A reaction–diffusion system modeling predator–prey with prey-taxis

Aïnseba, Bedr’Eddine; Bendahmane, Mostafa; Noussaïr, Ahmed

doi:10.1016/j.nonrwa.2007.06.017

Cited by 167 publications

(112 citation statements)

References 10 publications

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“…At last we mention the article [7] where the existence of weak solutions for chemotaxis equations with two-sidedly (twopoint) degenerate diffusion of cells was shown independently of [52] by means of slightly different method. In [3] the existence of weak solutions to predator prey system with prey taxis is proved. In this model predator plays formally the role of cells in the chemotaxis model and prey plays that of chemoattractant, the reaction terms reflect prey-predator interactions and the density of prey is assumed to be bounded due to the volume filling effects.…”

Section: That (26) Holds True Under the Assumptions (224)-(227) Amentioning

confidence: 99%

Volume Filling Effect in Modelling Chemotaxis

Wrzosek

2010

Math. Model. Nat. Phenom.

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Abstract. The oriented movement of biological cells or organisms in response to a chemical gradient is called chemotaxis. The most interesting situation related to self-organization phenomenon takes place when the cells detect and response to a chemical which is secreted by themselves. Since pioneering works of Patlak (1953) and Keller and Segel (1970) many particularized models have been proposed to describe the aggregation phase of this process. Most of efforts were concentrated, so far, on mathematical models in which the formation of aggregate is interpreted as finite time blow-up of cell density. In recently proposed models cells are no more treated as point masses and their finite volume is accounted for. Thus, arbitrary high cell densities are precluded in such description and a threshold value for cells density is a priori assumed. Different modeling approaches based on this assumption lead to a class of quasilinear parabolic systems with strong nonlinearities including degenerate or singular diffusion. We give a survey of analytical results on the existence and uniqueness of global-in-time solutions, their convergence to stationary states and on a possibility of reaching the density threshold by a solution. Unsolved problems are pointed as well.

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Section: That (26) Holds True Under the Assumptions (224)-(227) Amentioning

confidence: 99%

Volume Filling Effect in Modelling Chemotaxis

Wrzosek

2010

Math. Model. Nat. Phenom.

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“…It is worthwhile to mention that besides the competition model, the advection, which is referred to as the prey-taxis, has been studied in predator-prey models by various authors. See [7][8][9][10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Global solutions and uniform boundedness of attractive/repulsive LV competition systems

Zhang

2018

Adv Differ Equ

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“…Then, Mimura and Kawasaki [4] introduced the notion of cross-diffusion to describe numerically the segregation effects for large time of a two-component competitive system, starting from heterogeneous distributions initially (see also Mimura and Yamaguti [5]). Later, Galiano et al [6,7] perform the numerical analysis of solutions for a one-dimensional nonlinear cross-diffusion population model (see also [8][9][10][11][12][13][14]). …”

Section: Introductionmentioning

confidence: 99%

A convergent finite volume method for a model of indirectly transmitted diseases with nonlocal cross-diffusion

Anaya

Bendahmane

Langlais

et al. 2015

Computers & Mathematics with Applications

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a b s t r a c tIn this paper, we are concerned with a model of the indirect transmission of an epidemic disease between two spatially distributed host populations having non-coincident spatial domains with nonlocal and cross-diffusion, the epidemic disease transmission occurring through a contaminated environment. The mobility of each class is assumed to be influenced by the gradient of the other classes. We address the questions of existence of weak solutions and existence and uniqueness of classical solution by using, respectively, a regularization method and an interpolation result between Banach spaces. Moreover, we propose a finite volume scheme and proved the well-posedness, nonnegativity and convergence of the discrete solution. The convergence proof is based on deriving a series of a priori estimates and by using a general L p compactness criterion. Finally, the numerical scheme is illustrated by some examples.

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A reaction–diffusion system modeling predator–prey with prey-taxis

Cited by 167 publications

References 10 publications

Volume Filling Effect in Modelling Chemotaxis

Volume Filling Effect in Modelling Chemotaxis

Global solutions and uniform boundedness of attractive/repulsive LV competition systems

A convergent finite volume method for a model of indirectly transmitted diseases with nonlocal cross-diffusion

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