Bifurcation and Stability Analysis of a System of Fractional-Order Differential Equations for a Plant–Herbivore Model with Allee Effect

Ali Yousef,Fatma Bozkurt Yousef

doi:10.3390/math7050454

Abstract

This article concerns establishing a system of fractional-order differential equations (FDEs) to model a plant–herbivore interaction. Firstly, we show that the model has non-negative solutions, and then we study the existence and stability analysis of the constructed model. To investigate the case according to a low population density of the plant population, we incorporate the Allee function into the model. Considering the center manifold theorem and bifurcation theory, we show that the model shows flip bifurcation. Finally, the simulation results agree with the theoretical studies.

Highlights

Mathematical modeling for various biological problems is considered to be an exciting research area in the discipline of applied mathematics
Kangalgil and Kartal analyzed the host–parasite model that led to a system of differential equations of piecewise constant arguments at specific time t, as well as the stability of all obtained equilibrium points; they showed the conditions of flip and Neimark–Sacker bifurcation [6]
Model is mainly dependent on the plant population size and carrying capacity

Summary

Introduction

Mathematical modeling for various biological problems is considered to be an exciting research area in the discipline of applied mathematics. Kangalgil and Kartal analyzed the host–parasite model that led to a system of differential equations of piecewise constant arguments at specific time t, as well as the stability of all obtained equilibrium points; they showed the conditions of flip and Neimark–Sacker bifurcation [6]. Most of these studies are restricted to integer-order differential equations or differential equations with piecewise constant arguments. Fractional-order differential equations can model complex biological phenomena with non-linear behavior and long-term memory, which cannot be represented mathematically by integer-order differential equations (IDEs) [16,17].

Equilibrium Points

Local Stability

Existence and value problem can be written in the form

Analyzing the Plant–Herbivore Population at Low Density

Flip Bifurcation with Discretization Process

Conclusions