
The dynamics of biological exploitationOne thing is clear: the problem of cyclic fluctuations is fundamentally a problem of dynamics, the change of variables through time. We can explore the dynamics of biological exploitation by modelling the change in populations through time by means of differential equations. The earliest attempt to capture the dynamics associated with exploitation was that of Vito Volterra, the famous Italian mathematician. Interestingly, at about the same time, Alfred Lotka, the American chemist, studied the same set of equations in the context of a hypothetical chemical reaction. We recall that the LotkaVolterra equations for the densities P, V of predator and victim species, respectively, read where T is time, r is the victim species' Malthusian rate of increase, k the perpredator kill rate, a conversion factor from victims to baby predators, and m the predator's death rate. As each of the densities P, V change in time, we can plot each density against time. The two rightmost graphs show the densities of predator and victim against time. Which one is which? We can also plot one density against the other. The resulting graph (on the left) is called a phase portrait. For simple twovariable models such as this, the phase portrait gives us all the essential information about the model.
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