Fisher waves and front roughening in a two-species invasion model with preemptive competition

Abstract

We study front propagation when an invading species competes with a resident; we assume nearestneighbor preemptive competition for resources in an individual-based, two-dimensional lattice model. The asymptotic front velocity exhibits power-law dependence on the difference between the two species' clonal propagation rates (key ecological parameters). The mean-field approximation behaves similarly, but the power law's exponent slightly differs from the individual-based model's result. We also study roughening of the front, using the framework of non-equilibrium interface growth. Our analysis indicates that initially flat, linear invading fronts exhibit Kardar-Parisi-Zhang (KPZ) roughening in one transverse dimension. Further, this finding implies, and is also confirmed by simulations, that the temporal correction to the asymptotic front velocity is of O(t −2/3 ).

Spatial Variability Enhances Species Fitness in Stochastic Predator-Prey Interactions

“…The classical two-species Lotka-Volterra reactiondiffusion equations, i.e., essentially the mean-field rate equations supplemented with diffusive spreading, are well-known to support traveling wave solutions [4,17,18,19], whose minimal front speed can be established by standard mathematical tools [29,30,31]. Beyond the mean-field approximation, however, already in singlespecies systems the incorporation of intrinsic reaction noise in the computation of wave front propagation velocities is a rather difficult problem [32,33,34,35,36], and there are very few results available for two-species models [35,37].…”

Finite Volume Kolmogorov-Johnson-Mehl-Avrami Theory

“…Phase conversions are of importance in physics, chemistry and other fields. Examples are numerous and include crystal physics [1], metallurgy [2], polymer physics [3,4], ferroelectric domain switching [5], magnetization and metastability in statistical physics models [6,7], phase transitions in particle physics [8], as well as ecological landscapes [9].…”

Modelling of Tissue Invasion in Epithelial Monolayers

“…To interpret the numerical results obtained for the invasion velocity shown in Figure 5 , we compare the behavior of the CPM model to an analogous reaction–diffusion model with two competing species described by the following system of partial differential equations [ 35 ]: where and are the densities of the competing species, and are the reproduction rate constants, and are the mortality parameters, and D is the diffusion coefficient. The spatially uniform system has three steady states: , , and .…”