Starting with the general Morris-Varley survivorship model, delete for the present the expressions for competition for depletable resources and predation so we are left with a model for intraspecific competition for fixed resources

G_h = maximum production of offspring in a given environment when population density is very sparse (no competition for fixed resources).

S_m = probability of an individual surviving the effects of intraspecific competition for fixed resources. This usually means competition for some kind of spatial resource such as territories or nesting sites. The effects of competition are interpreted in their broadest sense to include reduction from the potential birth rate as well as direct mortality.

Royama (1992) approached the problem of modeling the survival of an individual organism from intraspecific competition for fixed resources by analyzing a spatially-defined geometric model. Following Royama's analysis, the probability of an individual surviving form the effects of intra-specific competition is given by

where b_h is a coefficient of intraspecific competition. Inserting this expression into the general survival model yields

This equation turns out to be identical to the well-known Ricker (1954) spawner-recruit model, commonly used by fisheries biologists to describe the relationship between the abundance of spawning fish and the subsequent number recruits to the fishery. Although less appreciated, the same equation was also suggested by Moran (1950). Converting the Moran-Ricker equation to natural logarithms

and setting (ln G_h) = a_h, the instantaneous per-capita rate of change, then

Then taking antilogs,

we arrive at a discrete form of the famous logistic equation (Cook 1965). Not only is Royama's intraspecific competition equation identical to the Moran, Ricker and logistic models, but it is also the same as Nicholson's (1933) "competition curve". It is undoubtedly the most widely used and most general model for intraspecific competition. Based on its origin and logical derivation, I believe it is the correct model to use to describe the competitive process.

Of course, this equation can also be transformed to natural logarithms

or, for convenience, to the R-function

with R_h the realized per-capita rate of change of population H.

References

Cook, L. M. 1965. Oscillation in the simple logistic growth model. Nature 207: 316.

Moran, P. A. P. 1950. Some remarks on animal population dynamics. Biometrics 6: 250-258.

Nicholson, A. J. 1933. The balance of animal populations. Journal of Animal Ecology 2: 132-178.

Ricker, W. E. 1954. Stock and recruitement. Journal of the Fisheries Research Board of Canada 11: 559-623.

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