The dynamics of a Leslie type predator\u2013prey model with fear and Allee effect

S Vinoth,K Sathiyanathan,R Vadivel,Nallappan Gunasekaran,Grienggrai Rajchakit,R Sivasamy,Bundit Unyong

doi:10.1186/s13662-021-03490-x

Abstract

In this article, we discuss the dynamics of a Leslie–Gower ratio-dependent predator–prey model incorporating fear in the prey population. Moreover, the Allee effect in the predator growth is added into account from both biological and mathematical points of view. We explore the influence of the Allee and fear effect on the existence of all positive equilibria. Furthermore, the local stability properties and possible bifurcation behaviors of the proposed system about positive equilibria are discussed with the help of trace and determinant values of the Jacobian matrix. With the help of Sotomayor’s theorem, the conditions for existence of saddle-node bifurcation are derived. Also, we show that the proposed system admits limit cycle dynamics, and its stability is discussed with the value of first Lyapunov coefficient. Moreover, the numerical simulations including phase portrait, one- and two-parameter bifurcation diagrams are performed to validate our important findings.

Highlights

One of the prominent topics for ecologists, biologists, and even mathematicians is the study of various mechanisms related to the interaction between prey and predators
The predation function depends on prey only such as Holling type classification [3, 4], whereas ratio-dependent [5], Beddington [6], and Crowley Martin [7] are the functions of both prey and predator
7 Conclusion In the present study, the fear effect in the prey population and the Allee effect in the predator population were considered in the modified Leslie type predator–prey model with ratio-dependent interaction term

Summary

Introduction

One of the prominent topics for ecologists, biologists, and even mathematicians is the study of various mechanisms related to the interaction between prey and predators. Many researchers have developed and implemented various forms of interaction terms to understand the more realistic situation. The predation function depends on prey only such as Holling type classification [3, 4], whereas ratio-dependent [5], Beddington [6], and Crowley Martin [7] are the functions of both prey and predator. In real-world situations, [8, 9] clearly confirm that the ratio-dependent interaction has a greater description of predation rates than the prey-dependent interaction. The authors in [10] assumed that the predator growth function is different from the predator’s predation function, and its reduction has a reciprocal relationship with per capita availability of its preferred food, the so-called Leslie

Objectives

Findings

Conclusion