Stability and Double-Hopf Bifurcations of a Gause–Kolmogorov-Type Predator–Prey System with Indirect Prey-Taxis

Zuo, Wangmeng; Song, Yongli

doi:10.1007/s10884-020-09878-9

Cited by 34 publications

(7 citation statements)

References 48 publications

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“…On the other hand, indirect prey taxis in Gauss-type predator-prey Fig. 14 Spatiotemporal spiky patterns for prey (u) and predator v a Emergence of taxis-driven interior spikes, b emergence of taxisdriven boundary spikes in two-dimensional chemo-attraction-induced model (5.1) models also annihilates Turing spatial patterns and gives rise to spatially inhomogeneous Hopf bifurcations [38]. Numerically we observe stable attractor and a stable limit cycle for the non-diffusive version of the system.…”

Section: Discussionmentioning

confidence: 99%

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Drivers of pattern formation in a predator–prey model with defense in fearful prey

Mishra

Tiwari

2021

Nonlinear Dyn

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Existence of predator is routinely used to induce fear and anxiety in prey which is well known for shaping entire ecosystem. Fear of predation restricts the development of prey and promotes inducible defense in prey communities for the survival. Motivated by this fact, we investigate the dynamics of a Leslie–Gower predator prey model with group defense in a fearful prey. We obtain conditions under which system possess unique global-in-time solutions and determine all the biological feasible states of the system. Local stability is analyzed by linearization technique and Lyapunov direct method has been applied for global stability analysis of steady states. We show the occurrence of Hopf bifurcation and its direction at the vicinity of coexisting equilibrium point for temporal model. We consider random movement in species and establish conditions for the stability of the system in the presence of diffusion. We derive conditions for existence of non-constant steady states and Turing instability at coexisting population state of diffusive system. Incorporating indirect prey taxis with the assumption that the predator moves toward the smell of prey rather than random movement gives rise to taxis-driven inhomogeneous Hopf bifurcation in predator–prey model. Numerical simulations are intended to demonstrate the role of biological as well as physical drivers on pattern formation that go beyond analytical conclusions.

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Section: Discussionmentioning

confidence: 99%

“…Please note that that Turing instability does not occur in Gauss-type indirect prey taxis systems for χ > 0 and inhomogeneous Hopf-type instability may exist. We refer authors to look at the paper [38] for detailed information on taxis-induced inhomogeneous Hopf bifurcation in the models of predator-prey interaction.…”

Section: Theorem 11mentioning

confidence: 99%

Drivers of pattern formation in a predator–prey model with defense in fearful prey

Mishra

Tiwari

2021

Nonlinear Dyn

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“…In 1987, Kareiva and Odell [12] first proposed an ecological model with prey-taxis to elucidate the directional movement of species. In the following decades, scholars have established numerous models with prey-taxis to investigate the dynamic relationship between predators and prey [13,14,15]. Recently, a novel form of predator-taxis has been developed, and distinguished from prey-taxis, in which prey escapes from areas with high predator density to avoid capture.…”

Section: Introductionmentioning

confidence: 99%

Dynamics behavior of a predator-prey diffusion model incorporating hunting cooperation and predator-taxis

Zhang

2024

Preprint

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In this paper, we consider a predator-prey diffusion model incorporating hunting cooperation and predator-taxis. Firstly, we establish the global existence of classical solution for the model in any spatial dimension. Secondly, we analyze the instability caused by predator-taxis, and we observed that negative predator-taxis play the key role in inducing stability changes. Finally, numerical simulations are employed to validate the obtained results. MCS: Primary 35A09, 35K59; Secondary 35Q92, 92D25.

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“…Authors have noted that in the case of weakly level prey taxis, the system cannot emerge spatial patterns, however; repulsive prey taxis can generate different kinds of spatial patterns. In [27], authors have shown the existence of double‐Hopf bifurcations in the Gause–Kolmogorov‐type predator–prey system with indirect prey taxis. They found that the presence of indirect prey taxis may induce the appearance of spatially inhomogeneous Hopf bifurcations.…”

Section: Introductionmentioning

confidence: 99%

A predator–prey system with prey social behavior and generalized Holling III functional response: Role of predator‐taxis on spatial patterns

Souna

Tiwari

Belabbas

et al. 2023

Math Methods in App Sciences

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In this paper, we investigate the intrinsic impact of the predator‐taxis coefficient on the formation of spatial patterns in a predator–prey system with prey social behavior subject to Neumann boundary conditions. By treating the predator‐taxis coefficient as a potential critical parameter for Turing bifurcation, we observe that the Turing pattern is fully captured by three distinct critical thresholds. Meanwhile, utilizing weakly nonlinear analysis and the amplitude equation, we establish the direction of Turing bifurcation. Our mathematical analysis reveals that the inclusion of the predator‐taxis coefficient in the predator–prey system may lead to the emergence of either subcritical or supercritical Turing bifurcation. Our numerical experiments confirm the theoretical findings and exhibit various spatial patterns with different values of predator‐taxis coefficients.

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Stability and Double-Hopf Bifurcations of a Gause–Kolmogorov-Type Predator–Prey System with Indirect Prey-Taxis

Cited by 34 publications

References 48 publications

Drivers of pattern formation in a predator–prey model with defense in fearful prey

Drivers of pattern formation in a predator–prey model with defense in fearful prey

Dynamics behavior of a predator-prey diffusion model incorporating hunting cooperation and predator-taxis

A predator–prey system with prey social behavior and generalized Holling III functional response: Role of predator‐taxis on spatial patterns

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