## Stability Caused by Asynchrony in Local Dynamics

Here let us start with a simple example that considers two spatial patches. In one patch prey are vulnerable to predation while the other patch is a complete refuge. We assume that up to S prey can be in the refuge and both vulnerable and invulnerable prey reproduce at the same positive rate r. This means that the refuge is always fully occupied and animals born in the refuge must disperse to the open patch. Population dynamics in the open patch are described by the Lotka-Volterra model dR

with equilibrium densities eA A m

The recruitment of prey to the open patch makes the per capita prey growth rate r(1 + S/R) in the open patch negatively density dependent (i.e., it decreases with increasing prey abundance R) similarly as in the case of the logistic growth. The stability conditions [5] and [6] hold and the above equilibrium is locally asymptotically stable, which supports the general conclusion from other theoretical studies that refugia that protect a constant number of prey have a strong stabilizing effect on prey-predator population dynamics.

This example nicely illustrates stabilizing mechanism of asynchronous oscillations in population densities. In this example, the abundance of prey in the refuge is constant while the abundance of vulnerable prey varies due to demographic changes and recruitment of prey from the refuge. This asynchrony, then leads to the negative density-dependent recruitment rate of prey to the vulnerable patch. While this mechanism is clear in this simple example, it is much less obvious in many models that consider space explicitly.

Now let us consider a more complex prey-predator model in a heterogeneous environment consisting of N patches. The simplest possible case assumes that all patches are identical and animal dispersal is unconditional (random). The question is whether animal dispersal can stabilize population dynamics that are unstable without dispersal. Because unconditional animal dispersal tends to equalize prey and predator abundance across patches, animal dispersal tends to synchronize animal population dynamics and the answer to the above question is negative. However, patch-dependent dispersal rates and/or differences in local population dynamics can lead to asynchrony in local population dynamics, thus to negative density-dependent recruitment rates that can stabilize prey-predator population dynamics on a global scale exactly as in the example with the refuge. Figure 9 shows the stabilizing effect of prey dispersal in a

(a) No dispersal Prey disperse

Time Time

"S 125 at

75 50

75 50

40 Time

40 Time

40 Time

40 Time

Figure 9 Thisfigureshowsthe stabilizing effect of dispersal. The left panel shows prey (solid line) and predator (dashed line) dynamics in patch 1 (top panel) and patch 2 (bottom panel) without any dispersal (e1 = e2 = 0). These dynamics assume the Holling type II functional response which excludes prey and predator coexistence in either patch. The right panel shows the same system where prey disperse between patches (e1 = e2 = 1). Parameters: r1 = 1, r2 = 0.2, A1 = A2 = 0.1, e1 = e2 = 0.2, h1 = h2 = 0.02, m1 = m2 = 1.

two-patch environment. Population dynamics are described by the following model:

dt dCi

1 1 + hiAiRi eiAiRi

AjRj

Ci dC2

i + hjAjRj ejAjRj

■Cj i + hjAjRj where " (i = i, 2) describes prey dispersal between patches. Without dispersal, the local prey-predator population dynamics are unstable due to the Holling type II functional response (Figure 9, left panel). Prey dispersal (Figure 9, right panel) can stabilize population dynamics at an equilibrium. Similarly, dispersal of predators (or both prey and predators) can (but does not necessarily) stabilize population densities. This mechanism is in the roots of deterministic metapopulation dynamics where populations can coexist on the global spatial scale despite local extinctions. The necessary conditions for such global stability are differences in patch or migration dynamics and dispersal rates that are not too high to synchronize local patch dynamics.

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