A diffusive predator–prey model with a protection zone

Du, Yihong; Shi, Junping

doi:10.1016/j.jde.2006.01.013

Cited by 137 publications

(90 citation statements)

References 43 publications

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“…See [7,11] and references therein for the stability of prey-predator model incorporating prey refuge. Du and Shi studied the effect of a protection zone in the diffusive Holling type II functional response prey-predator model in [40]. In [29,35,37], authors discussed the stability analysis of prey-predator model with Holling type II functional response in a two patch environment: one accessible to both prey and predator (patch 1) and the other one being a refuge for the prey (patch 2).…”

Section: Introductionmentioning

confidence: 99%

Dynamical analysis of a prey–predator model with Beddington–DeAngelis type function response incorporating a prey refuge

2014

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This paper discusses a prey-predator model with reserved area. The feeding rate of consumers (predators) per consumer (i.e., functional response) is considered to be Beddington-DeAngelis type. The Beddington-DeAngelis functional response is similar to the Holling type II functional response but contains an extra term describing mutual interference by predators. We investigate the role of reserved region and degree of mutual interference among predators in the dynamics of system. We obtain different conditions that affect the persistence of the system. We also discuss local and global asymptotic stability behavior of various equilibrium solutions to understand the dynamics of the model system. The global asymptotic stability of positive interior equilibrium solution is established using suitable Lyapunov functional. It is found that the Hopf bifurcation occurs when the parameter corresponding to reserved region (i.e., m) crosses some critical value. Our result indicates that the predator species exist so long as prey reserve value (m) does not cross a threshold value and after this value the predator species extinct. To mimic the real-world scenario, we also solve the inverse problem of estimation of model parameter (m) using the sampled data of the system. The results can also be interpreted in different contexts such as resource conservation, pest management and

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Section: Introductionmentioning

confidence: 99%

Dynamical analysis of a prey–predator model with Beddington–DeAngelis type function response incorporating a prey refuge

2014

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“…It may be noted here that under conditions (9) or (10), both the predator and prey species coexist, and they settle down at its equilibrium level.…”

Section: Theoremmentioning

confidence: 86%

“…Grieco et al [12] developed a hybrid numerical approach to study the transport processes in the GON and applied it to the dispersion of zooplankton and phytoplankton population dynamics. Du and Shi [9] studied the effects of a protection zone for the prey on a diffusive predator-prey model with Holling type II response and no-flux boundary condition. Ko and Ryu [14,15] investigated the existence and non-existence of non-constant positive steady-states of a diffusive predator-prey interaction system under homogeneous Neumann boundary condition and observed that the monotonicity of a prey isocline at the positive constant solution plays an important role.…”

Section: Introductionmentioning

confidence: 99%

Spatiotemporal pattern formation in a diffusive predator-prey system: an analytical approach

Dubey

Kumari

Upadhyay

2009

J. Appl. Math. Comput.

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In this paper, we propose and analyse a mathematical model to study the mathematical aspect of reaction diffusion pattern formation mechanism in a predatorprey system. An attempt is made to provide an analytical explanation for understanding plankton patchiness in a minimal model of aquatic ecosystem consisting of phytoplankton, zooplankton, fish and nutrient. The reaction diffusion model system exhibits spatiotemporal chaos causing plankton patchiness in marine system. Our analytical findings, supported by the results of numerical experiments, suggest that an unstable diffusive system can be made stable by increasing diffusivity constant to a sufficiently large value. It is also observed that the solution of the system converges to its equilibrium faster in the case of two-dimensional diffusion in comparison to the one-dimensional diffusion. The ideas contained in the present paper may provide a better understanding of the pattern formation in marine ecosystem.

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“…To conclude the paper, we discuss the stability of the unique positive solution when n = 1 by estimating the eigenvalues of the linearized equation. Similar arguments have been used in, for example, [14,28]. The local stability of the unique positive solution of (P ) k is important for a better understanding of the dynamics of the original reaction-diffusion system (1.2) when p = 1 and n = 1, but it is a challenging question in general.…”

Section: ) Then (0 0) Is the Unique Solution Of (42)mentioning

confidence: 93%

Qualitative analysis of an autocatalytic chemical reaction model with decay

Zhou¹,

Shi²

2014

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this paper, we revisit a reaction-diffusion autocatalytic chemical reaction model with decay. For higher-order reactions, we prove that the system possesses at least two positive steady-state solutions; hence, it has bistable dynamics similar to the system without decay. For the linear reaction, we determine the necessary and sufficient condition to ensure the existence of a solution. Moreover, in the one-dimensional case, we prove that the positive steady-state solution is unique. Our results demonstrate the drastic difference in dynamics caused by the order of chemical reactions.

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A diffusive predator–prey model with a protection zone

Cited by 137 publications

References 43 publications

Dynamical analysis of a prey–predator model with Beddington–DeAngelis type function response incorporating a prey refuge

Dynamical analysis of a prey–predator model with Beddington–DeAngelis type function response incorporating a prey refuge

Spatiotemporal pattern formation in a diffusive predator-prey system: an analytical approach

Qualitative analysis of an autocatalytic chemical reaction model with decay

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