SPATIAL HETEROGENEITY AND ANOMALOUS KINETICS - EMERGENT PATTERNS IN DIFFUSION-LIMITED PREDATORY PREY INTERACTION

The Lotka-Volterra model of predator-prey interaction is based on the assumption of mass action, a concept borrowed from the traditional the ory of chemical kinetics in which reactants are assumed to be homogene ously mixed. In order to explore the effect of spatial heterogeneity o n predator-prey dynamics, we constructed a lattice-based reaction-diff usion model corresponding to the Lotka-Volterra equations. Spatial het erogeneity was imposed on the system using percolation maps, gradient percolation maps, and fractional Brownian surfaces. In all simulations where diffusion distances were short, anomalously low reaction orders and aggregated spatial patterns were observed, including traveling wa ve patterns. In general, the estimated reaction order decreased with i ncreasing degrees of spatial heterogeneity. For simulations using perc olation maps with p-values varying between 1.0 (all cells available) t o 0.5 (50% available), order estimates varied from 1.27 to 0.47. Gradi ent percolation maps and fractional Brownian surfaces also resulted in anomalously low reaction orders. Increasing diffusion distances resul ted in reaction order estimates approaching the expected value of 2. A nalysis of the qualitative dynamics of the model showed little differe nce between simulations where individuals diffused locally and those w here individuals moved to random locations, suggesting that global den sity dependence is an important determinant of the overall model dynam ics. However, localized interactions did introduce time dependence in the system attractor owing to emergent spatial patterns. We conclude t hat individual-based spatially explicit models are important tools for modeling population dynamics as they allow one to incorporate fine-sc ale ecological data about localized interactions and then to observe e mergent patterns through simulation. When heterogeneous patterns arise , it can lead to anomalies with respect to the predictions of traditio nal mathematical approaches using global state variables.