  The current paper is devoted to the study 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\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
$(\mathcal{A}u)(t,x)=\int_{\RR}\kappa(y-x)u(t,y)dy-u(t,x)$ ($\kappa(\cdot)$ 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. We, in particular, 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.
