Suppose P predators each eat W prey per year resulting in WP prey being killed each year. The probability of an individual prey being killed in a population of N prey, or the per-capita death rate of prey, is then

Now suppose there are also other prey species present in the area. Under these conditions, the per-capita death rate of prey will depend on the density and the relative preference of the predators for these other prey - call this F, so that

The number of prey killed per predator per year, N_a, can be obtained by multiplying this expression by N and dividing by P

which gives us the cyrtoid behavioral response of the predator to prey density.

Let the per-capita rate of change of the prey population in the absence of predation be given by A_N and let this quantity be reduced in proportion to the per-capita death rate due to predation, D_N , then

where R_N is the realized per-capita rate of change of prey and N_t-1 is the density of prey at the beginning of a time interval (say a year). Note that the use of the time subscript identifies N as a dynamic variable while all the other units are assumed constant. The parameter A_N , the maximum per-capita rate of change, reflects the genetic potential of the prey species when living in a particular environment free of predators.

The escape threshold for the prey population can be determined by remembering that R_N = 0 when N = E and, substituting these values in the equation to obtain

Thompson's model for prey survival from parasitism.

Thompson (1924) seems to have been the first to propose a model for the survival of a prey population from attack by natural enemies. Although Thompson was thinking about insect parasitoids attacking insect hosts, the model can be generalized to predators with a little imagination and caution. Insect parasitoids lay their eggs on or in an insect host and the young larval parasitoids eventually devour and kill the host. Thompson assumed that the parasitoid population deposited its eggs at random among a population of 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 (click here for more information from Alexei Sharov)

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 W eggs on N hosts, then the total number of eggs laid by P parasitoids is WP and the mean number laid per host is WP/N. Substituting this quantity for we see that the probability of a host having no attacks and, therefore, surviving parasitism, is

Now the population in the absence of parasitism is described by the finite growth equation N_t = N_t-1G for unconstrained population growth. Multiplying this equation by the probability of surviving parasitoid attack provides us with a dynamic model for the growth of a prey population in the presence of a natural enemy population

Converting this equation to natural logarithms

with A_N = lnG, provides us with an R-function for a prey population attacked by natural enemies which is almost identical to that derived earlier (with no alternative prey). Notice that the per-capita rate of prey survival is converted to a per-capita death rate in logarithms. This can be demonstrated by the conversion ln(N_a / N) = lnN_a - lnN, where N_a is the number of prey attacked, N_a/N is the prey survival rate, and -(lnN_a - lnN) is the logarithmic death rate.

Thompson's prey survival function underlies many of the models used to study predator-prey relationships. For instance, it becomes the well-known Nicholson-Bailey (1935) model when W = aN (Royama 1971), and the Michaelis-Menten-Holling model when alternative prey are included (Berryman 1992). Royama (1971) presents a highly technical treatment of the various models for predators and prey interactions.

References:

• Berryman, A. A. 1992. The origins and evolution of predator-prey theory. Ecology 73: 1530-1535.
• Nicholson, A. J. and V. A. Bailey. 1935. The balance of animal populations. Part 1. Proceedings of the Zoological Society of London, Part 3: 551-598.
• Royama, T. 1971. A comparative study of models for predation and parasitism. Researches in Population Ecology, Supplement No.1, 91 pp.
• 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.

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