70 research outputs found

    Spatiotemporal pattern induced by self and cross-diffusion in a spatial Holling-Tanner model

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    In this paper, we have made an attempt to provide a unified framework to understand the complex spatiotemporal patterns induced by self and cross diffusion in a spatial Holling-Tanner model forphytoplankton-zooplankton-fish interaction. The effect of critical wave length which can drive the system to instability is investigated. We have examined the criterion between two cross-diffusivity (constant and timevarying)on the stability of the model system and for diffusive instability to occur. Based on these conditions and by performing a series of extensive simulations, we observed the irregular patterns, stationary strips, spots, and strips-spots mixture patterns. Numerical simulation results reveal that the regular strip-spot mixture patterns prevail over the whole domain on increasing the values of self- diffusion coefficients of phytoplankton and zooplankton and the dynamics of the system do not undergo any further changes

    Bifurcation on diffusive Holling–Tanner predator–prey model with stoichiometric density dependence

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    This paper studies a diffusive Holling–Tanner predator–prey system with stoichiometric density dependence. The local stability of positive equilibrium, the existence of Hopf bifurcation and stability of bifurcating periodic solutions have been obtained in the absence of diffusion. We also study the spatially homogeneous and nonhomogeneous periodic solutions through all parameters of the system, which are spatially homogeneous. In order to verify our theoretical results, some numerical simulations are carried out.&nbsp

    Existence of spatial patterns in reaction–diffusion systems incorporating a prey refuge

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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

    Cross-diffusion driven instability in a predator-prey system with cross-diffusion

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    In this work we investigate the process of pattern formation induced by nonlinear diffusion in a reaction-diffusion system with Lotka-Volterra predator-prey kinetics. We show that the cross-diffusion term is responsible of the destabilizing mechanism that leads to the emergence of spatial patterns. Near marginal stability we perform a weakly nonlinear analysis to predict the amplitude and the form of the pattern, deriving the Stuart-Landau amplitude equations. Moreover, in a large portion of the subcritical zone, numerical simulations show the emergence of oscillating patterns, which cannot be predicted by the weakly nonlinear analysis. Finally when the pattern invades the domain as a travelling wavefront, we derive the Ginzburg-Landau amplitude equation which is able to describe the shape and the speed of the wave.Comment: 15 pages, 5 figure

    Analytical detection of stationary and dynamic patterns in a prey-predator model with reproductive Allee effect in prey growth

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    Allee effect in population dynamics has a major impact in suppressing the paradox of enrichment through global bifurcation, and it can generate highly complex dynamics. The influence of the reproductive Allee effect, incorporated in the prey's growth rate of a prey-predator model with Beddington-DeAngelis functional response, is investigated here. Preliminary local and global bifurcations are identified of the temporal model. Existence and non-existence of heterogeneous steady-state solutions of the spatio-temporal system are established for suitable ranges of parameter values. The spatio-temporal model satisfies Turing instability conditions, but numerical investigation reveals that the heterogeneous patterns corresponding to unstable Turing eigen modes acts as a transitory pattern. Inclusion of the reproductive Allee effect in the prey population has a destabilising effect on the coexistence equilibrium. For a range of parameter values, various branches of stationary solutions including mode-dependent Turing solutions and localized pattern solutions are identified using numerical bifurcation technique. The model is also capable to produce some complex dynamic patterns such as travelling wave, moving pulse solution, and spatio-temporal chaos for certain range of parameters and diffusivity along with appropriate choice of initial conditions Judicious choices of parametrization for the Beddington-DeAngelis functional response help us to infer about the resulting patterns for similar prey-predator models with Holling type-II functional response and ratio-dependent functional response
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