Diffusion-Driven Instabilities and Spatio-Temporal Patterns in an Aquatic Predator–Prey System with Beddington–Deangelis Type 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.
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 in a spatial plankton system
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In this paper, we investigate the complex dynamics of a spatial plankton-fish system with Holling type III functional responses. We have carried out the analytical study for both one and two dimensional system in details and found out a condition for diffusive instability of a locally stable equilibrium. Furthermore, we present a theoretical analysis of processes of pattern formation that involves organism distribution and their interaction of spatially distributed population with local diffusion. The results of numerical simulations reveal that, on increasing the value of the fish predation rates, the sequences spots rightarrow\rightarrowrightarrow spot-stripe mixtures$\rightarrow$ stripes$\rightarrow$ hole-stripe mixtures holes$\rightarrow$ wave pattern is observed. Our study shows that the spatially extended model system has not only more complex dynamic patterns in the space, but also has spiral waves.
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Chaos and Pattern Formation in Spatial Phytoplankton Dynamics
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In this paper the dynamics of spatially extended infected phytoplankton with the Holling type II functional response and logistically growing susceptible phytoplankton system is studied. The proposed model is an extension of temporal model available [6], in spatiotemporal domain. The reaction diffusion system exhibits spatiotemporal chaos in phytoplankton dynamics. This is particularly important for the spatially extended systems that are studied in this paper as they display a wide spectrum of ecologically relevant behavior, including chaos. In this system stability of the system is studied with respect to disease contact rate and the growth fraction of infected phytoplankton indirectly rejoin the susceptible phytoplankton population. The results of numerical experiments in one dimension and two dimensions in space as well as time series in temporal models are presented using MATLAB simulation. Moreover, the stability of the corresponding temporal model is studied analytically. Fin...
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The spatiotemporal pattern formation and selection driven by nonlinear crossdiffusion of a toxic-phytoplankton-zooplankton model with Allee effect is investigated in this paper. We first perform the mathematical analysis of the corresponding non-spatial model and spatial model, which give us a complete picture of the global dynamics. Then the linear stability analysis shows that the nonlinear crossdiffusion is the key mechanism for the formation of spatial patterns. By taking crossdiffusion rate as bifurcation parameter, amplitude equations under nonlinear crossdiffusion are derived that describe the spatiotemporal dynamics, which interprets the structural transitions and stability of various forms of Turing patterns. Finally, numerical simulations illustrate the effectiveness of theoretical results. It is shown that the spatiotemporal distribution of the plankton is homogeneous when in the absence of cross-diffusion. However, when the cross-diffusivity is greater than its critical value, the spatiotemporal distribution of all the plankton species becomes inhomogeneous in spaces and results in different kinds of patterns: spot, stripe, and the mixture of spot and stripe depending on the nonlinear cross-diffusivity. Simultaneously, the impact of Allee effect and toxin-producing rate of toxicphytoplankton species on pattern selection is also explored.
The spatial patterns through diffusion-driven instability in a predator–prey model
Applied Mathematical Modelling, 2012
Studies on stability mechanism and bifurcation analysis of a system of interacting populations by the combined effect of self and cross-diffusion become an important issue in ecology. In the current investigation, we derive the conditions for existence and stability properties of a predator-prey model under the influence of self and cross-diffusion. Numerical simulations have been carried out in order to show the significant role of self and cross-diffusion coefficients and other important parameters of the system. Various contour pictures of spatial patterns through Turing instability are portrayed and analysed in order to substantiate the applicability of the present model. Finally, the paper ends with an extended discussion of biological implications of our findings.
Spatial dynamics in a predator-prey model with Beddington-DeAngelis functional response
In this paper spatial dynamics of the Beddington-DeAngelis predator-prey model is investigated. We analyze the linear stability and obtain the condition of Turing instability of this model. Moreover, we deduce the amplitude equations and determine the stability of different patterns. In Turing space, we found that this model has coexistence of H 0 hexagon patterns and stripe patterns, H π hexagon patterns, and H 0 hexagon patterns. To better describe the real ecosystem, we consider the ecosystem as an open system and take the environmental noise into account. It is found that noise can decrease the number of the patterns and make the patterns more regular. What is more, noise can induce two kinds of typical pattern transitions. One is from the H π hexagon patterns to the regular stripe patterns, and the other is from the coexistence of H 0 hexagon patterns and stripe patterns to the regular stripe patterns. The obtained results enrich the finding in the Beddington-DeAngelis predator-prey model well.