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Relaxation oscillations in a slow-fast modified Leslie-Gower model. (English) Zbl 1433.34069

The authors consider the singularly perturbed planar system \[\frac{dx}{dt}= x(1-x)- \frac{axy}{x+ e_1},\qquad \frac{dy}{dt}={\varepsilon y}\Biggl(1-\frac{y}{k+ e_2}\Biggr),\tag{\(*\)}\] where all parameters are positive, \(\varepsilon\) is small.
They derive conditions on the parameters such that \((*)\) for sufficiently small \(\varepsilon\) has a unique limit cycle in the positive orthant. The proof is based on the entry-exit function and on the geometric perturbation theory.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
92D25 Population dynamics (general)
34E15 Singular perturbations for ordinary differential equations
34C26 Relaxation oscillations for ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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