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Register a new userColonization of a territory by a stochastic population under a strong Allee effect and a low immigration pressure
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We study the dynamics of colonization of a territory by a stochastic population at low immigration pressure. We assume a sufficiently strong Allee effect that introduces, in deterministic theory, a large critical population size for colonization. At low immigration rates, the average pre-colonization population size is small thus invalidating the WKB approximation to the master equation. We circumvent this difficulty by deriving an exact zero-flux solution of the master equation and matching it with an approximate non-zero-flux solution of the pertinent Fokker-Planck equation in a small region around the critical population size. This procedure provides an accurate evaluation of the quasi-stationary probability distribution of population sizes in the pre-colonization state, and of the mean time to colonization, for a wide range of immigration rates. At sufficiently high immigration rates our results agree with WKB results obtained previously. At low immigration rates the results can be very different.
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We study the extinction risk of a fragmented population residing on a network of patches coupled by migration, where the local patch dynamics include the Allee effect. We show that mixing between patches dramatically influences the populations viability. Slow migration is shown to always increase the populations global extinction risk compared to the isolated case. At fast migration, we demonstrate that synchrony between patches minimizes the populations extinction risk. Moreover, we discover a critical migration rate that maximizes the extinction risk of the population, and identify an early-warning signal when approaching this state. Our theoretical results are confirmed via the highly-efficient weighted ensemble method. Notably, our analysis can also be applied to studying switching in gene regulatory networks with multiple transcriptional states.
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Many unicellular organisms allocate their key proteins asymmetrically between the mother and daughter cells, especially in a stressed environment. A recent theoretical model is able to predict when the asymmetry in segregation of key proteins enhances the population fitness, extrapolating the solution at two limits where the segregation is perfectly asymmetric (asymmetry $a$ = 1) and when the asymmetry is small ($0 leq a ll 1$). We generalize the model by introducing stochasticity and use a transport equation to obtain a self-consistent equation for the population growth rate and the distribution of the amount of key proteins. We provide two ways of solving the self-consistent equation: numerically by updating the solution for the self-consistent equation iteratively and analytically by expanding moments of the distribution. With these more powerful tools, we can extend the previous model by Lin et al. to include stochasticity to the segregation asymmetry. We show the stochastic model is equivalent to the deterministic one with a modified effective asymmetry parameter ($a_{rm eff}$). We discuss the biological implication of our models and compare with other theoretical models.
Environmental changes greatly influence the evolution of populations. Here, we study the dynamics of a population of two strains, one growing slightly faster than the other, competing for resources in a time-varying binary environment modeled by a carrying capacity switching either randomly or periodically between states of abundance and scarcity. The population dynamics is characterized by demographic noise (birth and death events) coupled to a varying environment. We elucidate the similarities and differences of the evolution subject to a stochastically- and periodically-varying environment. Importantly, the population size distribution is generally found to be broader under intermediate and fast random switching than under periodic variations, which results in markedly different asymptotic behaviors between the fixation probability of random and periodic switching. We also determine the detailed conditions under which the fixation probability of the slow strain is maximal.
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