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Stability analysis of a fractional order modified Leslie-Gower model with additive Allee effect. (English) Zbl 1462.34077

Summary: We analyze the dynamics of a fractional order modified Leslie-Gower model with Beddington-DeAngelis functional response and additive Allee effect by means of local stability. In this respect, all possible equilibria and their existence conditions are determined and their stability properties are established. We also construct nonstandard numerical schemes based on Grünwald-Letnikov approximation. The constructed scheme is explicit and maintains the positivity of solutions. Using this scheme, we perform some numerical simulations to illustrate the dynamical behavior of the model. It is noticed that the nonstandard Grünwald-Letnikov scheme preserves the dynamical properties of the continuous model, while the classical scheme may fail to maintain those dynamical properties.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
34D20 Stability of solutions to ordinary differential equations
34A08 Fractional ordinary differential equations
92D25 Population dynamics (general)
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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