Interaction between prey and mutually interfering predator in prey reserve habitat: Pattern formation and the Turing-Hopf bifurcation (original) (raw)
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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.
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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.
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An appropriate mathematical structure to describe the population dynamics is given by the partial differential equations of reaction-diffusion type. The spatiotemporal dynamics and bifurcations of a ratio-dependent Holling type II predator-prey model system with both the effect of linear prey harvesting and constant proportion of prey refuge are investigated. The existence of all ecologically feasible equilibria for the non-spatial model is determined, and the dynamical classifications of these equilibria are developed. The model system representing initial boundary value problem under study is subjected to zero flux boundary conditions. The conditions of diffusion-driven instability and the Turing bifurcation region in two parameter space are explored. The consequences of spatial pattern analysis in two-dimensional domain by means of numerical simulations reveal that the typical dynamics of population density variation is the formation of isolated groups, i.e. spotted or stripe-like patterns or coexistence of both the patterns or labyrinthine patterns and so on. The results around the unique interior feasible equilibrium solution indicate that the effect of refuge and harvesting plays a significant role on the control of spatial pattern formation of the species. Finally, the
Scientific Reports
The present paper is dealt with a predator–prey model in which the growth of the prey population is influenced by the Allee effect while the predator species are contended with the prey population following the Crowley–Martin type response function. The proposed model is comprehensively analyzed in terms of stability and manifestation of bifurcation of the system. The system unveils the bi-stability together with the existence of a separatrix. In view of the eminence of spatial ecology, the dynamical complexity emanating from the induction of the Allee effect in prey species of a Crowley–Martin reaction–diffusion predator–prey model is also investigated profoundly. The results of numerical simulations reveal that the present system dynamics is motivated by both the Allee effect and diffusion-controlled pattern formation growth to hot spots, stripe-hot spot mixtures, stripes, labyrinthine, stripe-cold spot mixtures, and cold spots replication. The theoretical consequences of the spat...
Patterns formations in a diffusive ratio-dependent predator–prey model of interacting populations
Physica A: Statistical Mechanics and its Applications, 2016
The present investigation deals with the analysis of the spatial pattern formation of a diffusive predator-prey system with ratio-dependent functional response involving the influence of intra-species competition among predators within two-dimensional space. The appropriate condition of Turing instability around the interior equilibrium point of the present model has been determined. The emergence of complex patterns in the diffusive predator-prey model is illustrated through numerical simulations. These results are based on the existence of bifurcations of higher codimension such as Turing-Hopf, Turing-Saddle-node, Turing-Transcritical bifurcation, and the codimension-3 Turing-Takens-Bogdanov bifurcation. The paper concludes with discussions of our results in ecology.
Spatiotemporal dynamics of a predator-prey model incorporating a prey refuge
Journal of Applied Analysis and Computation, 2013
In this paper, we investigate the spatiotemporal dynamics of a ratio-dependent predator-prey model with cross diffusion incorporating proportion of prey refuge. First we get the critical lines of Hopf and Turing bifurcations in a spatial domain by using mathematical theory. More specifically, the exact Turing region is given in a two parameter space. 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 show that it is useful to use the predator-prey model to reveal the spatial dynamics in the real world.
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.