Analytical solution of a stochastic model of risk spreading with global coupling

Abstract

We study a stochastic matrix model to understand the mechanics of risk-spreading (or bethedging) by dispersion. Such model has been mostly dealt numerically except for well-mixed case, so far. Here, we present an analytical result, which shows that optimal dispersion leads to Zipf's law. Moreover, we found that the arithmetic ensemble average of the total growth rate converges to the geometric one, because the sample size is finite.

Disadvantages of Preferential Dispersals in Fluctuating Environments

“…Therefore, for a individual changing the habitat, it is advantageous to choose populous sites than sparsely populated ones. However, what is good for individual is not necessarily good for a whole.In this paper, we study a stochastic metapopulation which is given as a discrete-time stochastic matrix model [11,12]. In these studies, to evaluate the population's fitness for this class model, the total growth was calculated approximately.…”

“…In this paper, we study a stochastic metapopulation which is given as a discrete-time stochastic matrix model [11,12]. In these studies, to evaluate the population's fitness for this class model, the total growth was calculated approximately.…”

“…If φ (also q) is near 0, the distribution of the relative population size y i (t) has a heavy tail and the fluctuation of m is not negligible. When the fluctuation of m(t) increases, the time average of log m(t) grows more smaller than the logarithm of the mean value of m(t) [12]. As a result, if the environment has time correlation (τ c > 0), there is a finite optimal value of φ.…”