Stability, Hopf bifurcations and spatial patterns in a delayed diffusive predator–prey model with herd behavior

Tang, Xiaosong; Song, Yongli

doi:10.1016/j.amc.2014.12.143

Cited by 61 publications

(39 citation statements)

References 35 publications

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“…which are derived by direct computation together with (11) and (12). Based on the above discussion and the qualitative theory of the dynamical systems, we have the following results.…”

Section: Substituting (8) Inmentioning

confidence: 99%

“…So, we should consider the spatial disperse associated with model (1). In [11], the authors mainly focused on the delay effect on the reactiondiffusion system corresponding to model (1) and investigated the stability/instability of the coexistence equilibrium and associated Hopf bifurcation, the instability of the Hopf bifurcation induced by diffusion and delay, respectively, which can lead to the emergence of spatial patterns.…”

mentioning

confidence: 99%

“…The nonnegative constants d 11 and d 22 are the diffusion coefficients of prey and predator, respectively, and d 12 and d 21 , called cross-diffusion coefficients, describe the respective population fluxes of preys and predators resulting from the presence of the other species, respectively. In [58], we have investigated the crossdiffusion-induced spatiotemporal patterns of model (2) in the Turing region only.…”

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confidence: 99%

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Turing–Hopf bifurcation analysis of a predator–prey model with herd behavior and cross-diffusion

2016

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In this paper, we consider a predator-prey model with herd behavior and cross-diffusion subject to homogeneous Neumann boundary condition. Firstly, the existence and priori bound of a solution for the model without cross-diffusion are shown. Then, by computing and analyzing the normal form on the center manifold associated with the Turing-Hopf bifurcation, we find a wealth of spatiotemporal dynamics near the Turing-Hopf bifurcation point under suitable conditions. Furthermore, some numerical simulations to illustrate the theoretical analysis are carried out.

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“…which are derived by direct computation together with (11) and (12). Based on the above discussion and the qualitative theory of the dynamical systems, we have the following results.…”

Section: Substituting (8) Inmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Turing–Hopf bifurcation analysis of a predator–prey model with herd behavior and cross-diffusion

2016

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“…D ynamics of predator-prey model is one of important subjects in ecology and mathematical ecology, and many researchers have studied it and derived some important results [1][2][3][4][5][6][7][8]. In predator-prey models, mortality rate of the predator is essential.…”

Section: Introductionmentioning

confidence: 99%

The effect of prey refuge and time delay on a diffusive predator‐prey system with hyperbolic mortality

Yang

Zhang

2016

Complexity

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In this article, we investigate the effect of prey refuge and time delay on a diffusive predator-prey system with Holling II functional response and hyperbolic mortality subject to Neumann boundary condition. More precisely, we study Turing instability of positive equilibrium by using refuge as parameter, instability and Hopf bifurcation induced by time delay. In addition, by the theory of normal form and center manifold, we derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution.

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“…cross‐)diffusion on the dynamics of predator‐prey models has been always scrutinized since the pioneering work of Turing . The presence of diffusion in general can generate a surprising complex dynamics, such as Hopf bifurcation, Turing instability, and Turing‐Hopf (T‐H) bifurcation, steady state bifurcation . For discussing some of this achievements, we reformulate the system in the presence of self‐diffusion subject to the homogeneous Neumann boundary conditions in the following structure:

({, \begin{matrix} left \\ array \frac{\partial N (x, t)}{\partial t} = d_{1} N_{x x} (x, t) + N (x, t) (1 - true \frac{N false(x, t false)}{k}) - \sqrt{N (x, t)} P (x, t) x \in (0, l π), t > 0, \\ array \frac{\partial P (x, t)}{\partial t} = d_{2} P_{x x} (x, t) - μ P (x, t) + \sqrt{N (x, t)} P (x, t) x \in (0, l π), t > 0, \\ array N_{x} (x, t) = P_{x} (x, t) = 0, x = 0, l π, t > 0, \\ array N (x, 0) = N_{0} (x) \geq 0, P (x, 0) = P \end{matrix})

…”

Section: Introductionmentioning

confidence: 99%

Pattern formation of a diffusive predator‐prey model with herd behavior and nonlocal prey competition

Djilali

2020

Math Methods in App Sciences

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In this paper, we study the influence of the nonlocal interspecific competition of the prey population on the dynamics of the diffusive predator‐prey model with prey social behavior. Using the linear stability analysis, the conditions for the positive constant steady state at which undergoes Hopf bifurcation, T‐H bifurcation (Turing‐Hopf bifurcation) are investigated. The Turing patterns occur in the presence of the nonlocal competition and cannot be found in the original system. For determining the dynamical behavior near T‐H bifurcation point, the normal form of the T‐H bifurcation has been used. Some graphical representations are provided to illustrate the theoretical results.

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Stability, Hopf bifurcations and spatial patterns in a delayed diffusive predator–prey model with herd behavior

Cited by 61 publications

References 35 publications

Turing–Hopf bifurcation analysis of a predator–prey model with herd behavior and cross-diffusion

Turing–Hopf bifurcation analysis of a predator–prey model with herd behavior and cross-diffusion

The effect of prey refuge and time delay on a diffusive predator‐prey system with hyperbolic mortality

Pattern formation of a diffusive predator‐prey model with herd behavior and nonlocal prey competition

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