Continuous Models for Interacting Populations

Abstract

When species interact the population dynamics of each species is affected. In general there is a whole web of interacting species, sometimes called a trophic web, which makes for structurally complex communities. We consider here systems involving 2 or more species, concentrating particularly on two-species systems. The book by Kot (2001) discusses such models (including age-structured interacting population systems) with numerous recent practical examples. There are three main types of interaction. (i) If the growth rate of one population is decreased and the other increased the populations are in a predator–prey situation. (ii) If the growth rate of each population is decreased then it is competition. (iii) If each population's growth rate is enhanced then it is called mutualism or symbiosis. All of the mathematical techniques and analytical methods in this chapter are di-rectly applicable to Chapter 6 on reaction kinetics, where similar equations arise; there the 'species' are chemical concentrations. 3.1 Predator–Prey Models: Lotka–Volterra Systems Volterra (1926) first proposed a simple model for the predation of one species by another to explain the oscillatory levels of certain fish catches in the Adriatic. If N (t) is the prey population and P(t) that of the predator at time t then Volterra's model is d N dt = N (a − bP), (3.1) d P dt = P(cN − d), (3.2) where a, b, c and d are positive constants. The assumptions in the model are: (i) The prey in the absence of any predation grows unboundedly in a Malthusian way; this is the a N term in (3.1). (ii) The effect of the predation is to reduce the prey's per capita growth rate by a term proportional to the prey and predator populations; this is the −bN P term. (iii) In the absence of any prey for sustenance the predator's death rate results in exponential decay, that is, the −d P term in (3.2). (iv) The prey's contribution to the predators' growth rate is cN P; that is, it is proportional to the available prey as well as to the size of the predator population. The N P terms can be thought of as representing the conversion of energy from one source