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Register a new userExtinction times for birth-death processes: exact results, continuum asymptotics, and the failure of the Fokker-Planck approximation
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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.
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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.
The question of whether a population will persist or go extinct is of key interest throughout ecology and biology. Various mathematical techniques allow us to generate knowledge regarding individual behaviour, which can be analysed to obtain predictions about the ultimate survival or extinction of the population. A common model employed to describe population dynamics is the lattice-based random walk model with crowding (exclusion). This model can incorporate behaviour such as birth, death and movement, while including natural phenomena such as finite size effects. Performing sufficiently many realisations of the random walk model to extract representative population behaviour is computationally intensive. Therefore, continuum approximations of random walk models are routinely employed. However, standard continuum approximations are notoriously incapable of making accurate predictions about population extinction. Here, we develop a new continuum approximation, the state space diffusion approximation, which explicitly accounts for population extinction. Predictions from our approximation faithfully capture the behaviour in the random walk model, and provides additional information compared to standard approximations. We examine the influence of the number of lattice sites and initial number of individuals on the long-term population behaviour, and demonstrate the reduction in computation time between the random walk model and our approximation.
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.
We investigate in detail the model of a trophic web proposed by Amaral and Meyer [Phys. Rev. Lett. 82, 652 (1999)]. We focused on small-size systems that are relevant for real biological food webs and for which the fluctuations are playing an important role. We show, using Monte Carlo simulations, that such webs can be non-viable, leading to extinction of all species in small and/or weakly coupled systems. Estimations of the extinction times and survival chances are also given. We show that before the extinction the fraction of highly-connected species (omnivores) is increasing. Viable food webs exhibit a pyramidal structure, where the density of occupied niches is higher at lower trophic levels, and moreover the occupations of adjacent levels are closely correlated. We also demonstrate that the distribution of the lengths of food chains has an exponential character and changes weakly with the parameters of the model. On the contrary, the distribution of avalanche sizes of the extinct species depends strongly on the connectedness of the web. For rather loosely connected systems we recover the power-law type of behavior with the same exponent as found in earlier studies, while for densely-connected webs the distribution is not of a power-law type.
We use methods from combinatorics and algebraic statistics to study analogues of birth-and-death processes that have as their state space a finite subset of the $m$-dimensional lattice and for which the $m$ matrices that record the transition probabilities in each of the lattice directions commute pairwise. One reason such processes are of interest is that the transition matrix is straightforward to diagonalize, and hence it is easy to compute $n$ step transition probabilities. The set of commuting birth-and-death processes decomposes as a union of toric varieties, with the main component being the closure of all processes whose nearest neighbor transition probabilities are positive. We exhibit an explicit monomial parametrization for this main component, and we explore the boundary components using primary decomposition.
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