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We study the stability loss delay phenomenon in a dynamic transcritical bifurcation in a class of three-dimensional prey and predator systems. The dynamics of the predator is assumed to be slow comparatively to the dynamics of the prey. As an application, a well-known model considered by Clark will be discussed.

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When populations are in competition, it often happens that one of them disappears. Harvesting may be used for the control and management of competing species to stabilize the populations at a persistent equilibrium. A three-dimensional model, where the harvesting effort is a dynamic variable, is studied in the case where the growth rate of the harvesting effort is very slow. The analysis shows that the system can have relaxation oscillations. Dynamic bifurcation theory is used to determine the maximal and minimal values of harvesting effort along this cycle of oscillations. Academic Press