Spatial dynamics in a predator-prey model with Beddington-DeAngelis functional response (original) (raw)
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Spatiotemporal pattern formation of Beddington-DeAngelis-type predator-prey model
In this paper, we investigate the emergence of a predator-prey model with Beddington-DeAngelistype functional response and reaction-diffusion. We derive the conditions for Hopf and Turing bifurcation on the spatial domain. Based on the stability and bifurcation analysis, we give the spatial pattern formation via numerical simulation, i.e., the evolution process of the model near the coexistence equilibrium point. We find that for the model we consider, pure Turing instability gives birth to the spotted pattern, pure Hopf instability gives birth to the spiral wave pattern, and both Hopf and Turing instability give birth to stripe-like pattern. Our results show that reaction-diffusion model is an appropriate tool for investigating fundamental mechanism of complex spatiotemporal dynamics. It will be useful for studying the dynamic complexity of ecosystems.
Pattern dynamics of a spatial predator–prey model with noise
Nonlinear Dynamics, 2012
A spatial predator-prey model with colored noise is investigated in this paper. We find that the number of the spotted pattern is increased as the noise intensity is increased. When the noise intensity and temporal correlation are in appropriate levels, the model exhibits phase transition from spotted to stripe pattern. Moreover, we show the number of the spotted and stripe pattern, with respect to both noise intensity and temporal correlation. These studies raise important questions on the role of noise in the pattern formation of the populations, which may well explain some data obtained in the ecosystems.
Two Dimensional Pattern Formation of Prey-predator System
We investigate the reaction-diffusion system of the classical Bazykin model in spatial two dimensional domain. In this paper, we derive the conditions for turing instability in detail and obtain the turing space, in which the spatial system can emerge turing pattern. Furthermore, we simulate the pattern formation using the periodical boundary condition. Our results show that the Bazykin system stabilizes to a striplike pattern structure when diffusion is present.
Pattern formation of a predator–prey model
Nonlinear Analysis: Hybrid Systems, 2009
In this paper, we analyze the spatial pattern of a predator–prey system. We get the critical line of Hopf and Turing bifurcation in a spatial domain. In particular, the exact Turing domain is given. Also we perform a series of numerical simulations. The obtained results reveal that this system has rich dynamics, such as spotted, stripe and labyrinth patterns, which shows that it is useful to use the reaction–diffusion model to reveal the spatial dynamics in the real world.
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
Results are reported concerning the formation of spatial patterns in the two-species ratiodependent predator-prey model driven by spatial colored-noise. The results show that there is a critical value with respect to the intensity of spatial noise for this system when the parameters are in the Turing space, above which the regular spatial patterns appear in two dimensions, but under which there are not regular spatial patterns produced. In particular, we investigate in twodimensional space the formation of regular spatial patterns with the spatial noise added in the side and the center of the simulation domain, respectively.
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.
Pattern Dynamics in a Spatial Predator-Prey System with Allee Effect
Abstract and Applied Analysis, 2013
We investigate the spatial dynamics of a predator-prey system with Allee effect. By using bifurcation analysis, the exact Turing domain is found in the parameters space. Furthermore, we obtain the amplitude equations and determine the stability of different patterns. In Turing space, it is found that predator-prey systems with Allee effect have rich dynamics. Our results indicate that predator mortality plays an important role in the pattern formation of populations. More specifically, as predator mortality rate increases, coexistence of spotted and stripe patterns, stripe patterns, spotted patterns, and spiral wave emerge successively. The results enrich the finding in the spatial predator-prey systems well.
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. *