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Local-nonlocal interaction and spatial-temporal patterns in single species population over a patchy environment. (English) Zbl 0859.34056

A system of functional differential equations is proposed to describe the dynamics of a single-species population distributed over a patchy environment. Of major concern is the impact of the interaction between local aggregation and global delayed competition on the dynamics and the spatial-temporal patterns of the considered system. It is shown that the spatially heterogeneous steady state solutions can bifurcate from a spatially homogeneous steady state solution if the dispersion rate is large. Moreover, Hopf bifurcation of periodic solutions including phase-locked oscillations and synchronous oscillations can occur when the time delay in the global intraspecies competition reaches a critical value. Examples are provided to exhibit the complexity of the dynamics and the co-existence of phase-locked oscillations and heterogeneous steady state solutions.
Reviewer: M.Lizana (Merida)

MSC:

34K13 Periodic solutions to functional-differential equations
34K18 Bifurcation theory of functional-differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations