A Non-local Cross-Diffusion Model of Population Dynamics I: Emergent Spatial and Spatiotemporal Patterns

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 sub-optimal 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 pattens 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.

Bespoke Turing Systems

“…Turing-type patterning is also studied in a variety of other settings, such as in nonlinear optics (Oppo 2009;Ardizzone et al 2013;Chembo et al 2017), geochemistry (Baurmann et al 2004;McBride and Picard 2004), astrophysics (Smolin 1996), reaction-advection-diffusion systems (Klika et al 2018;Krause et al 2018a;Van Gorder et al 2019), network-organised media (Nakao and Mikhailov 2010;Asllani et al 2014), spatial ecology (Sherratt 2012;Hata et al 2014;Taylor et al 2020), and social dynamics (Wakano et al 2009). Whilst it has been known for decades that chemical systems can be engineered to produce specific Turing patterns (Vanag and Epstein 2001), recently Tan et al (2018) employed such designed Turing systems to manufacture a porous filter for use in water purification.…”

A Non-local Cross-Diffusion Model of Population Dynamics II: Exact, Approximate, and Numerical Traveling Waves in Single- and Multi-species Populations

“…We also find interesting effects from the non-local interactions in the model, suggesting that single species invasions can be enhanced with stronger non-locality, but that invasion of a competitive species may be slowed due to this non-local effect. Finally we simulate pattern formation behind waves of invasion, showing that directed motion can have substantial impacts not only on wave speed but also on the existence and structure of emergent patterns, as predicted in the first part of our study (Taylor et al, 2020).…”

“…In (Taylor et al, 2020), we studied the dispersal and clustering of populations of multiple species interacting via an extension of the model proposed by Grindrod (1988) controlling the dispersal of individuals in single and multispecies communities (Grindrod, 1988(Grindrod, , 1991. The work of Grindrod (1988) built on simpler models by Gurtin and MacCamy (1977) and Bertsch et al (1985), by modelling behaviors where individuals can respond to threats of predation and starvation, and responses to the environment occur on a much shorter timescale than births and deaths of individuals.…”

A Multispecies Cross-Diffusion Model for Territorial Development

“…This is reasonable, since we would not expect the size of the gangs to change significantly over the course of time that we simulate the territorial dynamics. The birth and death neglect is in line with other territorial ecological models that concentrate on short term movement where the individual response is faster than the population changes [9,12]. We work on a periodic lattice, which we think of as tiling a very large or infinite space, since we do not have specific geographic constraints in mind.…”

“…Some, such as those in [9], are similar to the continuum systems derived in [10] and in Sections 1.1-3.2 of the current work, though there are several subtle but important differences. The interested reader is referred to [9,11,12], which provide an excellent overview of the current state of this ecological literature and important new results for crossdiffusion models in spatial population dynamics. Our work here provides a mechanistic underpinning for this type of PDE model, along with the formal derivation of convectiondiffusion systems.…”