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The current paper is devoted to the study of two species competition systemsof 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\RR\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\RR \end{cases} \end{equation*}where $\mathcal{A}u=u_{xx}$, or$(\mathcal{A}u)(t,x)=\int_{\RR}\kappa(y-x)u(t,y)dy-u(t,x)$ ($\kappa(\cdot)$ isa smooth non-negative convolution kernel supported on an interval centered atthe 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 spatiallyhomogeneous 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 $|x|\gg1$. Such a system can be viewed as a time periodic competition system subjectto certain localized spatial variations. We, in particular, study the effectsof localized spatial variations on the uniform persistence and spreading speedsof the system. Among others, it is proved that any localized spatial variationdoes not affect the uniform persistence of the system, does not slow down thespreading speeds of the system, and under some linear determinant condition,does not speed up the spreading speeds.

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