Cross-diffusion induced Turing and non-Turing patterns in Rosenzweig–MacArthur model (original) (raw)
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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. *
Ecological Complexity, 2018
Spatio-temporal chaos is an intriguing part of the spatio-temporal pattern formation, observed in many interacting population models when their heterogeneous distributions within their habitats and movement from one location to the other are taken care of within the modeling approach. When the homogeneous steady-states become unstable, the solutions of the corresponding reaction-diffusion systems never approach a stationary state rather exhibit an irregular nature with respect to both space and/or time. Cross-diffusion terms are incorporated in a system of reaction-diffusion equations to model the situation where presence, absence, abundance of one species influence the movement of another species and vice-versa. In this work, cross-diffusion is considered in a prey-predator model with ratiodependent functional response along with the self-diffusion terms. After deriving the Turing instability conditions in terms of cross-diffusion parameters, extensive numerical simulations are carried out to study the effect of cross-diffusion on the chaotic dynamics and stationary Turing patterns generated in the system containing self-diffusion terms only. Appropriate numerical tools are used to characterize the spatio-temporal chaos. Route to spatio-temporal chaos and its disappearance are discussed in detail. The chaotic dynamics of the self-diffusion model may be suppressed leading to a stationary state or preserved depending on the cross-diffusion coefficients. The stationary patches of both the species generated in the Turing domain remain stationary but their configurations may change due to the effect of densitydependent cross-diffusion. On the other hand, stationary patches generated in the Turing-Hopf domain change to spatio-temporal chaos for higher dispersal rates of predator avoidance by prey.
Pattern formation induced by cross-diffusion in a predator–prey system
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.
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...
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
Spatiotemporal dynamics of a diffusive predator-prey model with generalist predator
Discrete & Continuous Dynamical Systems - S, 2018
In this paper, we study the spatiotemporal dynamics of a diffusive predator-prey model with generalist predator subject to homogeneous Neumann boundary condition. Some basic dynamics including the dissipation, persistence and non-persistence(i.e., one species goes extinct), the local and global stability of non-negative constant steady states of the model are investigated. The conditions of Turing instability due to diffusion at positive constant steady states are presented. A critical value ρ of the ratio d 2 d 1 of diffusions of predator to prey is obtained, such that if d 2 d 1 > ρ, then along with other suitable conditions Turing bifurcation will emerge at a positive steady state, in particular so it is with the large diffusion rate of predator or the small diffusion rate of prey; while if d 2 d 1 < ρ, both the reaction-diffusion system and its corresponding ODE system are stable at the positive steady state. In addition, we provide some results on the existence and non-existence of positive non-constant steady states. These existence results indicate that the occurrence of Turing bifurcation, along with other suitable conditions, implies the existence of non-constant positive steady states bifurcating from the constant solution. At last, by numerical simulations, we demonstrate Turing pattern formation on the effect of the varied diffusive ratio d 2 d 1. As d 2 d 1 increases, Turing patterns change from spots pattern, stripes pattern into spots-stripes pattern. It indicates that the pattern formation of the model is rich and complex.
Pattern formation in a reaction-diffusion ratio-dependent predator-prey model
Miskolc Mathematical Notes, 2005
A. In this paper we study the effect of diffusion on the stability of the equilibria in a reaction-diffusion ratio-dependent predator-prey model and we explore under which parameter values Turing instability can occur giving rise to nonuniform stationary solutions. Moreover, their stability is studied.
Spatiotemporal pattern formation in a prey-predator model under environmental driving forces
Journal of Physics: Conference Series, 2015
Many existing studies on pattern formation in the reaction-diffusion systems rely on deterministic models. However, environmental noise is often a major factor which leads to significant changes in the spatiotemporal dynamics. In this paper, we focus on the spatiotemporal patterns produced by the predator-prey model with ratio-dependent functional response and density dependent death rate of predator. We get the reaction-diffusion equations incorporating the self-diffusion terms, corresponding to random movement of the individuals within two dimensional habitats, into the growth equations for the prey and predator population. In order to have to have the noise added model, small amplitude heterogeneous perturbations to the linear intrinsic growth rates are introduced using uncorrelated Gaussian white noise terms. For the noise added system, we then observe spatial patterns for the parameter values lying outside the Turing instability region. With thorough numerical simulations we characterize the patterns corresponding to Turing and Turing-Hopf domain and study their dependence on different system parameters like noise-intensity, etc.
Existence of spatial patterns in reaction–diffusion systems incorporating a prey refuge
Nonlinear Analysis: Modelling and Control, 2015
In real-world ecosystem, studies on the mechanisms of spatiotemporal pattern formation in a system of interacting populations deserve special attention for its own importance in contemporary theoretical ecology. The present investigation deals with the spatial dynamical system of a two-dimensional continuous diffusive predator-prey model involving the influence of intra-species competition among predators with the incorporation of a constant proportion of prey refuge. The linear stability analysis has been carried out and the appropriate condition of Turing instability around the unique positive interior equilibrium point of the present model system has been determined. Furthermore, the existence of the various spatial patterns through diffusion-driven instability and the Turing space in the spatial domain have been explored thoroughly. The results of numerical simulations reveal the dynamics of population density variation in the formation of isolated groups, following spotted or stripe-like patterns or coexistence of both the patterns. The results of the present investigation also point out that the prey refuge does have significant influence on the pattern formation of the interacting populations of the model under consideration.