Non-constant positive steady states of a predator-prey system with non-monotonic functional response and diffusion

Pang, Peter Y. H.; Wang, Mingxin

doi:10.1112/s0024611503014321

Cited by 118 publications

(83 citation statements)

References 35 publications

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“…This is often seen when microorganisms are used for waste decomposition or for water purification [14]. Such an inhibitory effect is modeled by a function called the MonodHaldane function [30], which is similar to the Monod (i.e., the Michaelis-Menten) function for low concentrations but includes the inhibitory effect at high concentrations. Collings [10] also used the response function in a mite predator-prey interaction model and called it a Holling type-IV function.…”

Section: The Model Systemsmentioning

confidence: 99%

Wave of Chaos and Pattern Formation in Spatial Predator-Prey Systems with Holling Type IV Predator Response

Upadhyay

Kumari

Rai

2008

Math. Model. Nat. Phenom.

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Abstract. The challenges to live in the open water and the diversity of habitats in the marine environments prompts phytoplankton to devise strategies which often involve production of toxins by Harmful Algal Bloom (HAB) and rapid production of metabolites from non-toxic precursor. The functional response of the predator is described by Holling type IV. We investigate wave phenomena and non-linear non-equilibrium pattern formation in a phytoplankton-zooplankton system with Holling type IV functional response. Type IV functional response yields to type II response in the event of large immunity from or tolerance of prey. It has been found that the Wave of Chaos (WoC) is still effective mechanism for the propagation of chaotic dynamics in predation and competitive systems. Fish predation has a significant role in the temporal evolution of spatial patterns of phytoplankton-zooplankton system. From a field ecologist's perspective, it is not only important to know the stable stationary patterns of an evolving phytoplankton-zooplankton system, but also to know at what point of time these observations are carried out. This has implications for ecological theory.

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Section: The Model Systemsmentioning

confidence: 99%

Wave of Chaos and Pattern Formation in Spatial Predator-Prey Systems with Holling Type IV Predator Response

Upadhyay

Kumari

Rai

2008

Math. Model. Nat. Phenom.

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“…The Gray-Scott equation is a kind of very important reaction-diffusion system, which arises from many chemical or biological systems [2][3][4][5]. This equation has been researched by many authors (see [2][3][4][5][6][7][8][9][10]).…”

Section: Introductionmentioning

confidence: 99%

“…This equation has been researched by many authors (see [2][3][4][5][6][7][8][9][10]). One of the most important problems in mathematical physics is the asymptotic behavior of dynamical system, which has been developed greatly in recent years.…”

Section: Introductionmentioning

confidence: 99%

Random attractors for stochastic two-compartment Gray-Scott equations with a multiplicative noise

2016

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Abstract:In this paper, we consider the existence of a pullback attractor for the random dynamical system generated by stochastic two-compartment Gray-Scott equation for a multiplicative noise with the homogeneous Neumann boundary condition on a bounded domain of space dimension n Ä 3. We first show that the stochastic Gray-Scott equation generates a random dynamical system by transforming this stochastic equation into a random one. We also show that the existence of a random attractor for the stochastic equation follows from the conjugation relation between systems. Then, we prove pullback asymptotical compactness of solutions through the uniform estimate on the solutions. Finally, we obtain the existence of a pullback attractor.

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“…Finally, we remark that in the past decades, there has been much work on the existence and non-existence of non-constant positive steady states of ecological models with diffusion or cross-diffusion under the homogeneous Neumann boundary conditions. One can refer to [4,11,13,[19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. The role of diffusion in modeling many physical, chemical and biological processes has been extensively studied.…”

Section: Introductionmentioning

confidence: 99%

“…Starting with Turing's seminal paper [34], diffusion and cross-diffusion have been observed as causes of the spontaneous emergence of ordered structures, called patterns in a variety of non-equilibrium situations. They include the Gierer-Meinhardt model [35][36][37][38], the Sel'kov model [26,39], the Lotka-Volterra competition model [40][41][42] and the Lotka-Volterra predator-prey model [20,23,24,[43][44][45] and so on.…”

Section: Introductionmentioning

confidence: 99%

Qualitative analysis of a strongly coupled predator–prey system with modified Holling–Tanner functional response

Zeng

2018

Bound Value Probl

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Non-constant positive steady states of a predator-prey system with non-monotonic functional response and diffusion

Cited by 118 publications

References 35 publications

Wave of Chaos and Pattern Formation in Spatial Predator-Prey Systems with Holling Type IV Predator Response

Wave of Chaos and Pattern Formation in Spatial Predator-Prey Systems with Holling Type IV Predator Response

Random attractors for stochastic two-compartment Gray-Scott equations with a multiplicative noise

Qualitative analysis of a strongly coupled predator–prey system with modified Holling–Tanner functional response

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