Effects of Demographic Noise on the Synchronization of a Metapopulation in a Fluctuating Environment

We use the theory of noise-induced phase synchronization to analyze the effects of demographic noise on the synchronization of a metapopulation of predator-prey systems within a fluctuating environment (Moran effect). Treating each local predator-prey population as a stochastic urn model, we derive a Langevin equation for the stochastic dynamics of the metapopulation. Assuming each local population acts as a limit cycle oscillator in the deterministic limit, we use phase reduction and averaging methods to derive the steady-state probability density for pairwise phase differences between oscillators, which is then used to determine the degree of synchronization of the metapopulation.

Figure 1

(a) Plot of components $Z_1$, $Z_2$ of PRC for a limit cycle solution of the Holling-Tanner equations with $r=1$, $A_0=0.1$, $s=0.1$, $h=2$, $m=3$, and $K=0.5$. (b) Corresponding functions $g(\varphi )$ and $h(\varphi )$. Inset shows limit cycle.

Figure 2

Comparison of numerical simulations and analytical results for $\mathcal{N}=200$ Holling-Tanner oscillators using the same parameters as Fig. 1. (a)–(c) Steady-state probability density $\Phi (\varphi )$ of pairwise phase differences for various levels of environmental and demographic noise: (a) $\sigma ={10}^{-3}$, $ϵ={10}^{-6}$; (b) $\sigma ={10}^{-2}$ and $ϵ={10}^{-3}$; (c) $\sigma ={10}^{-6}$ and $ϵ={10}^{-3}$. (d)–(f) Corresponding ensemble dynamics in state space showing a snapshot of all the oscillators (black dots) and a scatterplot of the averaged trajectory ${\overline{X}}_j(t)$ of the metapopulation over multiple realizations (blue or gray dots).

Figure 3

Plots of time-averaged order parameter $\overline{R}$ (black curve) and half-width $1-\Delta /(2\pi )$ (gray curve) as a function of system size $N$. Variance of $R(t)$ with respect to time is shown by error bars. Parameters as in Fig. 2 with $\sigma =0.01$.