Classification and stability of global inhomogeneous solutions of a macroscopic model of cell motion.

Many micro-organisms use chemotaxis for aggregation, resulting in stable patterns. In this paper, the amoeba Dictyostelium discoideum serves as a model organism for understanding the conditions for aggregation and classification of resulting patterns. To accomplish this, a 1D nonlinear diffusion equation with chemotaxis that models amoeba behavior is analyzed. A classification of the steady state solutions is presented, and a Lyapunov functional is used to determine conditions for stability of inhomogenous solutions. Changing the chemical sensitivity, production rate of the chemical attractant, or domain length can cause the system to transition from having an asymptotic steady state, to having asymptotically stable single-step solution and multi-stepped stable plateau solutions.

Macroscopic model of self-propelled bacteria swarming with regular reversals.

Periodic reversals in the direction of motion in systems of self-propelled rod-shaped bacteria enable them to effectively resolve traffic jams formed during swarming and maximize the swarming rate of the colony. In this paper, a connection is established between a microscopic one-dimensional cell-based stochastic model of reversing nonoverlapping bacteria and a macroscopic nonlinear diffusion equation describing the dynamics of cellular density. Boltzmann-Matano analysis is used to determine the nonlinear diffusion equation corresponding to the specific reversal frequency. Stochastic dynamics averaged over an ensemble is shown to be in very good agreement with the numerical solutions of this nonlinear diffusion equation. Critical density p(0) is obtained such that nonlinear diffusion is dominated by the collisions between cells for the densities p>p(0). An analytical approximation of the pairwise collision time and semianalytical fit for the total jam time per reversal period are also obtained. It is shown that cell populations with high reversal frequencies are able to spread out effectively at high densities. If the cells rarely reverse, then they are able to spread out at lower densities but are less efficient at spreading out at higher densities.

Evolutionary convergence to ideal free dispersal strategies and coexistence.

We study a two species competition model in which the species have the same population dynamics but different dispersal strategies and show how these dispersal strategies evolve. We introduce a general dispersal strategy which can result in the ideal free distributions of both competing species at equilibrium and generalize the result of Averill et al. (2011). We further investigate the convergent stability of this ideal free dispersal strategy by varying random dispersal rates, advection rates, or both of these two parameters simultaneously. For monotone resource functions, our analysis reveals that among two similar dispersal strategies, selection generally prefers the strategy which is closer to the ideal free dispersal strategy. For nonmonotone resource functions, our findings suggest that there may exist some dispersal strategies which are not ideal free, but could be locally evolutionarily stable and/or convergent stable, and allow for the coexistence of more than one species.

Establishment of new mutations under divergence and genome hitchhiking.

Theoretical models addressing genome-wide patterns of divergence during speciation are needed to help us understand the evolutionary processes generating empirical patterns. Here, we examine a critical issue concerning speciation-with-gene flow: to what degree does physical linkage (r < 0.5) of new mutations to already diverged genes aid the build-up of genomic islands of differentiation? We used simulation and analytical approaches to partition the probability of establishment for a new divergently selected mutation when the mutation (i) is the first to arise in an undifferentiated genome (the direct effect of selection), (ii) arises unlinked to any selected loci (r = 0.5), but within a genome that has some already diverged genes (the effect of genome-wide reductions in gene flow for facilitating divergence, which we term 'genome hitchhiking'), and (iii) arises in physical linkage to a diverged locus (divergence hitchhiking). We find that the strength of selection acting directly on a new mutation is generally the most important predictor for establishment, with divergence and genomic hitchhiking having smaller effects. We outline the specific conditions under which divergence and genome hitchhiking can aid mutation establishment. The results generate predictions about genome divergence at different points in the speciation process and avenues for further work.

Adaptive chromosomal divergence driven by mixed geographic mode of evolution.

Chromosomal inversions are ubiquitous in nature and of great significance for understanding adaptation and speciation. Inversions were the first markers used to investigate the genetic structure of natural populations, leading to the concept of coadapted gene complexes and theories concerning founder effects and genetic drift in small populations. However, we still lack elements of a general theory accounting for the origins and distribution of inversions in nature. Here, we use computer simulations to show that a "mixed geographic mode" of evolution involving allopatric separation of populations followed by secondary contact and gene flow generates chromosomal divergence by natural selection under wider conditions than previous hypotheses. This occurs because inversions arising in allopatry contain a full complement of locally adapted genes. Once gene flow ensues, reduced recombination within inversions keeps these favorable genotypic combinations intact, resulting in inverted genomic regions being favored over collinear regions. This process allows inversions to establish to high frequencies. Our model can account for several classic patterns in the geographic distribution of inversions and highlights how selection on standing genetic variation allows rapid chromosomal evolution without the waiting time for new mutations. As inversion differences often separate closely related taxa, mixed modes of divergence could be common.