Ecological invasion: spatial clustering and the critical radius

Question: When localized dispersal spatially aggregates an introduced species, so that an introduction’s success or failure corresponds to growth or decay of invader clusters respectively, how does the probability of cluster expansion vary with propagation rates of locally interacting invaders and residents, with their mortality rates, and with dispersal distance? Mathematical method: We apply the physical theory for nucleation in homogeneous spatial systems to invader–resident competition, and focus on an invader cluster’s critical radius. At the critical radius, growth and decay of an invader cluster have equal probability. Using this definition, analytically and computationally, we explore the effects of individual-level vital rates on the probability of invader-cluster growth. Key assumptions: The two species compete preemptively for space. Invader and resident have the same mortality rate, but the invader has the greater rate of local propagation. Invader clusters are (on average) circular, and advance or decline of the invader occurs at the cluster’s perimeter. Conclusions: The probability that an invader cluster grows, so that introduction succeeds, increases with an error function of the logarithm of the cluster’s radius. Variation in lattice-neighbourhood size has little effect on cluster-growth probability in simulation. For small differences in propagation rates, increasing the common mortality rate increases the invader’s critical radius, but may also increase the probability that large clusters invade the resident. That is, increased mortality hampers small invader clusters through chance extinction, but provides more opportunities for growth at the periphery of large clusters. The (approximate) critical radius of invader clusters scales as a power law of the difference in the two species’ propagation rates.