Analytical solution of metapopulation dynamics in a stochastic environment

We study a discrete stochastic linear metapopulation model to understand the effect of risk spreading by dispersion. We calculate analytically the stable distribution of populations in different habitats. The simultaneous distribution of populations in habitats has a complicated self-similar structure, but the population in each habitat follows a log-normal distribution. A class of discrete stochastic matrix models was mostly dealt with numerically. Our analytical predictions are robust in the wide range of parameters. Qualitative predictions of the current results should hold in the case of multiple habitats. We thus conclude that environmental stochasticity always promotes dispersal.

Figure 1

Stationary distribution of $r\left(t\right)=x_2\left(t\right)/x_1\left(t\right)$ for $m_{+}=3$, $m_{-}=0.01$, $c=0$, $p=0.3$, $q=0.2$, and $s=0.5$. The horizontal axis is a logarithmic scale. (a) Theoretical result obtained by solving the Frobenius-Perron equation (8) numerically. (b) Time distribution obtained by computer simulation over 100  000 time steps after a transient of 1000 time steps. We observe a good agreement between these two distributions.

Figure 2

Evolution of probability distribution of $\ln x_1\left(t\right)$ for (a) $q=0.2$ and (b) $q=0.65$, when $t=1000$, 2000, and 4000. The other parameters are $m_{+}=3$, $m_{-}=0.01$, $c=0$, $p=0.3$, and $s=0.5$. The crosses stand for the distribution obtained by 100 000 stochastic realizations. The lines stand for the approximation when neglecting the time correlation of the effective growth rates. (c) The temporal correlation functions of the effective growth rates of $x_1\left(t\right)$ in cases (a) and (b). We find a slight negative correlation at short time lags and rapid decay to zero. The negative correlation implies that the variance of the actual probability distribution tends to be smaller than the approximation that neglects the temporal correlation of the effective growth rate.

Figure 3

(a) Long-term population growth rate as a function of migration rate $q$ for three values of correlation ($c=-0.2,0,0.2$). The other parameters are set as $m_{+}=3$, $m_{-}=0.01$, $p=0.3$, and $s=0.5$. (b) The optimal migration rate as a function of survival rate through migration $s$ for $p=0.2,03,0.5,0.7,0.8$. The curve is obtained by Eq. (12) and the stationary distribution ${\rho }_{*}\left(r\right)$. The crosses represent the numerical result of Eq. (13) for $T=10000$, averaged over 10 000 stochastic realizations.