| Abstract: |
| The effects of spatial heterogeneity on population dynamics have been studied extensively. However, the
effects of temporal periodicity on the dynamics of general periodic-parabolic reaction-diffusion systems
remain largely unexplored. As a first attempt to understand such effects, we analyze the asymptotic behavior
of the principal eigenvalue for linear cooperative periodic-parabolic systems with small diffusion rates.
As an application, we show that if a cooperative system of periodic ordinary differential equations has a
unique positive periodic solution which is globally asymptotically stable, then the corresponding reactiondiffusion system with either the Neumann or regular oblique derivative boundary condition also has a unique
positive periodic solution which is globally asymptotically stable, provided that the diffusion coefficients
are sufficiently small. The role of temporal periodicity, spatial heterogeneity and their combined effects
with diffusion will be studied in subsequent papers for further understanding and illustration. |
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