Abstract:

The spreading speeds of partially degenerate reaction-diffusion systems with advection term and time-space periodic coefficients in multi-dimensional space has been studied in the current paper. In the direction of $\mathbf{e}\in S^{N-1}$, we use the the spreading properties of solution with front-like initial values to show that there is a finite spreading speed interval of such time-space periodic system in any direction and the interval admits a single spreading speed under certain special conditions. In the direction of $\mathbf{\eta}$, we introduce the concept of asymptotic spreading ray speed interval, and under the compact supported initial values, we show that such time-space periodic system exists an asymptotic spreading ray speed and an asymptotic spreading set. The results show that the Freidlin-G$\ddot{\rm a}$rtner's formula can be used to describe the asymptotic spreading ray speed for such partially degenerate systems. We also apply these results to some partially degenerate models in multi-dimensional time and space periodic media including the benthic-pelagic model, a dengue transmission model and man-environment-man epidemics model.

Key words: time-space periodic, partially degenerate system, spreading speed, multi-dimension space