A Ratio-Dependent Nonlinear Predator-Prey Model With Certain Dynamical Results

Asifa Tassaddiq,Qamar Din,Khalil Ahmad,Sabeena Kazi,Muhammad Sajjad Shabbir

doi:10.1109/access.2020.3030778

Asifa Tassaddiq, Qamar Din + Show 3 more

Open Access

https://doi.org/10.1109/access.2020.3030778

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Abstract

In order to see the dynamics of prey-predator interaction, differential or difference equations are frequently used for modeling of such interactions. In present manuscript, we explore some qualitative aspects of two-dimensional ratio-dependent predator-prey model. Taking into account the non-overlapping generations for class of predator-prey system, a novel consistency preserving scheme is proposed. Our study reveals that the implemented discretization is bifurcation preserving. Some dynamical aspects including local behavior of equilibria, phase-plane analysis and emergence of Hopf bifurcation for continuous predator-prey model are studied. Moreover, existence of biologically feasible fixed points, their local asymptotic behavior and phase-plane classification of interior (positive) fixed point are carried out. Furthermore, bifurcation theory of normal forms is implemented to prove that proposed discrete-time model undergoes Neimark-Sacker bifurcation around its unique positive fixed point. Taking into account the bifurcating and fluctuating behaviour of discrete system, three chaos control strategies are implemented. Numerical simulations are provided to illustrate the theoretical discussion and effectiveness of introduced chaos control methods.

Highlights

In case of continuous-time predator-prey interaction theory, differential equations are frequently used whenever it can be considered that the generation of population overlapped and fluctuate continuously in time
When population dynamics cannot be evaluated in terms of continuous functions, a discretization is more effective to develop difference equations for the sake of getting appropriate dynamical results
The functional response that indicate the rate of consumption per capita of the predator-prey models in case of system of differential equations which represents continuoustime models, is normally interpreted as an emphasizing behavior

Summary

INTRODUCTION

In case of continuous-time predator-prey interaction theory, differential equations are frequently used whenever it can be considered that the generation of population overlapped and fluctuate continuously in time. We consider the following continuous-time ratio-dependent predator-prey model which was studied by several authors [6, 9, 10]: dx axy dt = x(1 − x) − x + y , dy cxy (1). Din [40, 41, 42] discussed several classes of prey-predator interaction models to explore bifurcation and chaos control study by implementing piecewise constant arguments method All these investigations ensured that there exists a dynamical. We consider a general description related to local stability analysis of system (3) In this case, if there exists an arbitrary fixed point Η⋆ and assume that: J(Η⋆) = [ωω1211 ωω1222] be the Jacobian matrix calculated at Η⋆, the characteristic equation of J(Η⋆) is given: Ρ(ξ) = ξ2 − Τξ + Δ, where Τ = (ω11 + ω22), and Δ = ω11ω22 − ω12ω21. Our novel nonstandard finite difference scheme certainly increases the stability region

NEIMARK-SACKER BIFURCATION

CHAOS CONTROL ANALYSIS

NUMERICAL SIMULATION

CONCLUSION

VIII. FUTURE DIRECTION