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

Krause, Andrew L.; Gorder, Robert A. Van

doi:10.1007/s11538-020-00787-y

“…Niche partitioning in a spatial context has been observed in various ecosystems (Albrecht and Gotelli, 2001;Fitzsimons et al, 2008;Lorenzetti et al, 1997;Lyson and Longrich, 2010;Quillfeldt et al, 2013;Schuette et al, 2013;Winder, 2009), making the model (6) potentially attractive for modelling a variety of specific ecosystems. Krause et al (2020) shows that for heterogeneous reactiondiffusion systems, Turing-type instabilities will localize on regions that parametrically satisfy the criteria one would get from a linear stability analysis (under the assumption that the heterogeneity varies slowly compared to any possible emergent patterning). We anticipate that such a localization will also be true for these reaction-advection-diffusion systems, allowing a clearer decoupling between niches which exist due to Turing-type patterns and those which are due to spatial heterogeneity.…”

Section: Discussionmentioning

confidence: 99%

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

Taylor

¹

,

Kim

²

,

Krause

³

et al. 2020

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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.

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“…Here α and β are constants that are not equal to one. The autonomous form of (103) is known to admit solutions of the form u 1 = (1 + exp(x − c 0 t + x 0 )) −2 and u 2 = 1 − u 1 (see [57,41,40]) which can model the invasion of one species habitat by the other (depending on the sign of the wavespeed c 0 ). Making a similar assumption on the form of a solution pair for the non-autonomous managed case, equation (103) has a solution of the desired form when the constraint system (96) takes the form…”

Section: A43 Controlled Waves Of Invasion Under a Lotka-volterra Popu...mentioning

confidence: 99%

Adiabatic soliton management: Controlling solitary wave motion while keeping the wave envelope unchanged

Van Gorder

2021

Preprint

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Travelling waves arise in several areas of science, hence modification of travelling wave properties is of great interest. While many studies have demonstrated how to control the form or shape of a solitary travelling wave by employing soliton or dispersion management, far less is known about controlling the motion of a travelling wave while keeping its form unchanged. We present a technique for control of travelling wave motion using time-varying coefficients, which we refer to as wave management. The technique allows one to alter the trajectory of a travelling wave, slowing, stopping, or reversing the direction of the wave, all while ensuring that the wave form is unchanged, and we illustrate this through multiple examples. Our results suggest that wave management is a promising tool for applications where one needs to modify the motion of a wave while preserving its form, and we highlight several potential applications.

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“…We address this question by nondimensionalising the mathematical model and numerically exploring travelling wave solutions in one dimension. Not only does travelling wave analysis of the mathematical model has a direct link to the application in question, we note that travelling wave analysis provides mathematical insight into various models of invasion with applications including tissue engineering (Landman and Cai 2007 ), directed migration (Krause and Van Gorder 2020 ), disease progression (Strobl et al. 2020 ), and various applications in ecology (Hogan and Myerscough 2017 ; El-Hachem et al.…”

Section: Introductionmentioning

confidence: 99%

A Continuum Mathematical Model of Substrate-Mediated Tissue Growth

El‐Hachem

¹

,

McCue

²

,

Simpson

³

2022

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We consider a continuum mathematical model of biological tissue formation inspired by recent experiments describing thin tissue growth in 3D-printed bioscaffolds. The continuum model, which we call the substrate model, involves a partial differential equation describing the density of tissue, $${\hat{u}}(\hat{{\mathbf {x}}},{\hat{t}})$$ u ^ ( x ^ , t ^ ) that is coupled to the concentration of an immobile extracellular substrate, $${\hat{s}}(\hat{{\mathbf {x}}},{\hat{t}})$$ s ^ ( x ^ , t ^ ) . Cell migration is modelled with a nonlinear diffusion term, where the diffusive flux is proportional to $${\hat{s}}$$ s ^ , while a logistic growth term models cell proliferation. The extracellular substrate $${\hat{s}}$$ s ^ is produced by cells and undergoes linear decay. Preliminary numerical simulations show that this mathematical model is able to recapitulate key features of recent tissue growth experiments, including the formation of sharp fronts. To provide a deeper understanding of the model we analyse travelling wave solutions of the substrate model, showing that the model supports both sharp-fronted travelling wave solutions that move with a minimum wave speed, $$c = c_{\mathrm{min}}$$ c = c min , as well as smooth-fronted travelling wave solutions that move with a faster travelling wave speed, $$c > c_{\mathrm{min}}$$ c > c min . We provide a geometric interpretation that explains the difference between smooth and sharp-fronted travelling wave solutions that is based on a slow manifold reduction of the desingularised three-dimensional phase space. In addition, we also develop and test a series of useful approximations that describe the shape of the travelling wave solutions in various limits. These approximations apply to both the sharp-fronted and smooth-fronted travelling wave solutions. Software to implement all calculations is available at GitHub.

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

Cited by 11 publications

References 66 publications

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

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

Adiabatic soliton management: Controlling solitary wave motion while keeping the wave envelope unchanged

A Continuum Mathematical Model of Substrate-Mediated Tissue Growth

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