A predator–prey interaction model with self and cross-diffusion

Dubey, B.; Das, Barun; Hussain, Jamal

doi:10.1016/s0304-3800(01)00255-1

Cited by 127 publications

(53 citation statements)

References 41 publications

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“…However, if predators recognize prey, this recognition may affect the rate or direction of their movement, thereby helping the predators find prey. This phenomenon, known as cross-diffusion, has recently received significant attention, as described in [16] [17] [18] [19]. Value of the cross-diffusion coefficient may be positive, negative, or zero.…”

Section: Cross-diffusion Modelmentioning

confidence: 99%

Mathematical Model for Two-Spotted Spider Mites System: Verification and Validation

Kuang¹,

Ben‐Arieh²,

Zhao³

et al. 2017

OJMSi

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This paper presents and compares four mathematical models with unique spatial effects for a prey-predator system, with Tetranychus urticae as prey and Phytoseiulus persimilis as predator. Tetranychus urticae, also known as two-spotted spider mite, is a harmful plant-feeding pest that causes damage to over 300 species of plants. Its predator, Phytoseiulus persimilis, a mite in the Family Phytoseiidae, effectively controls spider mite populations. In this study, we compared four mathematical models using a numerical simulation. These models include two known models: self-diffusion, and cross-diffusion, and two new models: chemotaxis effect model, and integro diffusion model, all with a Beddington-De Angelis functional response. The modeling results were validated by fitting experimental data. Results demonstrate that interaction scheme plays an important role in the prey-predator system and that the crossdiffusion model fits the real system best. The main contribution of this paper is in the two new models developed, as well as the validation of all the models using experimental data.

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Section: Cross-diffusion Modelmentioning

confidence: 99%

Mathematical Model for Two-Spotted Spider Mites System: Verification and Validation

Kuang¹,

Ben‐Arieh²,

Zhao³

et al. 2017

OJMSi

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“…£ ÓÂÃÑÕÇ ¥ÂÃÇâ Ô ÔÑÂÄÕÑÓÂÏË [24] ÓÂÔÔÏÑÕÓÇÐÂ ÔÎÇAEÖáÜÂâ ÏÑAEÇÎß ÕËÒÂ ØËÜÐËÍ ë ÉÇÓÕÄÂ Ô ÔÂÏÑ-Ë ÍÓÑÔÔ-AEË××ÖÊËÇÌ: …”

Section: ¥äöïçóðââ ôóçAeâunclassified

Волны В Кросс-Диффузионных Системах — Особый Класс Нелинейных Волн

Tsyganov¹,

Biktashev²,

Бриндли³

et al. 2007

Uspekhi Fizicheskikh Nauk

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“…It is easy to see that, if a > b, then system (1.3) has a unique positive constant equilibrium (ũ,ṽ), wherẽ u is the unique positive root of equation 4) and satisfiesũ…”

Section: Introductionmentioning

confidence: 99%

“…The authors also introduced the same cross-diffusion term in the papers [4,5,7,13]. In this paper, we shall adopt some of the mathematical techniques, which are used in the papers [9,14].…”

Section: Introductionmentioning

confidence: 99%

Global stability and stationary pattern of a diffusive prey-predator model with modified Leslie-Gower term and Holling II functional response

Li¹,

Zhang²,

Liu³

2016

J. Nonlinear Sci. Appl.

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This paper is concerned with a diffusive prey-predator model with modified Leslie-Gower term and Holling II functional response subject to the homogeneous Neumann boundary condition. Firstly, by upper and lower solutions method, we prove the global asymptotic stability of the unique positive constant steady state solution. Secondly, introducing the cross diffusion, we obtain the existence of non-constant positive solutions. The results demonstrate that under certain conditions, even though the unique positive constant steady state is globally asymptotically stable for the model with self-diffusion, the non-constant positive steady states can exist due to the emergency of cross-diffusion, that is to say, cross-diffusion can create stationary pattern. Finally, using the bifurcation theory and treating cross diffusion as a bifurcation parameter, we obtain the existence of positive non-constant solutions.

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A predator–prey interaction model with self and cross-diffusion

Cited by 127 publications

References 41 publications

Mathematical Model for Two-Spotted Spider Mites System: Verification and Validation

Mathematical Model for Two-Spotted Spider Mites System: Verification and Validation

Волны В Кросс-Диффузионных Системах — Особый Класс Нелинейных Волн

Global stability and stationary pattern of a diffusive prey-predator model with modified Leslie-Gower term and Holling II functional response

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