Delay induced multiple stability switch and chaos in a predator–prey model with fear effect

AbstractIn this article, we discuss the dynamics of a Leslie–Gower ratio-dependent predator–prey model incorporating fear in the prey population. Moreover, the Allee effect in the predator growth is added into account from both biological and mathematical points of view. We explore the influence of the Allee and fear effect on the existence of all positive equilibria. Furthermore, the local stability properties and possible bifurcation behaviors of the proposed system about positive equilibria are discussed with the help of trace and determinant values of the Jacobian matrix. With the help of Sotomayor’s theorem, the conditions for existence of saddle-node bifurcation are derived. Also, we show that the proposed system admits limit cycle dynamics, and its stability is discussed with the value of first Lyapunov coefficient. Moreover, the numerical simulations including phase portrait, one- and two-parameter bifurcation diagrams are performed to validate our important findings.

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Corrigendum to “Multiple stability and uniqueness of the limit cycle in a Gause-type predator–prey model considering the Allee effect on prey” [Nonlinear Anal. RWA 12 (2011) 2931–2942]

Nonlinear Analysis Real World Applications ◽

10.1016/j.nonrwa.2012.05.008 ◽

2013 ◽

Vol 14 (1) ◽

pp. 888-891

Author(s):

Eduardo González-Olivares ◽

Héctor Meneses-Alcay ◽

Betsabé González-Yañez ◽

Jaime Mena-Lorca ◽

Alejandro Rojas-Palma ◽

...

Keyword(s):

Limit Cycle ◽

Allee Effect ◽

Predator Prey ◽

Predator Prey Model ◽

Nonlinear Anal ◽

Prey Model ◽

Multiple Stability

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Qualitative analysis of a diffusive predator–prey model with Allee and fear effects

International Journal of Biomathematics ◽

10.1142/s1793524521500376 ◽

2021 ◽

pp. 2150037

Author(s):

Jia Liu

Keyword(s):

Allee Effect ◽

Sufficient Conditions ◽

Allee Effects ◽

Predator Prey ◽

Predator Prey Model ◽

Fear Effect ◽

Prey Model ◽

Strong Allee Effect ◽

Spatially Homogeneous ◽

Fear Effects

In this study, we consider a diffusive predator–prey model with multiple Allee effects induced by fear factors. We investigate the existence, boundedness and permanence of the solution of the system. We also discuss the existence and non-existence of non-constant solutions. We derive sufficient conditions for spatially homogeneous (non-homogenous) Hopf bifurcation and steady state bifurcation. Theoretical and numerical simulations show that strong Allee effect and fear effect have great effect on the dynamics of system.

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Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect

Mathematics ◽

10.3390/math8081280 ◽

2020 ◽

Vol 8 (8) ◽

pp. 1280

Author(s):

Liyun Lai ◽

Zhenliang Zhu ◽

Fengde Chen

Keyword(s):

Allee Effect ◽

Sufficient Conditions ◽

Positive Equilibrium ◽

Predator Species ◽

Predator Prey ◽

Predator Prey Model ◽

Fear Effect ◽

Prey Model ◽

Strong Allee Effect ◽

Theoretical Results

We proposed and analyzed a predator–prey model with both the additive Allee effect and the fear effect in the prey. Firstly, we studied the existence and local stability of equilibria. Some sufficient conditions on the global stability of the positive equilibrium were established by applying the Dulac theorem. Those results indicate that some bifurcations occur. We then confirmed the occurrence of saddle-node bifurcation, transcritical bifurcation, and Hopf bifurcation. Those theoretical results were demonstrated with numerical simulations. In the bifurcation analysis, we only considered the effect of the strong Allee effect. Finally, we found that the stronger the fear effect, the smaller the density of predator species. However, the fear effect has no influence on the final density of the prey.

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Survival analysis of a stochastic predator–prey model with prey refuge and fear effect

Journal of Biological Dynamics ◽

10.1080/17513758.2020.1853832 ◽

2020 ◽

Vol 14 (1) ◽

pp. 871-892

Author(s):

Yixiu Xia ◽

Sanling Yuan

Keyword(s):

Survival Analysis ◽

Prey Refuge ◽

Predator Prey ◽

Predator Prey Model ◽

Fear Effect ◽

Prey Model

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Dynamical Behaviour of an Infected Predator-Prey Model with Fear Effect

Iranian Journal of Science and Technology Transactions A Science ◽

10.1007/s40995-020-01014-y ◽

2020 ◽

Author(s):

Dipesh Barman ◽

Jyotirmoy Roy ◽

Shariful Alam

Keyword(s):

Dynamical Behaviour ◽

Predator Prey ◽

Predator Prey Model ◽

Fear Effect ◽

Prey Model

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Age-selective harvesting in a delayed predator–prey model with fear effect

Zeitschrift für Naturforschung A ◽

10.1515/zna-2021-0217 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Ashok Mondal ◽

Amit K. Pal

Keyword(s):

Hopf Bifurcation ◽

Predator Population ◽

Ecological Management ◽

Predator Prey ◽

Selective Harvesting ◽

Predator Prey Model ◽

Elapsed Time ◽

Fear Effect ◽

Prey Model ◽

Fear Effects

Abstract In this article, we discussed the dynamic behavior of a delay-induced harvested predator–prey model with fear effects (perceived by the prey). We then considered selective harvesting terms for both species which provide some fixed elapsed time to the prey and for the predator population before they are harvested. In other words, we are limiting the harvesting of species below a certain age so that they can grow to a certain specific size or age and thus protect juvenile populations. Reproduction of the prey population can also be greatly impeded due to the influence of the fear effect. The consideration of selective harvesting together with the effect of fear on the proposed system to show stable coexistence to the oscillatory mode and vice versa via Hopf-bifurcation. For better ecological management of the community, our study reveals the fact that collection delays and intensities should be maintained. Numerical simulations were performed to validate our analytical results.

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