Well it’s time to start the mathematics. There is something we must discuss before we begin though. Differential equations are often used to describe biological populations. However, they are in my opinion inapplicable and often result in the wrong description. They were invented for the purpose of describing processes that operate continuously, like the motion of planets. Life is not at all like that, for organisms are born (a very small probability) and die (a certainty). They are therefore discrete entities. Animals like human beings, with completely overlapping generations and no particular breeding or dying times, can admittedly be approximated by a continuous system. However, many animals and plants breed once a year and, for some species, die once a year also. Continuous systems are not designed to describe such events. What is really needed is a new mathematics but, until such can be built, we have to make do with what we have. The best is to use a discrete mathematics, such as difference equations, with a timing of once a year (the circling of the planet around the sun). I will use these in my discussion (by the way, I have been as guilty as anyone of using differential equations; e.g., Berryman et al. 1995 – the differential equations in this paper should be read as difference equations).

The second important question is our continuity from first order to second order interactions (that is, first order interactions within a species and second order interactions between two species). We do not want to change our thought process during this shift because it has caused many of the problems in population modeling. Thus, we will identify two basic elements in our discussion of first order events; N the number of prey and P the number of consumers (or predators). We will derive the two basic models of first order dynamics (one for cooperation and one for competition) by reference to these two groups. Then we will build predator-prey (two species) interaction models by using the single-species models as a starting point. This is completely different from the usual way that second order models are derived.

The first thing we need to discuss is the first principle. This principle merely describes the exponential growth of a population with a known per-capita rate of change. On a logarithmic scale this is a constant rate. Thus, we only need it for computing the actual numbers of organisms in the future population for a given per-capita rate of change. We also cannot say very much about cooperation (second principle) since little theoretical work has been done on this process. Thus, I will begin by considering simple mathematical models of competition, the third principle of population dynamics. Using this principle we will then consider the structure of cooperation models.

Berryman, A. A., Michalski, J., Gutierrez, A. P. and Arditi, R. 1995. Logistic theory of food web dynamics. Ecology 76: 336-343.