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 theory of chemical kinetics in which reactants are assumed to be homogeneously mixed. In order to explore the effect of spatial heterogeneity on predator-prey dynamics, we constructed a lattice-based reaction-diffusion model corresponding to the Lotka-Volterra equations. Spatial heterogeneity 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 wave patterns. In general, the estimated reaction order decreased with increasing degrees of spatial heterogeneity. For simulations using percolation maps with p-values varying between 1.0 (all cells available) to 0.5 (50% available), order estimates varied from 1.27 to 0.47. Gradient percolation maps and fractional Brownian surfaces also resulted in anomalously low reaction orders. Increasing diffusion distances resulted in reaction order estimates approaching the expected value of 2. Analysis of the qualitative dynamics of the model showed little difference between simulations where individuals diffused locally and those where individuals moved to random locations, suggesting that global density dependence is an important determinant of the overall model dynamics. However, localized interactions did introduce time dependence in the system attractor owing to emergent spatial patterns. We conclude that individual-based spatially explicit models are important tools for modeling population dynamics as they allow one to incorporate fine-scale ecological data about localized interactions and then to observe emergent patterns through simulation. When heterogeneous patterns arise, it can lead to anomalies with respect to the predictions of traditional mathematical approaches using global state variables.