Instability of Spatial Patterns and Its Ambiguous Impact on Species Diversity

Self-arrangement of individuals into spatial patterns often accompanies and promotes species diversity in ecological systems. Here, we investigate pattern formation arising from cyclic dominance of three species, operating near a bifurcation point. In its vicinity, an Eckhaus instability occurs, leading to convectively unstable “blurred” patterns. At the bifurcation point, stochastic effects dominate and induce counterintuitive effects on diversity: Large patterns, emerging for medium values of individuals’ mobility, lead to rapid species extinction, while small patterns (low mobility) promote diversity, and high mobilities render spatial structures irrelevant. We provide a quantitative analysis of these phenomena, employing a complex Ginzburg-Landau equation.

Figure 1

Snapshots of the biodiverse state for $D=1\times {10}^{-5}$. (a) For large rates $\gamma$, entangled and stable spiral waves form. (b) A convective (Eckhaus) instability occurs at ${\gamma }_E\approx 2$; below this value, the spiral patterns blur. (c) At the bifurcation point $\gamma =0$, only very weak spatial modulations emerge; we have amplified them by a factor of 2 for better visibility. The snapshots stem from numerical solution of the SPDE (5) with initially homogeneous densities $a=b=c=1/4$.

Figure 2

The dependence of the wavelength $\lambda$ on the rate $\gamma$. Analytical predictions from the CGLE (7) (divided by a factor 1.55), red (solid) line, are compared to numerical results (□). For $\gamma <{\gamma }_E\approx 2$ spirals are convectively instable. When approaching the bifurcation point, $\gamma \to 0$, computation of the correlation length (⊙) shows that the spatial structures are no longer determined by the (diverging) wavelength, but reach a constant size $l_c$, see text.

Figure 3

Regimes of stable, unstable, and neutrally stable biodiversity. They are revealed by computing the extinction probability for $\gamma =0$ and $\rho =1$, depending on $D$, after a waiting time $t=N$, for increasing system size $N$. We show curves for $N=20\times 20$ (□), $N=25\times 25$ (○), $N=40\times 40$ (△), and $N=100\times 100$ (◇). The transitions occur at $D_c^{(1)}\approx 3.5\times {10}^{-3}$ and $D_c^{(2)}\approx 0.2$.