MIO UM 110, Aix-Marseille Univ., Univ. Toulon, CNRS, IRD, 13288, Marseille, France.
Faculty of Science, VU Amsterdam, De Boelelaan 1085, 1081 HV, Amsterdam, The Netherlands.
J Math Biol. 2020 Jan;80(1-2):39-60. doi: 10.1007/s00285-019-01337-4. Epub 2019 Feb 20.
We study a predator-prey model with different characteristic time scales for the prey and predator populations, 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 we prove that when this bifurcation occurs, a canard phenomenon arises. We provide an analytic expression 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. Numerical simulations illustrate the result and allow to see the canard explosion phenomenon.
我们研究了一个具有不同特征时间尺度的捕食者-猎物模型，假设捕食者的动态比猎物的动态慢得多。几何奇异摄动理论为分析模型的动力学特性提供了数学框架。该模型表现出Hopf 分岔，我们证明了当这种分岔发生时，会出现一个canard 现象。我们提供了一个解析表达式，以获得发生最大 canard 解的分岔参数值的近似值。该模型是著名的 Rosenzweig-MacArthur 捕食者-猎物微分系统。一个具有稳定和不稳定分支的不变流形出现，并且使用几何方法来显式确定这些分支交点处的解。用于执行此分析的方法基于 Blow-up 技术。在平衡点上对 Blow-up 对象上的向量场进行分析，其中 Hopf 分岔发生，零摄动参数表示时间尺度比，这允许证明结果。数值模拟说明了结果，并允许看到 canard 爆炸现象。
