A mathematic model to account for density-dependent regulation of population growth was developed by Verhulst in 1838 and again, independently, by Pearl and Reed (1920). This logistic model (see Fig. 6.7) often is called the Pearl-Verhulst equation (Berryman 1981, Price 1997). The logistic equation is as follows:

where K is the carrying capacity of the environment. This model describes a sigmoid (S-shaped) curve (see Fig. 6.7) that reaches equilibrium at K. If N < K, then the population will increase up to N = K. If the ecosystem is disturbed in a way that N > K, then the population will decline to N = K.

General models such as the Pearl-Verhulst model usually do not predict the dynamics of real systems accurately. For example, the use of the logistic growth model is limited by several assumptions. First, individuals are assumed to be equal in their reproductive potential. Clearly, immature insects and males do not produce offspring, and females vary in their productivity, depending on nutrition, access to oviposition sites, etc. Second, population adjustment to changing density is assumed to be instantaneous, and effects of density-dependent factors are assumed to be a linear function of density. These assumptions ignore time lags, which may control dynamics of some populations and obscure the importance of density dependence (Turchin 1990). Finally, r and K are assumed to be constant. In fact, factors (including K) that affect natality, mortality, and dispersal affect r. Changing environmental conditions, including depletion by dense populations, affect K. Therefore, population size fluctuates with an amplitude that reflects variation in both K and the life history strategy of particular insect species. Species with the r strategy (high reproductive rates and low competitive ability) tend to undergo boom-and-bust cycles because of their tendency to overshoot K, deplete resources, and decline rapidly, often approaching their extinction threshold, whereas species with the K strategy (low reproductive rates and high competitive ability) tend to approach K more slowly and maintain relatively stable population sizes near K (Boyce 1984). Modeling real populations of interest, then, requires development of more complex models with additional parameters that correct these shortcomings, some of which are described as follows.

Nonlinear density-dependent processes and delayed feedback can be addressed by allowing r to vary as follows:

where rmax is the maximum per capita rate of increase, s represents the strength of interaction between individuals in the population, and T is the time delay in the feedback response (Berryman 1981). The sign and magnitude of s also can vary, depending on the relative dominance of competitive and cooperative interactions:

where sp is the maximum benefit from cooperative interactions, and sm is the competitive effect, assuming that s is a linear function of population density at time t (Berryman 1981). The extinction threshold, E, can be incorporated by adding a term forcing population change to be negative below this threshold:

Similarly, the effect of factors influencing natality, mortality, and dispersal can be incorporated into the model to improve representation of r.

The effect of other species interacting with a population was addressed first by Lotka (1925) and Volterra (1926). The Lotka-Volterra equation for the effect of a species competing for the same resources includes a term that reflects the degree to which the competing species reduces carrying capacity:

where N1 and N2 are populations of two competing species, and a is a competition coefficient that measures the per capita inhibitive effect of species 2 on species 1.

Similarly, the effects of a predator on a prey population can be incorporated into the logistic model (Lotka 1925, Volterra 1926) as follows:

where N1 is prey population density, N2 is predator population density, and p is a predation constant. This equation assumes random movement of prey and predator, prey capture and consumption for each encounter with a predator, and no self-limiting density effects for either population (Pianka 1974, Price 1997).

Pianka (1974) suggested that competition among prey could be incorporated by modifying the Lotka-Volterra competition equation as follows:

,.T ,.T r1N2t r1N1ta12N2t

K1 K1

where a12 is the per capita effect of the predator on the prey population. The prey population is density limited as carrying capacity is approached.

May (1981) and Dean (1983) modified the logistic model to include effects of mutualists on population growth. Species-interaction models are discussed more fully in Chapter 8.

Gutierrez (1996) and Royama (1992) discussed additional population-modeling approaches, including incorporation of age and mass structure and population refuges from predation. Clearly, the increasing complexity of these models, as more parameters are included, requires computerization for prediction of population trends.

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