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Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey. (English) Zbl 1228.35037
The dynamics of a predator-prey reaction-diffusion system with the functional response of Holling type II and Allee effect in the prey population is investigated. The global existence of its solutions and in various situations their global asymptotic behavior is determined. It is shown that a large amount of predators initially always drive both population into extinction, which is characteristic for predator-prey systems with Allee effect. Energy estimates are used for obtaining a priori bounds for the dynamic and steady state solutions. On this base nonexistence of nonconstant positive steady state solutions is shown for identifying the ranges of parameters of spatial pattern formation. With the aid of global bifurcation theory developed in [J. Shi and X. Wang, J. Differ. Equations 246, No. 7, 2788–2812 (2009; Zbl 1165.35358)], the existence of nonconstant steady state solutions with certain eigenmodes is obtained. Also following to [F. Yi, J. Wei and J. Shi, J. Differ. Equations 246, No. 5, 1944–1977 (2009; Zbl 1203.35030)] the existence of spatially nonhomogeneous time-periodic orbits is proved together with their bifurcation analysis.

35B32 Bifurcations in context of PDEs
35K57 Reaction-diffusion equations
35B36 Pattern formations in context of PDEs
92D25 Population dynamics (general)
35B10 Periodic solutions to PDEs
Full Text: DOI
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