Analyzing the spatial dynamics of a prey–predator lattice model with social behavior (original) (raw)
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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 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...
Spatial patterns of a predator-prey model with cross diffusion
Nonlinear Dynamics, 2012
In this paper, spatial patterns of a Holling-Tanner predator-prey model subject to cross diffusion, which means the prey species exercise a self-defense mechanism to protect themselves from the attack of the predator are investigated. By using the bifurcation theory, the conditions of Hopf and Turing bifurcation critical line in a spatial domain are obtained. A series of numerical simulations reveal that the typical dynamics of population density variation is the formation of isolated groups, such as spotted, stripe-like, or labyrinth patterns. Our results confirm that cross dif-fusion can create stationary patterns, which enrich the finding of pattern formation in an ecosystem.
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.
Journal of Theoretical Biology, 1995
The Lotka-Volterra model of predator-prey interaction is based on the assumption of mass action, a concept borrowed from the traditional theory of chemical kinetics in which reactants are assumed to be homogeneously mixed. In order to explore the effect of spatial heterogeneity on predator-prey dynamics, we constructed a lattice-based reaction-diffusion model corresponding to the Lotka-Volterra equations. Spatial heterogeneity was imposed on the system using percolation maps, gradient percolation maps, and fractional Brownian surfaces. In all simulations where diffusion distances were short, anomalously low reaction orders and aggregated spatial patterns were observed, including traveling wave patterns. In general, the estimated reaction order decreased with increasing degrees of spatial heterogeneity. For simulations using percolation maps with p-values varying between 1.0 (all cells available) to 0.5 (50% available), order estimates varied from 1.27 to 0.47. Gradient percolation maps and fractional Brownian surfaces also resulted in anomalously low reaction orders. Increasing diffusion distances resulted in reaction order estimates approaching the expected value of 2. Analysis of the qualitative dynamics of the model showed little difference between simulations where individuals diffused locally and those where individuals moved to random locations, suggesting that global density dependence is an important determinant of the overall model dynamics. However, localized interactions did introduce time dependence in the system attractor owing to emergent spatial patterns. We conclude that individual-based spatially explicit models are important tools for modeling population dynamics as they allow one to incorporate fine-scale ecological data about localized interactions and then to observe emergent patterns through simulation. When heterogeneous patterns arise, it can lead to anomalies with respect to the predictions of traditional mathematical approaches using global state variables.
Dynamical complexity of a spatial predator–prey model with migration
Ecological Modelling, 2008
Travelling pattern Synchrony a b s t r a c t Spatial component of ecological interactions has been identified as an important factor in how ecological communities are shaped. Thus, in this paper, spatial pattern of a spatial predator-prey model, which is combined with migration and diffusion, is investigated. We present the emergence of spatial pattern by both theoretical analysis and numerical simulations. Our study reveals that migration has great influence on the pattern formation. More specifically, travelling pattern can be induced by it. Furthermore, wavelength of the travelling pattern with respect to the diffusion coefficient of the predator is calculated. The obtained results show that the modeling by both migration and diffusion is useful for studying the dynamic complexity of ecosystems, which well extend the finding of pattern formation in the ecosystems and may well explain the field observed in some areas.
Pattern formation of a spatial predator–prey system
There are random and directed movements of predator and prey populations in many natural systems which are strongly influenced and modified by spatial factors. To investigate how these migration (directed movement) and diffusion (random movement) affect predator-prey systems, we have studied the spatiotemporal complexity in a predator-prey system with Holling-Tanner form. A theoretical analysis of emerging spatial pattern is presented and wavelength and pattern speed are calculated. At the same time, we present the properties of pattern solutions. The results of numerical simulations show that migration has prominent effect on the pattern formation of the population, i.e., changing Turing pattern into traveling pattern. This study suggests that modelling by migration and diffusion in predator-prey systems can account for the dynamical complexity of ecosystems.
Influence of Spatial Dispersal among Species in a Prey–Predator Model with Miniature Predator Groups
Symmetry
Dispersal among species is an important factor that can govern the prey–predator model’s dynamics and cause a variety of spatial structures on a geographical scale. These structures form when passive diffusion interacts with the reaction part of the reaction–diffusion system in such a way that even if the reaction lacks symmetry-breaking capabilities, diffusion can destabilize the symmetry and allow the system to have them. In this article, we look at how dispersal affects the prey–predator model with a Hassell–Varley-type functional response when predators do not form tight groups. By considering linear stability, the temporal stability of the model and the conditions for Hopf bifurcation at feasible equilibrium are derived. We explored spatial stability in the presence of diffusion and developed the criterion for diffusion-driven instability. Using amplitude equations, we then investigated the selection of Turing patterns around the Turing bifurcation threshold. The examination of...
Oscillations and patterns in interacting populations of two species
Physical Review E, 2008
Interacting populations often create complicated spatiotemporal behavior, and understanding it is a basic problem in the dynamics of spatial systems. We study the two-species case by simulations of a host-parasitoid model. In the case of coexistence , there are spatial patterns leading to noisesustained oscillations. We introduce a new measure for the patterns, and explain the oscillations as a consequence of a timescale separation and noise. They are linked together with the patterns by letting the spreading rates depend on instantaneous population densities. Applications are discussed.