The Lotka Volterra model
The simplest differential equation model is known (like the model of interspecific competition) by the name of its originators: LotkaVolterra (Volterra, 1926; Lotka, 1932). This will serve as a useful point of departure. The model has two components: P, the numbers present in a predator (or consumer) population, and N, the numbers or biomass present in a prey or plant population.
We assume initially that in the absence of consumers the prey population increases exponentially (see Section 5.9):
increase with the numbers of predators (P) and the numbers of prey (N). However, the exact number encountered and successfully consumed will depend on the searching and attacking efficiency of the predator: a, sometimes also called the 'attack rate'. The consumption rate of prey will thus be aPN, and overall:
the LotkaVolterra prey equation
In the absence of prey, predator numbers in the model are assumed to decline exponentially through starvation:
But prey individuals are removed by predators at a rate that depends on the frequency of predatorprey encounters. Encounters will where q is the predator mortality rate. This is counteracted by predator birth, the rate of which is assumed to depend on only two things: the rate at which food is consumed, aPN, and the predator's efficiency, f at turning this food into predator offspring. Predator birth rate is therefore faPN, and overall:
the LotkaVolterra predator equation dP/dt = faPN  qP.
Equations 10.2 and 10.4 constitute the LotkaVolterra model.
The properties of this model can be investigated by finding zero isoclines. Zero isoclines were described for models of twospecies competition in Section 8.4.1. Here, there are separate zero isoclines for the predators and prey, both of which are drawn on a graph of prey density (xaxis) against predator density (yyaxis). Each is a line joining those combinations of predator and prey density that lead either to an unchanging prey population (dN/dt = 0; prey zero isocline) or an unchanging predator population (dP/dt = 0; predator zero isocline). Having drawn, say, a prey zero isocline, we know that combinations to one side of it lead to prey decrease, and combinations to the other to prey increase. Thus, as we shall see, if we plot the prey and predator zero isoclines on the same figure, we can begin to determine the pattern of the dynamics of the joint predatorprey populations.
In the case of the prey (Equation 10.2), when:
properties revealed by zero isoclines
Thus, since r and a are constants, the prey zero isocline is a line for which P itself is a constant (Figure 10.2a). Below it, predator abundance is low and the prey increase; above it, predator abundance is high and the prey decrease.
Likewise, for the predators (Equation 10.4), when:
The predator zero isocline is therefore a line along which N is constant (Figure 10.2b). To the left, prey abundance is low and the predators decrease; to the right, prey abundance is high and the predators increase.
Putting the two isoclines together (Figure 10.2c) shows the behavior ofjoint populations. Predators increase in abundance when there are large numbers of prey, but this leads to an increased predation pressure on the prey, and thus to a decrease in prey abundance. This then leads to a food shortage for predators and a decrease in predator abundance, which leads to a relaxation of an underlying tendency towards coupled oscillations which are structurally unstable in this case predation pressure and an increase in prey abundance, which leads to an increase in predator abundance, and so on (Figure 10.2d). Thus, predator and prey populations undergo 'coupled oscillations' in abundance, which continue indefinitely.
The LotkaVolterra model, then, is useful in pointing to this underlying tendency for predatorprey interactions to generate fluctuations in the prey population tracked by fluctuations in the predator population. The detailed behavior of the model, however, should not be taken seriously, because the cycles it exhibits are 'structurally unstable', showing 'neutral stability'. That is, the populations would follow precisely the same cycles indefinitely, but only until some external influence shifted them to new values, after which they would follow new cycles indefinitely (Figure 10.2e). In practice, of course, environments are continually changing, and populations would continually be 'shifted to new values'. A population following the LotkaVolterra model would, therefore, not exhibit regular cycles, but, because of repeated disturbance, fluctuate erratically. No sooner would it start one cycle than it would be diverted to a new one.
For a population to exhibit regular and recognizable cycles, the cycles must themselves be stable: when an external influence changes the population level, there must be a tendency to return to the original cycle. In fact, as we shall see, predatorprey models (once we move beyond the very limiting assumptions of the LotkaVolterra model) are capable of generating a whole range of abundance patterns: stablepoint equilibria, multigeneration cycles, onegeneration cycles, chaos, etc.  a range repeated in surveys of real populations. The challenge is to discover what light the models can throw on the behavior of real populations.
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