Existence of periodic solutions for a predator–prey system with sparse effect and functional response on time scales

Tong, Yu Xia; Liu, Zhenjie; Gao, Zhiying; Wang, Yonghong

doi:10.1016/j.cnsns.2011.12.003

Cited by 5 publications

(2 citation statements)

References 15 publications

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“…Until now, many works have concerned discrete-time two-species predator-prey models (Jing and Yang, 2006;Guckenheimer and Holmes, 1983;Wiggins, 1990;He and Lai, 2011;Liu and Xiao, 2007;Hu et al, 2011;Tong et al, 2012). In these articles, the main studied topics are the existence and local stability of an equilibrium, flip bifurcation and Hopf bifurcation by using the center manifold theorem and bifurcation theory.…”

Section: Introductionmentioning

confidence: 99%

“…In these articles, the main studied topics are the existence and local stability of an equilibrium, flip bifurcation and Hopf bifurcation by using the center manifold theorem and bifurcation theory. Tong et al (2012) discussed the existence of periodic solutions for discrete semi-ratio-dependent prey-predator models with functional response. The permanence in a delayed discrete prey-predator model with Holling-III functional response was discussed by Fan and Li (2004).…”

Section: Introductionmentioning

confidence: 99%

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Bifurcation and control for a discrete-time prey–predator model with Holling-IV functional response

Chen¹,

Teng²,

Hu³

2013

International Journal of Applied Mathematics and Computer Science

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The dynamics of a discrete-time predator-prey model with Holling-IV functional response are investigated. It is shown that the model undergoes a flip bifurcation, a Hopf bifurcation and a saddle-node bifurcation by using the center manifold theorem and bifurcation theory. Numerical simulations not only exhibit our results with the theoretical analysis, but also show the complex dynamical behaviors, such as the period-3, 6, 9, 12, 20, 63, 70, 112 orbits, a cascade of period-doubling bifurcations in period-2, 4, 8, 16, quasi-periodic orbits, an attracting invariant circle, an inverse period-doubling bifurcation from the period-32 orbit leading to chaos and a boundary crisis, a sudden onset of chaos and a sudden disappearance of the chaotic dynamics, attracting chaotic sets and non-attracting sets. We also observe that when the prey is in chaotic dynamics the predator can tend to extinction or to a stable equilibrium. Specifically, we stabilize the chaotic orbits at an unstable fixed point by using OGY chaotic control.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bifurcation and control for a discrete-time prey–predator model with Holling-IV functional response

Chen¹,

Teng²,

Hu³

2013

International Journal of Applied Mathematics and Computer Science

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Spatiotemporal dynamics in a ratio-dependent predator-prey model with time delay near the Turing–Hopf bifurcation point

Chen

Liu

et al. 2019

Communications in Nonlinear Science and Numerical Simulation

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Spatio-Temporal Patterns of Predator–Prey Model with Allee Effect and Constant Stocking Rate for Predator

Tian

Guo

2021

Int. J. Bifurcation Chaos

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A diffusive predator–prey model with Allee effect and constant stocking rate for predator is investigated and it is shown that Allee effect is the decisive factor driving the formation of Turing pattern. Furthermore, it is observed that Turing pattern appears only when the diffusion rate of the prey is faster than that of the predator, which is just opposite to the condition of Turing pattern in the classical predator–prey system. Some sufficient conditions are obtained to ensure the asymptotical stability of a spatially homogeneous steady-state solution. The existence and nonexistence of positive nonconstant steady-state solutions are investigated to understand the mechanisms of generating spatiotemporal patterns. Furthermore, Hopf and steady-state bifurcations are analyzed in detail by using Lyapunov–Schmidt reduction.

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Existence of periodic solutions for a predator–prey system with sparse effect and functional response on time scales

Cited by 5 publications

References 15 publications

Bifurcation and control for a discrete-time prey–predator model with Holling-IV functional response

Bifurcation and control for a discrete-time prey–predator model with Holling-IV functional response

Spatiotemporal dynamics in a ratio-dependent predator-prey model with time delay near the Turing–Hopf bifurcation point

Spatio-Temporal Patterns of Predator–Prey Model with Allee Effect and Constant Stocking Rate for Predator

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