title: Co-operation, Competition and Crowding: A Discrete Framework Linking Allee Kinetics, Nonlinear Diffusion, Shocks and Sharp-Fronted Travelling Waves.
abstract: Invasion processes are ubiquitous throughout cell biology and ecology. During invasion, individuals can become isolated from the bulk population and behave differently. We present a discrete, exclusion-based description of the birth, death and movement of individuals. The model distinguishes between individuals that are part of, or are isolated from, the bulk population by imposing different rates of birth, death and movement. This enables the simulation of various co-operative or competitive mechanisms, where there is either a positive or negative benefit associated with being part of the bulk population, respectively. The mean-field approximation of the discrete process gives rise to 22 different classes of partial differential equation, which can include Allee kinetics and nonlinear diffusion. Here we examine the ability of each class of partial differential equation to support travelling wave solutions and interpret the long time behaviour in terms of the individual-level parameters. For the first time we show that the strong Allee effect and nonlinear diffusion can result in shock-fronted travelling waves. We also demonstrate how differences in group and individual motility rates can influence the persistence of a population and provide conditions for the successful invasion of a population.
authors.affiliation: Mathematical Sciences, Queensland University of Technology (QUT), Brisbane, Australia.|Tissue Repair and Regeneration Program, Institute of Health and Biomedical Innovation, QUT, Brisbane, Australia.
authors.forename: Stuart T
authors.initials: ST
authors.lastname: Johnston
authors.affiliation: Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, United Kingdom.
authors.forename: Ruth E
authors.initials: RE
authors.lastname: Baker
authors.affiliation: Mathematical Sciences, Queensland University of Technology (QUT), Brisbane, Australia.|Tissue Repair and Regeneration Program, Institute of Health and Biomedical Innovation, QUT, Brisbane, Australia.
authors.forename: D L Sean
authors.initials: DL
authors.lastname: McElwain
authors.affiliation: Mathematical Sciences, Queensland University of Technology (QUT), Brisbane, Australia.|Tissue Repair and Regeneration Program, Institute of Health and Biomedical Innovation, QUT, Brisbane, Australia.
authors.forename: Matthew J
authors.initials: MJ
authors.lastname: Simpson
mesh_terms: D000818:Animals; D001932:Brain Neoplasms / Q000473:pathology; D003585:Cell Biology / Q000706:statistics & numerical data*; D002465:Cell Movement; D003163:Competitive Behavior; D003299:Cooperative Behavior; D003441:Crowding; D004058:Diffusion; D004463:Ecology / Q000706:statistics & numerical data*; D005347:Fibroblasts / Q000166:cytology / Q000502:physiology; D005910:Glioma / Q000473:pathology; D006801:Humans; D007700:Kinetics; D015233:Models, Statistical; D009360:Neoplastic Cells, Circulating; D011157:Population Dynamics / Q000706:statistics & numerical data; D014945:Wound Healing / Q000502:physiology