Effects of localized spatial variations on the uniform persistence and spreading speeds of time periodic two species competition systems

This paper is concerned with the asymptotic dynamics of two species competition systems of the form

$ \begin{equation*} \begin{cases} u_t(t,x) = \mathcal{A} u+u(a_1(t,x)-b_1(t,x)u-c_1(t,x)v),\quad x\in {\mathbb{R}} \cr v_t(t,x) = \mathcal{A} v+ v(a_2(t,x)-b_2(t,x)u-c_2(t,x) v),\quad x\in {\mathbb{R}} \end{cases} \end{equation*} $

where $ (\mathcal{A}u)(t,x) = u_{xx}(t,x) $, or $ (\mathcal{A}u)(t,x) = \int_{ {\mathbb{R}} }\kappa(y-x)u(t,y)dy-u(t,x) $ ($ \kappa(\cdot) $ is a smooth non-negative convolution kernel supported on an interval centered at the origin), $ a_i(t+T,x) = a_i(t,x) $, $ b_i(t+T,x) = b_i(t,x) $, $ c_i(t+T,x) = c_i(t,x) $, and $ a_i $, $ b_i $, and $ c_i $ ($ i = 1,2 $) are spatially homogeneous when $ |x|\gg 1 $, that is, $ a_i(t,x) = a_i^0(t) $, $ b_i(t,x) = b_i^0(t) $, $ c_i(t,x) = c_i^0(t) $ for some $ a_i^0(t) $, $ b_i^0(t) $, $ c_i^0(t) $, and $ |x|\gg 1 $. Such a system can be viewed as a time periodic competition system subject to certain localized spatial variations. In particular, we study the effects of localized spatial variations on the uniform persistence and spreading speeds of the system. Among others, it is proved that any localized spatial variation does not affect the uniform persistence of the system, does not slow down the spreading speeds of the system, and under some linear determinant condition, does not speed up the spreading speeds. We also study a relevant problem, that is, the continuity of the spreading speeds of time periodic two species competition systems with respect to time periodic perturbations, and prove that the spread speeds of such systems are lower semicontinuous with respect to time periodic perturbations.