We begin our discussion of first order equations by discussing the third principle of competition. This is because competition gives rise to a persistent equilibrium while the process of cooperation (second principle) produces a transient, unstable equilibrium. For this reason, most of the theoretical work has been done on competition. We begin with a simple model for predators competing with each other for food (= prey). Verhulst (1838) modeled the continuous form of first order population growth, which I rewrite as a discrete (Cook 1965) non-linear logistic equation. Note that Verhulst (1838) regarded the linear logistic as a simplified version and suggested ways in which non-linear terms could be added, which is mostly unacknowledged in the literature nowadays. I write the logistic in terms of the per-capita rate of change (Figure 1)

where R_p is the observed rate of change of the average predator (= consumer), A_p is the maximum per-capita rate of change in a particular physical environment, P_t is the density of the predator population at time t, K is the equilibrium density of the population as determined by conditions in the environment, and Q is a non-linear parameter.

Figure 1. The non-linear logistic model showing two scenarios with different values of K (lateral perturbation as discussed later)

Remember that, according to the first principle, R_p= ln P_t – ln P_t-1, which will only be used hereafter when we want to calculate the actual number of predators produced. The logistic is the simplest model for the growth of a consumer population in a fixed environment, and has been derived, in the linear form, from first principles by Royama (1992; pp. 144-147).

Leslie (1948) seems to have been the first to notice that the logistic can be generalized to include a variable carrying capacity by letting K = CN_t-1 where N_t-1 is equal to the variable amount of prey (or any other limiting factor that sets the carrying capacity, such as territoriality) and C is a proportionality constant related to the size of the prey. Now the variable amount of food (or variable K in equation 1) has quite a different effect than most factors that influence population growth. Here, food acts on the per-capita growth rate as a joint factor and causes, what Royama (1992) calls, a lateral perturbation effect; i.e., causing variation along the P axis but not on the R axis (Figure 1). The more usual pattern assumed for exogenous variability is a vertical perturbation which causes changes in both P and R . Recently, Lindstrom et al. (2005) showed that the food supply for an insectivorous bird had a strong effect on the per-capita rate of change when modeled as a lateral perturbation, while it had almost no effect when modeled as a vertical perturbation. This study showed that the kind of perturbation (lateral vs vertical) was critical in determining the effect of food on the per-capita rate of change. Until recently, almost all the studies on external variables have assumed that they operate as vertical perturbations. However, with the recent importance of climate change, the type of perturbation has received increasing interest, and more studies are recognizing the importance of lateral perturbations (Elmberg et al. 2003, Einum 2005, Berryman and Lima 2006, Lima et al. 2006, Finstad et al. 2009). As far as I am aware, the logistic is the only R-function that automatically includes lateral perturbations and, therefore, I believe that it is the correct formulation from the point of view of usage and derivation (e.g., it has been derived from first principles by Royama, 1992). Notice that all first order negative feedback processes are called, for convenience, competition. Some may find this too general. However, it is true that the process can be viewed as competition in the broad sense. For instance, intra-specific cannibalism, as mentioned previously, can be thought of as the outcome of competition – the losers being directly fed upon by their competitors rather than just dying and not being consumed.

Berryman, A. A. and Lima, M. 2006. Deciphering the effects of climate on animal populations: diagnostic analysis provides new interpretation of Soay sheep dynamics. Amer. Natur. 168: 784-795.

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

Einum, S. 2005. Salmonid population dynamics: stability under weak density dependence? Oikos 110: 630-633.

Elmberg, J., Nummi, P., Poysa, H. and Sjoberg, K. 2003. Breeding success of sympatric dabbling ducks in relation to population density and food resources. Oikos 100: 333-341.

Finstad, A. G., Einum, S., Ugedal, O. and Forseth, T. 2009. Spatial distribution of limited resources and local density regulation in juvenile Atlantic salmon. J. Anim. Ecol. 78: 226–235.

Leslie, P. H. 1948. Some further notes on the use of matrices in population mathematics. Biometrika 35: 213-245.

Lima, M., Previtali, M. A. and Meserve, P. L. 2006. Climate and small rodent dynamics in semi-arid Chile: the role of lateral and vertical perturbations and intra-specific processes. Clim. Res. 30: 125–132.

Lindstrom, A., Enemar, A., Anderson, G., von Proschwitz, T. and Nyholm, N. E. I. 2005. Density-dependent reproductive output in relation to a drastically varying food supply: getting the density measure right. Oikos 110: 155-163.

Royama, T. 1992. Analytical Population Dynamics. Chapman & Hall.

Verhulst, P.-F. 1838 Notice sur la loi que la population suit dans son accrossement. Corr. Math. Phys. 10: 113-121.