On global bifurcation for a cross-diffusion predator–prey system with prey-taxis

For a predator-prey system, cross-diffusion has been confirmed to emerge Turing patterns. However, in the real world, the tendency for prey and predators moving along the direction of lower density of their own species, called self-diffusion, should be considered. For this, we investigate Turing instability for a predator-prey system with nonlinear diffusion terms including the normal diffusion, cross-diffusion, and self-diffusion. A sufficient condition of Turing instability for this system is obtained by analyzing the linear stability of spatial homogeneous equilibrium state of this model. A series of numerical simulations reveal Turing parameter regions of the interaction of diffusion parameters. According to these regions, we further demonstrate dispersion relations and spatial patterns. Our results indicate that self-diffusion plays an important role in the spatial patterns.

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Weak and classical solutions to predator–prey system with cross-diffusion

Nonlinear Analysis ◽

10.1016/j.na.2010.06.021 ◽

2010 ◽

Vol 73 (8) ◽

pp. 2489-2503 ◽

Cited By ~ 14

Author(s):

Mostafa Bendahmane

Keyword(s):

Classical Solutions ◽

Predator Prey ◽

Cross Diffusion ◽

Prey System

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Turing pattern formation in a predator–prey system with cross diffusion

Applied Mathematical Modelling ◽

10.1016/j.apm.2014.04.015 ◽

2014 ◽

Vol 38 (21-22) ◽

pp. 5022-5032 ◽

Cited By ~ 13

Author(s):

Zhi Ling ◽

Lai Zhang ◽

Zhigui Lin

Keyword(s):

Pattern Formation ◽

Turing Pattern ◽

Predator Prey ◽

Cross Diffusion ◽

Prey System

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Global existence of solutions to a cross-diffusion predator–prey system with Holling type-II functional response

Computers & Mathematics with Applications ◽

10.1016/j.camwa.2013.02.007 ◽

2013 ◽

Vol 65 (8) ◽

pp. 1152-1162 ◽

Cited By ~ 5

Author(s):

Chenglin Li

Keyword(s):

Global Existence ◽

Functional Response ◽

Existence Of Solutions ◽

Type Ii ◽

Predator Prey ◽

Cross Diffusion ◽

Global Existence Of Solutions ◽

Prey System ◽

Type Ii Functional Response

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SPATIAL PATTERN IN A PREDATOR-PREY SYSTEM WITH BOTH SELF- AND CROSS-DIFFUSION

International Journal of Modern Physics C ◽

10.1142/s0129183109013467 ◽

2009 ◽

Vol 20 (01) ◽

pp. 71-84 ◽

Cited By ~ 29

Author(s):

GUI-QUAN SUN ◽

ZHEN JIN ◽

YI-GUO ZHAO ◽

QUAN-XING LIU ◽

LI LI

Keyword(s):

Spatial Patterns ◽

Temporal Dynamics ◽

Spatial Scales ◽

Natural Populations ◽

Spatial Dynamics ◽

Density Variation ◽

Predator Prey ◽

Cross Diffusion ◽

Prey System ◽

Dynamics Of Population

The vast majority of models for spatial dynamics of natural populations assume a homogeneous physical environment. However, in practice, dispersing organisms may encounter landscape features that significantly inhibit their movement. And spatial patterns are ubiquitous in nature, which can modify the temporal dynamics and stability properties of population densities at a range of spatial scales. Thus, in this paper, a predator-prey system with Michaelis-Menten-type functional response and self- and cross-diffusion is investigated. Based on the mathematical analysis, we obtain the condition of the emergence of spatial patterns through diffusion instability, i.e., Turing pattern. A series of numerical simulations reveal that the typical dynamics of population density variation is the formation of isolated groups, i.e., stripe-like or spotted or coexistence of both. The obtained results show that the interaction of self-diffusion and cross-diffusion plays an important role on the pattern formation of the predator-prey system.

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