The general survivorship model

has been defined by the explicit model

Alternatively, the general R-function

has the explicit form

This model is composed of four basic components:

1. a_h = ln G_h, the maximum rate of production of herbivore offspring in a given environment when population density is very sparse (no competition for resources) and when predators are absent.
2. b_h H_t-1 = ln S_m, the reduction from the maximum rate of reproduction due to intraspecific competition for fixed resources, with b_h the coefficient of intraspecific competition.
3. c_h H_t-1 / (P_t-1 + w_h) = ln S_p , the reduction from the maximum rate of reproduction due to intraspecific competition for depletable resources, with c_h a coefficient of intraspecific competition for a unit of depletable resource in the lower trophic level.
4. d_c C_t-1 / (H_t-1 + w_c) = ln S_c , the reduction from the maximum rate of reproduction due to attack from enemies in the upper trophic level. This can also be perceived as intraspecific competition for enemy-free-space.

Remember that all the survival terms should be interpreted broadly to include the effects of competition and/or predation on the reproductive capacity of individuals as well as their death rates. It is also important to realize that the terms in the numerators of the survival terms (i.e., b_hH_t-1 and d_cC_t-1) represent the demand for resources by consumer populations while the terms in the denominators (w_h + P_t-1 and w_c + H_t-1) represent the total supply of resources, including alternative food. Hence, the terms in the model represent demand/supply ratios and the dynamics are regulated by demand/supply relationships. For this reason we can also consider the survivorship models to be ratio-dependent. However, demand/supply ratios of survival models are the reverse of the supply/demand ratios of the other ratio-dependent models (e.g., the Lotka-Volterra-Arditi-Ginzburg and Ivlev-Watt-Gutierrez ratio-dependent equations have the resource in the numerator and consumer in the denominator).

The individual survival equation developed above meets all the conditions for a plausible food chain model (Berryman et al. 1995a,b). Because the model emerges from generalizations of the logistic equation, the oldest and most widely known population model, it has been referred to as a logistic food chain model (Berryman 1995b). It is fairly straightforward, though by no means trivial, to extend the model to food webs of arbitrary complexity by expanding the alternative prey constant, w_i, to one or more specific prey populations, and by adding other species of predators to the numerator of the attack function (Berryman et al. 1995b). In this course, however, models of complex webs are not considered.

Reference

Berryman, A. A., A. P. Gutierrez and R. Arditi. 1995a. Credible, parsimonious and useful predator-prey models -- a reply to Abrams, Gleeson, and Sarnelle. Ecology 76: 1980-1985.

Berryman, A. A., Michalski, J., Gutierrez, A. P. and Artiditi, R. 1995b. Logistic theory of food dynamics. Ecology 76: 336-343.

©1997 Alan A. Berryman