An analytic and numerical study of a population density- dependent migration of a predator-prey system

Abstract

We formulate a population-density-dependent migration of a predator and prey between two patches. The migration is influenced by prey and predator densities such that when the predator population is huge, the migration of prey population is more likely to occur. We also assume that migration is dependent on factors other than predation. Each patch display a limit cycle behaviour of the predator and prey interaction. Migration is on a fast timescale as compared to the inter-species interaction within a patch, thus we reduce the system from four ordinary differential equation to two by aggregation. Furthermore, we non- dimensionalize the aggregated model to reduce the number of parameters for ease of analysis. The analysis of the aggregated model shows three fixed points. Two are unstable, the third where there is co-existence of both species, its instability or stability is parameterdependent. For some certain parameter values, we get a limit cycle. This limit cycle is a synchronous state, that is observable in both patches. Key words: Aggregation; Non-dimensionalization; Limit cycle; Synchronization Mathematics Subject classification(2020): 34D25, 34D20, 37N