Persistence, Nonlocal Competition and Evolution of Movement: The Role of Principal Eigenvalues (CSIAM Transactions on Life Sciences, 2025)
How does the movement of individuals influence the persistence of a single species and the competition of multiple populations? Studies of such questionsoften involve the principal eigenvalues of the associated linear differential operators.We explore the significant roles of the principal eigenvalue by investigating two typesof mathematical models for arbitrary but finite number of competing populations inspatially heterogeneous and temporally periodic environment. The interaction termsin these models are assumed to depend on the population sizes of all species in thewhole habitat, representing some kind of nonlocal competition. For both models, thesingle 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 toinvade when rare is to minimize the associated principal eigenvalue. For multiplepopulations, the global dynamics can also be completely characterized by the associated principal eigenvalues. Specifically, our results reveal that the species with thesmallest principal eigenvalue among all competing populations, will gain a competitive advantage and competitively exclude other populations. This suggests that themovement strategies minimizing the corresponding principal eigenvalue are evolutionarily stable, echoing the persistence criteria for the single species.