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![]() The dynamics of biological exploitation![]() One 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 Lotka-Volterra 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 per-predator kill rate,
We can also plot one density against the other. The resulting graph (on the left) is called a phase portrait. For simple two-variable models such as this, the phase portrait gives us all the essential information about the model.
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