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Relaxation oscillations in a class of predator-prey systems. (English) Zbl 1094.34025

We consider a class of three-dimensional, singularly perturbed predator-prey systems having two predators competing exploitatively for the same prey in a constant environment. By using dynamical system techniques and the geometric singular perturbation theory, we give precise conditions which guarantee the existence of stable relaxation oscillations for systems within the class. Such result shows the coexistence of the predators and the prey with quite diversified time response which typically happens when the prey population grows much faster than those of predators. As an application, a well-known model is discussed in detail by showing the existence of stable relaxation oscillations for a wide range of parameter values of the model.

MSC:

34C26 Relaxation oscillations for ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
92D25 Population dynamics (general)
34A26 Geometric methods in ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
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