The Morris-Varley Key Factor Approach

One of the traditional ways of modeling population dynamics is to view population change in terms of the reproduction and survival of individual organisms. For example, we could express the number of insects in one generation, H_t , as being equal to the number in the previous generation, H_t-1 , times the per-capita reproductive rate, G_h , times the probabilities of surviving different causes of mortality (Morris 1959, Berryman 1999, Berryman and Gutierrez 1999)

where 0 < S_m < 1 could be the probability of an individual surviving the effects of intraspecific competition for fixed resources (usually competition for some kind of spatial resource such as territories or nesting sites), 0 < S_p < 1 the probability of surviving intraspecific competition for depletable resources in the lower trophic level (in this case plant food, P), and 0 < S_c < 1 the probability of surviving the attack of natural enemies in the upper trophic level, C (or intraspecific competition for enemy-free-space). Of course the S-values could also refer to the survival of different stages of development, age classes, or other kinds of mortality factors. However, as we are mainly interested in the feedback structure of trophic webs, we will ignore these details.

When this equation is converted to logarithms we obtain the k-value model of Varley et al. (1973)

where k_i (i = m, p, c) = log₁₀ S_i. Notice that the killing power of the mortality factor, k_i,, is defined formally as the logarithm of the initial numbers on which the i^th mortality factor or process acted minus the logarithm of the number surviving that mortality factor. Although logarithms to the base 10 were used by Varley et al., it is probably more correct to use natural logarithms, as is done in this course, because population growth is an exponential (geometric) process.

References

Berryman, A. A. 1999. Alternative perspectives on consumer-resource dynamics: a reply to Ginzburg. Journal of Animal Ecology 68: 1263-1266.

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.

Morris, R. F. 1959. Single-factor analysis in population dynamics. Ecology 40: 580-588.

Varley, G. C., G. R. Gradwell and M. P. Hassell. 1973. Insect Population Ecology: an analytical approach. Blackwell, London (read pages 1-9).