Cross Diffusion Induced Turing Patterns in a Tritrophic Food Chain Model with Crowley-Martin Functional Response (original) (raw)
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Mathematical Biosciences, 2015
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Chinese Physics B, 2008
This paper considers the Holling-Tanner model for predator-prey with self and cross-diffusion. From the Turing theory, it is believed that there is no Turing pattern formation for the equal self-diffusion coefficients. However, combined with cross-diffusion, it shows that the system will exhibit spotted pattern by both mathematical analysis and numerical simulations. Furthermore, asynchrony of the predator and the prey in the space. The obtained results show that cross-diffusion plays an important role on the pattern formation of the predator-prey system.
Turing Patterns in a Predator-Prey System with Self-Diffusion
Abstract and Applied Analysis, 2013
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. *
International Journal of Dynamics and Control, 2016
The study of spatial pattern formation through diffusion-driven instability of reaction-diffusion models of interacting species has long been one of the fundamental problems in mathematical ecology. The present article is concerned with interacting predator-prey reactiondiffusion model with Beddington-DeAngelis type functional response. The essential conditions for Hopf and Turing bifurcations are derived on the spatial domain. The parameter space for Turing spatial structure is established. Based on the bifurcation analysis, the spatial pattern formation in Turing space through numerical simulations is carried out in order to study the evolution procedure of the proposed model system in the vicinity of coexistence equilibrium point. The consequences of the results obtained reveal that the effects of selfand cross-diffusion play significant role on the steady state spatiotemporal pattern formation of the reaction-diffusion predator-prey model system which concerns the influence of intra-species competition among predators. Finally, ecological implications of the present results obtained are discussed at length towards the end in order to validate the applicability of the model under consideration. Keywords Beddington-DeAngelis predator-prey model • Pursuit and evasion • Self-and cross-diffusion • Turing bifurcation • Spatiotemporal pattern formation B Santabrata Chakravarty
Spatial Pattern in a Predator-Prey System with Both Self- and Cross-Diffusion
International Journal of Modern Physics C, 2009
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 patte...
In this paper, we propose a diffusive prey-predator system with mutually interfering predator (Crowley-Martin functional response) and prey reserve. In particular, we develop and analyze both spatially homogeneous model based on ordinary differential equations and reaction-diffusion model. We mainly investigate the global existence and boundedness of positive solution, stability properties of homogeneous steady state, non-existence of non-constant positive steady state, conditions for Turing instability and Hopf bifurcation of the diffusive system analytically. Conventional stability properties of the non-spatial counterpart of the system are also studied. The analysis ensures that the prey reserve leaves stabilizing effect on the stability of temporal system. The prey reserve and diffusive parameters leave parallel impact on the stability of the spatio-temporal system. Furthermore, we illustrate the spatial patterns via numerical simulations, which show that the model dynamics exhibits diffusion controlled pattern formation by different interesting patterns.