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Global dynamics of a di usive single species model with periodic delay

School of Mathematics and Statistics, Xidian University, Xi’an, Shaanxi 710071, P.R. China

The growth of the species population is greatly influenced by seasonally varying environments. By regarding the maturation age of the species as a periodic developmental process, we propose a time periodic and diffusive model in bounded domain. To analyze this model with periodic delay, we first define the basic reproduction ratio $\mathcal {R}_0$ of the spatially homogeneous model and then show that the species population will be extinct when $\mathcal {R}_0\leq 1$ while remains persistent and tends to periodic oscillation if $\mathcal {R}_0>1$. Finally, combining the comparison principle with the fact that solutions of the spatially homogeneous model are also solutions of our model subject to Neumann boundary condition, we establish the global dynamics of a threshold type for PDE model in terms of $\mathcal {R}_0$.
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