A dynamically consistent nonstandard finite difference scheme for a predator–prey model

Advances in Difference Equations

卷:2019

Khalil Ahmad¹  Muhammad Sajjad Shabbir¹  Muhammad Asif Khan²  Qamar Din²  Muhammad Safeer² 

[1] Department of Mathematics, Air University;

[2] Department of Mathematics, University of the Poonch Rawalakot;

关键词: Predator–prey model;    Nonstandard finite difference scheme;    Persistence;    Stability;    Neimark–Sacker bifurcation;   

DOI  :  10.1186/s13662-019-2319-6

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown