The effect of asymmetry in the contact rate is shown in Figs. 3(a) and 3(b) for two migration rates, namely ν = 0.02 (black) and ν = 0.05 (gray). The value of the critical parameter ( Fig. 3(a) ) or φ (Fig. 3(b) ) at which the Hopf bifurcation occurs is plotted against the varying contact rate. The symmetric case is at the bottom of the curve. We see that two identical patches have the same bifurcation point as a single, well-mixed population even when migration is present (cf. Fig. 2a) . However, a striking difference in the dynamics appears when we make the two patches weakly asymmetric. The values of φ and at which the Hopf bifurcation occurs are dramatically different from the symmetric case. The steady state stability persists well into the parameter regime where a single patch would display chaotic oscillations (Billings et al. 2007; Bianco et al. 2009 ). The location of the single patch Hopf point does not depend strongly on β, and decreasing the contact rate is actually destabilizing (Fig. 2a) , so the stabilization observed here is clearly a result of the coupling between asymmetric systems. Slightly increasing the migration rate (from ν = 0.02 to ν = 0.05 in Fig. 3 ) further increases the stability of the system. Fig. 4 Critical values of the parameters (a) and (b) φ at which the Hopf bifurcation occurs for coupled patches, as a function of the birth/death rate in patch 1, for ν = 0.02. The birth/death rate in patch 2 is fixed at μ 2 = 0.02. In (a), φ = 1 (no ADE). In (b), = 0 (no cross immunity). β 1 = β 2 = 200 and other parameters are as in Table 1 Similar results are obtained if the asymmetry occurs in the other social parameter, the birth rate, as depicted in Fig. 4 . Again, for asymmetric patches either stronger cross immunity or stronger ADE is needed to destabilize the steady state through a Hopf bifurcation.