Turing instability and pattern induced by cross-diffusion in a predator-prey system with Allee effect

Peng, Yahong; Zhang, Tonghua

doi:10.1016/j.amc.2015.11.067

Cited by 36 publications

(18 citation statements)

References 52 publications

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“…We know that functional response function that reflects predator-prey interaction relationships is a crucial component of predator-prey model. In order to describe the features of the predator-prey interaction, many types of usual functional response functions, such as Holling I-IV types, ratio-dependent type, Hassell-Varley type, Beddington-DeAngelis type, Crowley-Martin type and the ones with Allee effect [3][4][5][6], have been proposed and investigated widely.…”

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confidence: 99%

“…The diffusion-driven instability of the equilibrium leads to a spatially inhomogeneous distribution of species concentration, which is the so-called Turing instability. Although Turing instability was first investigated in a morphogenesis, it has quickly spread to ecological systems [3,4,6,[13][14][15][16][17][18][19][20][21][22][23][24], chemical reaction system [25][26][27][28][29][30] and other reaction-diffusion system [31][32][33][34][35][36][37][38]. From [39], we know that the phenomenon of spatial pattern formation in (1) with diffusion can not occur under all possible diffusion rates.…”

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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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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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“…(3)(4)(5) describes the space-and timediscrete predator-prey model. All the parameters used in the discrete model are positive and the values of ( Next, we use the bifurcation theory of discrete model to study the Hopf bifurcation and the Turing instability of the discrete predator model.…”

Section: The Time-and Space-discrete Predator-prey Modelmentioning

confidence: 99%

“…Since Lotka and Volterra have put forward the basic model, it has attract more and more researchers' attention [1][2]. With the development of the predator-prey model, the dynamics behaviors are becoming more and more abundant and complex, such as bifurcations, Turing instability, chaos and some other phenomenon [3][4][5].…”

Section: Introductionmentioning

confidence: 99%

Bifurcation Analysis of a Time- and Space-Discrete Predator-Prey Model with Holling II Function Response

Zhang¹,

Ye²,

Ye³

2018

Proceedings of the 2018 International Conference on Mathematics, Modelling, Simulation and Algorithms (MMSA 2018)

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Abstract-In this paper, a space-and time-discrete predatorprey model with Holling type II function response is investigated. The mortality of predator is variable, which is related to the population density. The model is given by a coupled map lattice framework, it takes a nonlinear relationship between predatorprey reaction stage and dispersal stage. The stability of equilibrium point and the parameter conditions for the Hopf bifurcation are obtained when the diffusion is absent. After adding the diffusion, we obtained the parameter conditions for the Turing instability. Numerical simulations verify the theoretical analysis and show a series of spatial patterns with the change of the parameters.

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“…In [25], Song et al investigated pattern dynamics in a Gierer-Meinhardt model with a saturating term by the linear stability analysis, the multiple scales methods, and the numerical simulations. Peng and Zhang [26] investigated Turing instability and pattern induced by cross-diffusion in a predator-prey system with Allee effect by the multiple time scales and the weakly nonlinear analysis. For more research on Turing instability in a cross-diffusive system, please refer to [27][28][29][30] and relevant references therein.…”

Section: Introductionmentioning

confidence: 99%

Persistence and Turing instability in a cross-diffusive predator–prey system with generalist predator

Bai-qi

2018

Adv Differ Equ

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In this paper, we propose and investigate persistence and Turing instability of a cross-diffusion predator-prey system with generalist predator. First, by virtue of the comparison principle, we obtain sufficient conditions of persistence for a corresponding reaction-diffusion system without self-and cross-diffusion. Second, by using the linear stability analysis, we prove that under some conditions the unique positive equilibrium solution is locally asymptotically stable for the corresponding ODE system and the corresponding reaction-diffusion system without cross-diffusion and self-diffusion. Hence it does not belong to the classical Turing instability. Third, under some appropriate sufficient conditions, we obtain that the uniform positive equilibrium solution is linearly unstable for the cross-reaction-diffusion and partial self-diffusion system. The results indicate that cross-diffusion and partial self-diffusion play an important role in the study of Turing instability about reaction-diffusion systems with generalist predator. Fourth, we elaborate on the relations between the theoretical results and the cross-diffusion predator-prey system by relying on some examples. In the end, we conclude our findings and give a brief discussion. MSC: 35B36; 35B51

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Turing instability and pattern induced by cross-diffusion in a predator-prey system with Allee effect

Cited by 36 publications

References 52 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

Bifurcation Analysis of a Time- and Space-Discrete Predator-Prey Model with Holling II Function Response

Persistence and Turing instability in a cross-diffusive predator–prey system with generalist predator

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