SPATIAL PATTERN OF A RATIO-DEPENDENT PREDATOR-PREY MODEL WITH CROSS-DIFFUSION

2016 ◽  
Vol 100 (10) ◽  
pp. 1551-1563
Author(s):  
Zhi Ling ◽  
Jingjing Li
Keyword(s):  
Spatial Pattern ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Ratio Dependent

Spatial Pattern of Ratio-Dependent Predator–Prey Model with Prey Harvesting and Cross-Diffusion

2019 ◽  
Vol 29 (03) ◽  
pp. 1950036 ◽  
Author(s):  
R. Sivasamy ◽  
M. Sivakumar ◽  
K. Balachandran ◽  
K. Sathiyanathan
Keyword(s):  
Temporal Dynamics ◽  
Intake Rate ◽  
Diffusion Term ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Nonlinear Harvesting ◽  
Ratio Dependent ◽  
The Cross

This study focuses on the spatial-temporal dynamics of predator–prey model with cross-diffusion where the intake rate of prey is per capita predator according to ratio-dependent functional response and the prey is harvested through nonlinear harvesting strategy. The permanence analysis and local stability analysis of the proposed model without cross-diffusion are analyzed. We derive the conditions for the appearance of diffusion-driven instability and global stability of the considered model. Also the parameter space for Turing region is specified by keeping the cross-diffusion coefficient as one of the crucial parameters. Numerical simulations are given to justify the proposed theoretical results and to show that the cross-diffusion term plays a significant role in the pattern formation.

scholarly journals Pattern Formation in a Cross-Diffusive Ratio-Dependent Predator-Prey Model

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Xinze Lian ◽  
Yanhong Yue ◽  
Hailing Wang
Keyword(s):  
Evolutionary Process ◽  
Spiral Wave ◽  
Neumann Boundary ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Reaction Diffusion Model ◽  
Ratio Dependent

This paper presents a theoretical analysis of evolutionary process that involves organisms distribution and their interaction of spatial distribution of the species with self- and cross-diffusion in a Holling-III ratio-dependent predator-prey model. The diffusion instability of the positive equilibrium of the model with Neumann boundary conditions is discussed. Furthermore, we present novel numerical evidence of time evolution of patterns controlled by self- and cross-diffusion in the model and find that the model dynamics exhibits a cross-diffusion controlled formation growth to spots, stripes, and spiral wave pattern replication, which show that reaction-diffusion model is useful to reveal the spatial predation dynamics in the real world.

Spatial pattern in a diffusive predator–prey model with sigmoid ratio-dependent functional response

2014 ◽  
Vol 07 (05) ◽  
pp. 1450047 ◽  
Author(s):  
Lakshmi Narayan Guin ◽  
Prashanta Kumar Mandal
Keyword(s):  
Spatial Pattern ◽  
Numerical Simulations ◽  
Functional Response ◽  
Turing Instability ◽  
Spatial Pattern Analysis ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Prey Model ◽  
Ratio Dependent

In this paper, spatial patterns of a diffusive predator–prey model with sigmoid (Holling type III) ratio-dependent functional response which concerns the influence of logistic population growth in prey and intra-species competition among predators are investigated. The (local and global) asymptotic stability behavior of the corresponding non-spatial model around the unique positive interior equilibrium point in homogeneous steady state is obtained. In addition, we derive the conditions for Turing instability and the consequent parametric Turing space in spatial domain. The results of spatial pattern analysis through numerical simulations are depicted and analyzed. Furthermore, we perform a series of numerical simulations and find that the proposed model dynamics exhibits complex pattern replication. The feasible results obtained in this paper indicate that the effect of diffusion in Turing instability plays an important role to understand better the pattern formation in ecosystem.

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