This paper addresses the question of how heterogeneity may evolve due to interactions between the dynamics and movement of three–species systems involving hosts, parasites and hyperparasites in homogeneous environments. The models are motivated by the spread of soil–borne parasites within plant populations, where the hyperparasite is used as a biological control agent but where patchiness in the distribution of the parasite occurs, even when environmental conditions are apparently homogeneous. However, the models are introduced in generic form as three–species reaction–diffusion systems so that they have broad applicability to a range of ecological systems. We establish necessary criteria for the occurrence of population–driven patterning via diffusion–driven instability. Sufficient conditions are obtained for restricted cases with no host movement. The criteria are similar to those for the well–documented two–species reaction–diffusion system, although more possibilities arise for spatial patterning with three species. In particular, temporally varying patterns, that may be responsible for the apparent drifting of hot–spots of disease and periodic occurrence of disease at a given location, are possible when three species interact. We propose that the criteria can be used to screen population interactions, to distinguish those that cannot cause patterning from those that may give rise to population–driven patterning. This establishes a basic dynamical ‘landscape’ against which other perturbations, including environmentally driven variations, can be analysed and distinguished from population–driven patterns. By applying the theory to a specific model example for host–parasite–hyperparasite interactions both with and without host movement, we show directly how the evolution of spatial pattern is related to biologically meaningful parameters. In particular, we demonstrate that when there is strong density dependence limiting host growth, the pattern is stable over time, whereas with less stable underlying host growth, the pattern varies with time.
A Nonlocal Eigenvalue Problem and the Stability of Spikes for Reaction–Diffusion Systems with Fractional Reaction Rates
