Spatial effects in an individual-based model of the interaction of species at different trophic levels

A spatial individual-based model of producerconsumer interaction is described. The model includes a two-dimensional grid of cells in which a producer organism grows according to a logistic equation, and individual herbivores which move towards cells high in producer biomass to consume the producer. Herbivores die of starvation if their energy reserve is too low or they may reproduce asexually if they have eaten sufficiently. The individuals in the model are the herbivores and the matrix cells— the producers are represented only as an attribute of the individual cells. The model was built as simply as possible with parsimonious producer-consumer interaction and consumer behavior. In this paper we consider spatial effects in the model. We test the effect of three independent variables: a) the distance over which the herbivores may move per time step, b) a torus matrix compared with an edged matrix and c) the size of the space matrix. We also comment on the inclusion of space itself into the model algorithm. The effect of the three independent variables on population size is tested by using a 3-factor ANOVA for herbivore and producer populations. The model is executed ten times for each of 18 combinations of levels of the three factors. We find a significant Distance × Edge × Matrix interaction for the herbivores. The variation in population among the three matrix sizes at each move distance differed depending on whether an edge was present or not. For the producers the ANOVA showed a significant Distance × Matrix interaction which arose because the variation in population among the three matrix sizes was not consistent across the three move distances. The effects of two factors were clearly evident in the graphical display of the model, but not highly significant in the statistical procedure. The effect of an edge is evident when the populations are graphed over time or when the graphical display of the software is observed during a simulation. The herbivore population changes over time with an edge present are not as smooth as without an edge. Watching the graphical display reveals a cycle of near population extinctions as a majority of herbivores move towards high populations of producers only to be trapped by the edge and die of starvation. Observation of the graphical output reveals that such large falls in population in a small number of time steps do not occur to the same extent when an edge is not present. The herbivore maximum move distance also has an effect on the model dynamics which is not evident in the statistical analysis. An increasing move distance results in greater fluctuations in the herbivore population and an increased likelihood of extinction. Observing the model graphical output as the move distance increases, the herbivores move to areas of greater producer biomass more quickly, resulting in greater fluctuations in both populations. If the move distance is too great, the herbivore population is able to consume too much of the producer population in too few time steps and becomes extinct. This danger of extinction during steady state dynamics increases as the matrix size decreases. We also find metapopulation behavior emerging from the interactions between the individual herbivores and cells, where patches and transient barriers emerge and disappear, with local extinctions and recruitment occurring. This behavior was not explicitly coded into the model. The contribution of this paper to the modeling of ecological systems is threefold. First, we demonstrate that the details of distance moved, the existence of an edge and the size of the space involved do affect model dynamics and we argue that these details should be included in model descriptions in the literature. Second, we demonstrate that a simple algorithm of producer-consumer interaction can exhibit behavior characteristic of metapopulation theory. Third, we argue that graphical output of a spatial model can reveal details of behavior which are not evident in the population means.