The parasitoids are required to destroy 98% of the hosts and so to cover 0.98 of the area occupied by the animals.
To do this it is necessary for the parasitoids to traverse an area of
3.91, as can be seen from the competition curve. The required density of
parasitoids, therefore, is 3.91 / 0.04, i.e., 98 approximately. But in order
that the density of the parasitoids may be maintained exactly, each
parasitoid is required to find on the average one host. Therefore, the
parasitoids are required to find 98 hosts in the area of 0.98 they cover, so
that the steady density is 98 / 0.98, i.e., 100. Of course the steady densities calculated are
the numbers of animals per unit of area. It is always convenient to choose a
large unit for the measurement of area, so that the areas of discovery of the
animals are represented by fractions, for the densities of animals can then
be given in whole numbers. If small units of measurement are used, the
character of the results obtained is actually unaffected, but the densities
calculated have to be expressed as small fractions of an animal per unit of
area, which is not desirable. It should also be noticed that the densities
calculated are those within the areas in which the animals interact, and not
necessarily within the whole countryside. Thus, if the animals can live only
in areas containing a certain kind of vegetation, then the calculated
densities are those within such areas, while the intervening area in which
the vegetation is unsuitable for animals are ignored. Other things being equal, the density of species
within the whole countryside varies directly with the fraction of the
countryside that provides suitable conditions for the species. In this
considering this further, Nicholson (1933) concluded that this is however
only approximately true. GENERALITIES ON
MODELING ARTHROPOD POPULATIONS The subject of modeling of arthropod
populations has been recently reexamined by A. P. Gutierrez (personal
commun.). It was concluded that modeling should be regarded as but another
tool in an increasing arsenal of methods for examining prey-predator
interactions. The strength of the method lies in the ease with which one can
capture the relevant biology in a mathematically simple form, and the utility
of the model for examining field problems and theory (Gutierrez 1992). The
major deficiencies are the possible lack of mathematical rigor in the
formulation of many simulation models and the tendency to add too much
detail, both of which may impair utility for examining population theory. The
question posed may not have a simple answer, as many factors may affect the
outcome making interpretation of the results difficult. For example, the
cassava mealybug model has age structure, invulnerable age classes, age and
time varying fecundity and death rates, relationships to higher and lower
trophic levels, and other factors which interact. Gutierrez (1992) states that simulation
models, however, provide good summaries of our current knowledge of a system,
and furnish a mechanism for examining this knowledge in a dynamic manner.
This capability may stimulate further questions and help guide research. At
their best, simulation models are good tools for explaining components of
interactions not readily amenable to field experimentation and for the
development of simpler models designed to answer specific questions,
including those concerning theory. Most important, model predictions may be
compared with field data and may be used to help evaluate the economic impact
of pests and of introduced natural enemies. We might even be able to evaluate
possible candidate biological control agents before they are introduced.
However, Gutierrez (1992) stresses that only the introduction and release of
a species will provide the definitive answer concerning its potential as a
biological control agent. REFERENCES: <bc-71.ref.htm> [ Additional references may be found
at MELVYL Library ] |