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Analysis of a predator-prey model with specific time scales: a geometrical approach proving the occurrence of canard solutions. (English) Zbl 1475.92138

A predator-prey model with different characteristic time scales for the prey and predator populations was studied, assuming that the predator dynamics is much slower than the prey one. Geometrical singular perturbation theory provides the mathematical framework for analyzing the dynamical properties of the model. This model exhibits a Hopf bifurcation and it was proved that when this bifurcation occurs, a canard phenomenon arises. An analytic expression was provided to get an approximation of the bifurcation parameter value for which a maximal canard solution occurs. The model is the well-known Rosenzweig-MacArthur predator-prey differential system. An invariant manifold with a stable and an unstable branches occurs and a geometrical approach is used to explicitly determine a solution at the intersection of these branches. The method used to perform this analysis is based on blow-up techniques. The analysis of the vector field on the blown-up object at an equilibrium point where a Hopf bifurcation occurs with zero perturbation parameter representing the time scales ratio, allows to prove the result. Predator-prey models with two time scales have already be the topics of several papers and slow-fast limit cycles have already been highlighted in these models; Moreover, canard phenomenons in predator-prey systems have already be mentioned or conjectured in previous works but never been analyzed in detail with a generic method. Note that in dimension 2, this is more a mathematical curiosity than an ecological feature. However, the method used here is general and can be applied to more realistic situations. In higher dimension (e.g. food chain), the method still applies and can actually explain different types of fluctuations. In this case, the canard phenomenon would help to understand food chain dynamics and then contribute to explain ecological properties. Numerical simulations illustrate the result and allow to see the canard explosion phenomenon

MSC:

92D25 Population dynamics (general)
34C40 Ordinary differential equations and systems on manifolds
37G10 Bifurcations of singular points in dynamical systems
37N25 Dynamical systems in biology
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