Stability and bifurcation analysis of a discrete predator–prey system with modified Holling–Tanner functional response

In this paper, we study a discrete predator–prey system with modified Holling–Tanner functional response. We derive conditions of existence for flip bifurcations and Hopf bifurcations by using the center manifold theorem and bifurcation theory. Numerical simulations including bifurcation diagrams, m...

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Bibliographic Details
Published in:Advances in difference equations 2018-10, Vol.2018 (1), p.1-18, Article 402
Main Authors: Zhao, Jianglin, Yan, Yong
Format: Article
Language:English
Subjects:
Analysis
Bifurcation theory
Chaos
Computer simulation
Difference and Functional Equations
Discrete-time predator–prey system
Flip bifurcation
Functional Analysis
Hopf bifurcation
Integrals
Liapunov exponents
Mathematical models
Mathematics
Mathematics and Statistics
Ordinary Differential Equations
Partial Differential Equations
Predator-prey simulation
Stability analysis
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Summary:In this paper, we study a discrete predator–prey system with modified Holling–Tanner functional response. We derive conditions of existence for flip bifurcations and Hopf bifurcations by using the center manifold theorem and bifurcation theory. Numerical simulations including bifurcation diagrams, maximum Lyapunov exponents, and phase portraits not only illustrate the correctness of theoretical analysis, but also exhibit complex dynamical behaviors and biological phenomena. This suggests that the small integral step size can stabilize the system into the locally stable coexistence. However, the large integral step size may destabilize the system producing far richer dynamics. This also implies that when the intrinsic growth rate of prey is high, the model has bifurcation structures somewhat similar to the classic logistic one.
ISSN:1687-1847
1687-1839
1687-1847
DOI:10.1186/s13662-018-1819-0