A dynamically consistent nonstandard finite difference scheme for a predator\u2013prey model

Muhammad Sajjad Shabbir,Muhammad Asif Khan,Qamar Din,Muhammad Safeer,Khalil Ahmad

doi:10.1186/s13662-019-2319-6

Abstract

The interaction between prey and predator is one of the most fundamental processes in ecology. Discrete-time models are frequently used for describing the dynamics of predator and prey interaction with non-overlapping generations, such that a new generation replaces the old at regular time intervals. Keeping in view the dynamical consistency for continuous models, a nonstandard finite difference scheme is proposed for a class of predator–prey systems with Holling type-III functional response. Positivity, boundedness, and persistence of solutions are investigated. Analysis of existence of equilibria and their stability is carried out. It is proved that a continuous system undergoes a Hopf bifurcation at its interior equilibrium, whereas the discrete-time version undergoes a Neimark–Sacker bifurcation at its interior fixed point. A numerical simulation is provided to strengthen our theoretical discussion.

Highlights

Many real-life biological models including prey–predator interactions often are governed by nonlinear differential equations
For a similar type of investigations related to predator–prey systems the interested reader is referred to [6,7,8,9,10,11,12,13,14,15,16,17]. All these studies reveal that the discrete predator–prey models with implementation of Euler approximation are dynamically inconsistent with their continuous counterparts
All parameters r, k, α, β, s, h are positive constants. He and Lai [6] investigated stability, period-doubling bifurcation, Neimark–Sacker bifurcation and chaos control for a discrete counterpart of (1) with application of Euler forward approximation. Their investigation reveals the dynamical inconsistency between the discrete-time and continuous-time system because there is no chance of flip bifurcation in system (1)

Summary

Introduction

Many real-life biological models including prey–predator interactions often are governed by nonlinear differential equations. For a similar type of investigations related to predator–prey systems the interested reader is referred to [6,7,8,9,10,11,12,13,14,15,16,17] All these studies reveal that the discrete predator–prey models with implementation of Euler approximation are dynamically inconsistent with their continuous counterparts. All parameters r, k, α, β, s, h are positive constants He and Lai [6] investigated stability, period-doubling bifurcation, Neimark–Sacker bifurcation and chaos control for a discrete counterpart of (1) with application of Euler forward approximation. Keeping in view the dynamical consistency of model (1), the following discrete-time counterpart of (1) is proposed by implementing a Mickens type nonstandard finite difference scheme: xn+1 – xn δ. The variational matrix at interior equilibrium (x∗, y∗) is computed as follows:

According to the

The eigenvalues of

We assume that s