Abstract:The study of basic mechanisms and patterns underlying species spread in space are of central importance in the theory of biological invasions. Homogeneous environments frequently allow the establishment of classic reaction-diffusion models that show invasive species spreading at a linear rate and on continuous circular population fronts. However, there is growing evidence that in some cases invasion can take place via the formation, interaction and propagation of non-continuous patches of high species density that are separated by regions of nearly zero density. This type of spread is called ‘patchy spread’ or ‘patchy invasion’. It was shown by Petrovskii et al. that patchy spread can arise in deterministic reaction-diffusion models such as predator-prey systems with the Allee effect and multi-species systems. In this paper, a discrete probability cellular automata model (a discrete model widely used in mathematics, physics, complexity science and theoretical biology) is established to investigate the spreading patterns of a spatially implicit predator-prey system with the Allee effect in prey that exhibit patchy spread. We address the following questions: (1) Under what circumstances can patchy spread be observed? (2) Are there essential differences in spread speed under different spread patterns?
Extensive simulations were used to study parameter conditions under which patchy spread can be found, and the differences between different spreading regimes. We found that under patchy spread conditions, slight variations of parameters were sufficient to destroy this regime and either restore the propagation of continuous fronts or drive the species to extinction. Thus, patchy spread can be qualified as an invasion at the edge of extinction. In this paper, predator mortality was chosen as a control factor and all other parameters treated as fixed. When predator mortality was large, prey were not fully controlled by predators, continuous circular waves were formed, and prey densities were distributed homogeneously behind the wave front. A reduction in predator mortality resulted in a pulse of expanding circular fronts. Further decreases in predator mortality led to increasing break-ups of the inside of the spread wave and inhibited the formation of continuous circular waves, ultimately resulting in the complete break-up of continuous spread waves and the establishment of a patchy spread pattern. The transition from a continuous front propagation pattern to a patchy spread pattern was characterized by a remarkable drop in invasion speed.
Our results provide a theoretical basis for biological invasion control and exotic species monitoring and suggest the consideration of an exotic species management and control strategy based on the deliberate introduction of a specialist predator. While an exotic species is distributed in a patchy pattern at high densities, it poses the risk of an outbreak in the absence of natural enemies or environmental restrictions. The intensity of population management measures should be increased when an invasive species enters a patchy spread pattern until effective biological control with the aim of extermination can be put into practice.