The next step is to include intraspecific competition for depletable resources in the general survivorship model. Omitting survival for predation for the moment, the model can be written

G_h = maximum production of offspring in a given environment when population density is very sparse (no competition for resources) and when predators are absent. Remember that (ln G_h ) = a_h.

S_m = probability of an individual surviving the effects of intraspecific competition for fixed resources. Remember that (ln S_m) = (-b_hH), where b_h is a coefficient of intraspecific competition.

S_p = probability of an individual surviving the effects of intraspecific competition for depletable resources. This usually means competition for food in the lower trophic level because food, once assimilated, is removed permanently from the environment. Remember that the effects of competition are interpreted in their broadest sense to include reduction from the potential birth rate as well as direct mortality from starvation.

In order to develop a model for survival of an individual organism from the effects of intraspecific competition for depletable resources, first return to the survival model resulting from Royama's geometric analysis of sessile organisms competing for a constant supply of resources

Following Royama (1992), the parameter b_h can be decomposed into b_h = 4n(1 - k) / D, where n is the minimal per-capita resource requirement of the individual, k is the marginal effect of a competitor on the reproductive rate of an individual, and D is the fixed density of resources (see also Berryman et al. 1995). Setting c_h = 4n(1 - k), to be the effect of intraspecific competition for a unit of resource, then

It is now possible to relax the assumption of fixed resources by letting D = P_t-1 + w_h, where P_t-1 is the density of a particular food population and w_h is a constant density of alternative foods (Berryman 1992). Hence, the equation for intraspecific competition for depletable resources is

or

This expression can now be added to the general survival model to give

or the R-function

References

Berryman, A. A. 1992. The origins and evolution of predator-prey theory. Ecology 73: 1530-1535.

Berryman, A. A. and A. P. Gutierreez. 1999. Dynamics of insect predator-prey interactions. In C. B. Huffaker and A. P. Gutierrez (Eds.). Ecological Entomology (pp. 389-423). John Wiley and Sons, New York.

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

Royama, T. 1992. Analytical Population Dynamics. Chapman and Hall, London.