The interfaces between pifion-juniper canopies and grasslands in the southwestern USA present an opportunity to use the modern theory of spatial phase transitions as a formal characterization of ecotone structure. The theory requires an estimation of a critical value of tree cover at which the woodlands switch abruptly from a fragmented collection of small patches of trees to a network of connected canopies. Presumably, this transition is associated with critical environmental conditions that regulate the ecologies of trees vs. grasses. We developed and tested a new method to estimate the critical cover value of woodlands on complex terrain. The method was based on multiscale assessments of the associations between local tree coverage and two types of patch edge. Tests on artificial gradient percolation maps revealed an interaction between the type of edge used (''hull edge,'' which is based on only the orthogonal connections between canopy-occupied cells, vs. `'accessible edge,'' which is based on both orthogonal and diagonal connections beween canopy-occupied cells) and the neighborhood rule used to define a cluster (von Neumann 5-cell or Moore 9-cell). When applied to digitized, geographically referenced aerial photographs, the method indicated that areas less than or equal to 6.6 ha exhibited the theoretical critical value of 59.3% cover predicted for square lattices and the 5-cell neighborhood. Construction of both edge types on a given map can reveal locations of steep environmental gradients that may be buffered against modest climate fluctuations. The edges can be used in the calibration of independent variables to predict tree cover. The agreement between the expected and observed critical densities will motivate extensions of phase transition theory to studies of ecotones in real landscapes.
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.