Ants in a labyrinth : diffusion-limited population dynamics in spatially complex environments
Empirical studies have shown that individual movement and ecological interactions can be affected by the spatial complexity of the environment. However, classical population models typically assume that individuals are well-mixed at small spatial scales. I claim that interactions between individuals in populations can be considered diffusion-limited, and therefore mixing at small scales is not always an appropriate assumption. I show that reproductive rates and predation rates are reduced in simple Euclidean environments when individuals have limited mobility. These reductions in interaction rates alter the predicted dynamics and equilibrium densities of a predator-prey system and a 2-species competitive system. In individual-based simulations, predator-prey dynamics are less likely to display sustained oscillations, while coexistence of competitors can be found when exclusion is predicted by a mean-field model. Interaction rates are further reduced in spatially complex environments because of the anomalously slow rates of diffusion associated with fractal spaces. In simulated predator-prey systems, population dynamics are generally more stable in environments with low fractal dimension. However, for some parameter values, predators will go extinct in spatially complex spaces, when stable coexistence is predicted by the mean-field model. In the competitive system, equilibrium densities are a function of the fractal dimension of the environment. In some cases, I find that an inferior competitor can coexist at higher densities than a superior competitor in a spatially complex environment, even though a mean-field model predicts extinction of the inferior species. I use the results from these individual-based models, and theory from models of diffusion-limited chemical reactions, to create spatially implicit descriptions of the effects of limited mobility and spatial dimension on populations. I show that pseudospatial models give the same predictions for total abundance as the individual-based simulations. I apply standard phase plane analysis to these simple models, and demonstrate that we can express population dynamics as a function of spatial dimension and individual mobility.