Persistence, Nonlocal Competition and Evolution of Movement: The Role of Principal Eigenvalues

How does the movement of individuals influence the persistence of a single species and the competition of multiple populations? Studies of such questions often involve the principal eigenvalues of the associated linear differential operators. We explore the significant roles of the principal eigenvalue by investigating two types of mathematical models for arbitrary but finite number of competing populations in spatially heterogeneous and temporally periodic environment. The interaction terms in these models are assumed to depend on the population sizes of all species in the whole habitat, representing some kind of nonlocal competition. For both models, the single species can persist if and only if the principal eigenvalue for the linearized operator is of negative sign, suggesting that the best strategy for the single species to invade when rare is to minimize the associated principal eigenvalue. For multiple populations, the global dynamics can also be completely characterized by the associated principal eigenvalues. Specifically, our results reveal that the species with the smallest principal eigenvalue among all competing populations, will gain a competitive advantage and competitively exclude other populations. This suggests that the movement strategies minimizing the corresponding principal eigenvalue are evolutionarily stable, echoing the persistence criteria for the single species.