Pattern formation in a model of competing populations with nonlocal interactions

Segal, Benjamin L.; Volpert, Vitaly; Bayliss, A.

doi:10.1016/j.physd.2013.02.006

Cited by 62 publications

(42 citation statements)

References 8 publications

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“…These different types of kernel functions include top-hat kernel, Gaussian kernel, Laplacian kernel, triangular kernel, parabolic kernel, etc. [40][41][42]. Among these kernel functions, the simplest one is the symmetric top-hat kernel which is basically a step function, and incorporation of it in the modeling approach can lead to very rich and complex spatiotemporal pattern formations [40,42].…”

Section: Introductionmentioning

confidence: 99%

Dynamics of a Diffusive Two-Prey-One-Predator Model with Nonlocal Intra-Specific Competition for Both the Prey Species

2020

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Investigation of interacting populations is an active area of research, and various modeling approaches have been adopted to describe their dynamics. Mathematical models of such interactions using differential equations are capable to mimic the stationary and oscillating (regular or irregular) population distributions. Recently, some researchers have paid their attention to explain the consequences of transient dynamics of population density (especially the long transients) and able to capture such behaviors with simple models. Existence of multiple stationary patches and settlement to a stable distribution after a long quasi-stable transient dynamics can be explained by spatiotemporal models with nonlocal interaction terms. However, the studies of such interesting phenomena for three interacting species are not abundant in literature. Motivated by these facts here we have considered a three species prey–predator model where the predator is generalist in nature as it survives on two prey species. Nonlocalities are introduced in the intra-specific competition terms for the two prey species in order to model the accessibility of nearby resources. Using linear analysis, we have derived the Turing instability conditions for both the spatiotemporal models with and without nonlocal interactions. Validation of such conditions indicates the possibility of existence of stationary spatially heterogeneous distributions for all the three species. Existence of long transient dynamics has been presented under certain parametric domain. Exhaustive numerical simulations reveal various scenarios of stabilization of population distribution due to the presence of nonlocal intra-specific competition for the two prey species. Chaotic oscillation exhibited by the temporal model is significantly suppressed when the populations are allowed to move over their habitat and prey species can access the nearby resources.

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Section: Introductionmentioning

confidence: 99%

Dynamics of a Diffusive Two-Prey-One-Predator Model with Nonlocal Intra-Specific Competition for Both the Prey Species

2020

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“…6 In previous investigations, chimera patterns have been observed in numerical simulations of nonlocally coupled systems, in which the oscillators are coupled beyond nearest neighbors. [7][8][9][10][11] Nonlocal coupling leading to chimera states has been achieved in experiments on an optical coupled map lattice 12 and a photosensitive reactiondiffusion system. 13 Chimera structures do not require genuine non-local coupling but can also arise in reaction-diffusion systems under special conditions.…”

Section: Introductionmentioning

confidence: 99%

Novel type of chimera spiral waves arising from decoupling of a diffusible component

Tang

Yang

Epstein

et al. 2014

The Journal of Chemical Physics

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Spiral waves composed of coherent traveling waves surrounding a core containing stochastically distributed stationary areas are found in numerical simulations of a three-variable reaction-diffusion system with one diffusible species. In the spiral core, diffusion of this component (w) mediates transitions between dynamic states of the subsystem formed by the other two components, whose dynamics is more rapid than that of w. Diffusive coupling between adjacent sites can be "on" or "off" depending on the subsystem state. The incoherent structures in the spiral core are produced by this decoupling of the slow diffusive component from the fast non-diffusing subsystem. The phase diagram reveals that the region of incoherent behavior in chimera spirals grows drastically, leading to modulation and breakup of the spirals, in the transition zones between 1(n-1) and 1(n) local mixed-mode oscillations.

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“…Here we consider the step function for simplicity of mathematical calculations [49]. This step function means that the nonlocal consumption is confined within the range 2M, and the efficacy of consumption inside this range is constant.…”

Section: Nonlocal Modelmentioning

confidence: 99%

Prey-Predator Model with a Nonlocal Bistable Dynamics of Prey

2018

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Spatiotemporal pattern formation in integro-differential equation models of interacting populations is an active area of research, which has emerged through the introduction of nonlocal intra-and inter-specific interactions. Stationary patterns are reported for nonlocal interactions in prey and predator populations for models with prey-dependent functional response, specialist predator and linear intrinsic death rate for predator species. The primary goal of our present work is to consider nonlocal consumption of resources in a spatiotemporal prey-predator model with bistable reaction kinetics for prey growth in the absence of predators. We derive the conditions of the Turing and of the spatial Hopf bifurcation around the coexisting homogeneous steady-state and verify the analytical results through extensive numerical simulations. Bifurcations of spatial patterns are also explored numerically.

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Pattern formation in a model of competing populations with nonlocal interactions

Cited by 62 publications

References 8 publications

Dynamics of a Diffusive Two-Prey-One-Predator Model with Nonlocal Intra-Specific Competition for Both the Prey Species

Dynamics of a Diffusive Two-Prey-One-Predator Model with Nonlocal Intra-Specific Competition for Both the Prey Species

Novel type of chimera spiral waves arising from decoupling of a diffusible component

Prey-Predator Model with a Nonlocal Bistable Dynamics of Prey

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