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Register a new userThe fixation time of a strongly beneficial allele in a structured population
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For a beneficial allele which enters a large unstructured population and eventually goes to fixation, it is known that the time to fixation is approximately $2log(alpha)/alpha$ for a large selection coefficient $alpha$. For a population that is distributed over finitely many colonies, with migration between these colonies, we detect various regimes of the migration rate $mu$ for which the fixation times have different asymptotics as $alpha to infty$. If $mu$ is of order $alpha$, the allele fixes (as in the spatially unstructured case) in time $sim 2log(alpha)/alpha$. If $mu$ is of order $alpha^gamma, 0leq gamma leq 1$, the fixation time is $sim (2 + (1-gamma)Delta) log(alpha)/alpha$, where $Delta$ is the number of migration steps that are needed to reach all other colonies starting from the colony where the beneficial allele appeared. If $mu = 1/log(alpha)$, the fixation time is $sim (2+S)log(alpha)/alpha$, where $S$ is a random time in a simple epidemic model. The main idea for our analysis is to combine a new moment dual for the process conditioned to fixation with the time reversal in equilibrium of a spatial version of Neuhauser and Krones ancestral selection graph.
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If we follow an asexually reproducing population through time, then the amount of time that has passed since the most recent common ancestor (MRCA) of all current individuals lived will change as time progresses. The resulting MRCA age process has been studied previously when the population has a constant large size and evolves via the diffusion limit of standard Wright--Fisher dynamics. For any population model, the sample paths of the MRCA age process are made up of periods of linear upward drift with slope +1 punctuated by downward jumps. We build other Markov processes that have such paths from Poisson point processes on $mathbb{R}_{++}timesmathbb{R}_{++}$ with intensity measures of the form $lambdaotimesmu$ where $lambda$ is Lebesgue measure, and $mu$ (the family lifetime measure) is an arbitrary, absolutely continuous measure satisfying $mu((0,infty))=infty$ and $mu((x,infty))<infty$ for all $x>0$. Special cases of this construction describe the time evolution of the MRCA age in $(1+beta)$-stable continuous state branching processes conditioned on nonextinction--a particular case of which, $beta=1$, is Fellers continuous state branching process conditioned on nonextinction. As well as the continuous time process, we also consider the discrete time Markov chain that records the value of the continuous process just before and after its successive jumps. We find transition probabilities for both the continuous and discrete time processes, determine when these processes are transient and recurrent and compute stationary distributions when they exist.
The aim of this paper is to tackle part of the program set by Diekmann et al. in their seminal paper Diekmann et al. (2001). We quote It remains to investigate whether, and in what sense, the nonlinear determin-istic model formulation is the limit of a stochastic model for initial population size tending to infinity We set a precise and general framework for a stochastic individual based model : it is a piecewise deterministic Markov process defined on the set of finite measures. We then establish a law of large numbers under conditions easy to verify. Finally we show how this applies to old and new examples.
In the present article, we investigate the effects of dormancy on an abstract population genetic level. We first provide a short review of seed bank models in population genetics, and the role of dormancy for the interplay of evolutionary forces in general, before we discuss two recent paradigmatic models, referring to spontaneous resp. simultaneous switching of individuals between the active and the dormant state. We show that both mechanisms give rise to non-trivial mathematical objects, namely the (continuous) seed bank diffusion and the seed bank diffusion with jumps, as well as their dual processes, the seed bank coalescent and the seed bank coalescent with simultaneous switching.
Classical ecological theory predicts that environmental stochasticity increases extinction risk by reducing the average per-capita growth rate of populations. To understand the interactive effects of environmental stochasticity, spatial heterogeneity, and dispersal on population growth, we study the following model for population abundances in $n$ patches: the conditional law of $X_{t+dt}$ given $X_t=x$ is such that when $dt$ is small the conditional mean of $X_{t+dt}^i-X_t^i$ is approximately $[x^imu_i+sum_j(x^j D_{ji}-x^i D_{ij})]dt$, where $X_t^i$ and $mu_i$ are the abundance and per capita growth rate in the $i$-th patch respectivly, and $D_{ij}$ is the dispersal rate from the $i$-th to the $j$-th patch, and the conditional covariance of $X_{t+dt}^i-X_t^i$ and $X_{t+dt}^j-X_t^j$ is approximately $x^i x^j sigma_{ij}dt$. We show for such a spatially extended population that if $S_t=(X_t^1+...+X_t^n)$ is the total population abundance, then $Y_t=X_t/S_t$, the vector of patch proportions, converges in law to a random vector $Y_infty$ as $ttoinfty$, and the stochastic growth rate $lim_{ttoinfty}t^{-1}log S_t$ equals the space-time average per-capita growth rate $sum_imu_iE[Y_infty^i]$ experienced by the population minus half of the space-time average temporal variation $E[sum_{i,j}sigma_{ij}Y_infty^i Y_infty^j]$ experienced by the population. We derive analytic results for the law of $Y_infty$, find which choice of the dispersal mechanism $D$ produces an optimal stochastic growth rate for a freely dispersing population, and investigate the effect on the stochastic growth rate of constraints on dispersal rates. Our results provide fundamental insights into ideal free movement in the face of uncertainty, the persistence of coupled sink populations, the evolution of dispersal rates, and the single large or several small (SLOSS) debate in conservation biology.
We derive and apply a partial differential equation for the moment generating function of the Wright-Fisher model of population genetics.
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