A Non-local Cross-Diffusion Model of Population Dynamics I: Emergent Spatial and Spatiotemporal Patterns
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A Non-local Cross-Diffusion Model of Population Dynamics I: Emergent Spatial and Spatiotemporal Patterns Nick P. Taylor1 · Hyunyeon Kim2 · Andrew L. Krause2 · Robert A. Van Gorder3 Received: 4 August 2019 / Accepted: 31 July 2020 © Society for Mathematical Biology 2020
Abstract We extend a spatially non-local cross-diffusion model of aggregation between multiple species with directed motion toward resource gradients to include many species and more general kinds of dispersal. We first consider diffusive instabilities, determining that for directed motion along fecundity gradients, the model permits the Turing instability leading to colony formation and persistence provided there are three or more interacting species. We also prove that such patterning is not possible in the model under the Turing mechanism for two species under directed motion along fecundity gradients, confirming earlier findings in the literature. However, when the directed motion is not along fecundity gradients, for instance, if foraging or migration is suboptimal relative to fecundity gradients, we find that very different colony structures can emerge. This generalization also permits colony formation for two interacting species. In the advection-dominated case, aggregation patterns are more broad and global in nature, due to the inherent non-local nature of the advection which permits directed motion over greater distances, whereas in the diffusion-dominated case, more highly localized patterns and colonies develop, owing to the localized nature of random diffusion. We also consider the interplay between Turing patterning and spatial heterogeneity in resources. We find that for small spatial variations, there will be a combination of Turing patterns and patterning due to spatial forcing from the resources, whereas for large resource variations, spatial or spatiotemporal patterning can be modified greatly from what is predicted on homogeneous domains. For each of these emergent behaviors, we outline the theoretical mechanism leading to colony formation and then provide numerical simulations to illustrate the results. We also discuss implications this model has for studies of directed motion in different ecological settings. Keywords Aggregation · Directed motion · Turing instability · Colony formation
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Robert A. Van Gorder [email protected]
Extended author information available on the last page of the article 0123456789().: V,-vol
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1 Introduction Since Turing’s famous work on morphogenesis (Turing 1952), the emergence of structure in reaction–diffusion models in and beyond developmental biology has been well studied (Kishimoto 1982; Kishimoto et al. 1983; Kishimoto and Weinberger 1985; Riaz et al. 2005) and in particular can provide a useful modeling framework for spatial ecology. Reaction–diffusion models can also exhibit complex spatiotemporal structures of more complexity than stationary patterned states, and these have been implicated as important hallmarks of ecological complex
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