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Register a new userImmigration-induced phase transition in a regulated multispecies birth-death process
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Power-law-distributed species counts or clone counts arise in many biological settings such as multispecies cell populations, population genetics, and ecology. This empirical observation that the number of species $c_{k}$ represented by $k$ individuals scales as negative powers of $k$ is also supported by a series of theoretical birth-death-immigration (BDI) models that consistently predict many low-population species, a few intermediate-population species, and very high-population species. However, we show how a simple global population-dependent regulation in a neutral BDI model destroys the power law distributions. Simulation of the regulated BDI model shows a high probability of observing a high-population species that dominates the total population. Further analysis reveals that the origin of this breakdown is associated with the failure of a mean-field approximation for the expected species abundance distribution. We find an accurate estimate for the expected distribution $langle c_k rangle$ by mapping the problem to a lower-dimensional Moran process, allowing us to also straightforwardly calculate the covariances $langle c_k c_ell rangle$. Finally, we exploit the concepts associated with energy landscapes to explain the failure of the mean-field assumption by identifying a phase transition in the quasi-steady-state species counts triggered by a decreasing immigration rate.
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A feature of human creativity is the ability to take a subset of existing items (e.g. objects, ideas, or techniques) and combine them in various ways to give rise to new items, which, in turn, fuel further growth. Occasionally, some of these items may also disappear (extinction). We model this process by a simple stochastic birth--death model, with non-linear combinatorial terms in the growth coefficients to capture the propensity of subsets of items to give rise to new items. In its simplest form, this model involves just two parameters $(P, alpha)$. This process exhibits a characteristic hockey-stick behaviour: a long period of relatively little growth followed by a relatively sudden explosive increase. We provide exact expressions for the mean and variance of this time to explosion and compare the results with simulations. We then generalise our results to allow for more general parameter assignments, and consider possible applications to data involving human productivity and creativity.
We consider extinction times for a class of birth-death processes commonly found in applications, where there is a control parameter which determines whether the population quickly becomes extinct, or rather persists for a long time. We give an exact expression for the discrete case and its asymptotic expansion for large values of the population. We have results below the threshold, at the threshold, and above the threshold (where there is a quasi-stationary state and the extinction time is very long.) We show that the Fokker-Planck approximation is valid only quite near the threshold. We compare our analytical results to numerical simulations for the SIS epidemic model, which is in the class that we treat. This is an interesting example of the delicate relationship between discrete and continuum treatments of the same problem.
We discovered a dynamic phase transition induced by sexual reproduction. The dynamics is a pure Darwinian rule with both fundamental ingredients to drive evolution: 1) random mutations and crossings which act in the sense of increasing the entropy (or diversity); and 2) selection which acts in the opposite sense by limiting the entropy explosion. Selection wins this competition if mutations performed at birth are few enough. By slowly increasing the average number m of mutations, however, the population suddenly undergoes a mutational degradation precisely at a transition point mc. Above this point, the bad alleles spread over the genetic pool of the population, overcoming the selection pressure. Individuals become selectively alike, and evolution stops. Only below this point, m < mc, evolutionary life is possible. The finite-size-scaling behaviour of this transition is exhibited for large enough chromosome lengths L. One important and surprising observation is the L-independence of the transition curves, for large L. They are also independent on the population size. Another is that mc is near unity, i.e. life cannot be stable with much more than one mutation per diploid genome, independent of the chromosome length, in agreement with reality. One possible consequence is that an eventual evolutionary jump towards larger L enabling the storage of more genetic information would demand an improved DNA copying machinery in order to keep the same total number of mutations per offspring.
The dynamics of populations is frequently subject to intrinsic noise. At the same time unknown interaction networks or rate constants can present quenched uncertainty. Existing approaches often involve repeated sampling of the quenched disorder and then running the stochastic birth-death dynamics on these samples. In this paper we take a different view, and formulate an effective jump process, representative of the ensemble of quenched interactions as a whole. Using evolutionary games with random payoff matrices as an example, we develop an algorithm to simulate this process, and we discuss diffusion approximations in the limit of weak intrinsic noise.
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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