The final task is to develop a model for the survival of individual organisms from attack by their natural enemies. Returning to the general survivorship model

G_h = the 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 = the probability of an individual surviving the effects of intra-specific 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. Remember that (ln S_p) = [- c_hH_t-1 / (P_t-1 + w_h)] where is c_h is a coefficient of intraspecific competition for a unit of depletable resource in the lower trophic level, and w_h represents the abundance of, and relative preference for, alternative foods present in the environment.

S_c = the probability of an individual surviving the attack of enemies. This should also be interpreted broadly to include the effects of predators on the reproductive capacity of the individual as well as direct mortality; i.e., reduced mating frequency and egg production due to constant harassment by predators.

Thompson (1924) seems to have been the first to propose a model for the survival of a prey population from attack by natural enemies (see also Berryman and Gutierrez 1999). Thompson was thinking about insect parasitoids attacking insect hosts but the model can be extended to the general case with a little imagination. Insect parasitoids lay their eggs on or in an insect host, and the young larval parasitoid eventually devours and kills its host. Thompson assumed that female parasitoids deposited their eggs indiscriminately and at random among a population of H hosts. If the populations are large enough, then the probability of 0, 1, 2,...... attacks on a particular host is given by the Poisson distribution (external link to Poisson distribution)

where Pr(i) is the probability of exactly i attacks on a host and is the mean number of attacks per host. Now if each parasitoid lays d_c eggs on H hosts, then the total number of eggs laid by C parasitoids is d_cC and the mean number laid per host is d_cC / H. Berryman (1992) suggested that this concept be generalized to parasitoids that also attack a constant number of other hosts, w_c, by adding this number to the denominator. In this case, the mean number of eggs laid per host becomes d_cC/(H + w_c). Substituting this quantity for in the above equation, then the probability of a host having zero attacks and, therefore, surviving parasitism, is given by the zero term of the Poisson distribution (i = 0)

or in logarithmic form

Incorporating this expression to the general survival model yields

or the R-function

Notice that the R-function creates a linear model which can be fit to data with simple linear regression techniques. As we will see later, this gives the general survivorship model considerable practical value.

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.

Thompson, W. R. 1924. La theory mathematique de l'action des parasites entomophages et le facteur du hassard. Annales Faculte des Sciences de Marseille 2: 69-89.