In this paper, the authors study the reaction-diffusion equation of Keener-Tyson type in the whole space $\Bbb{R}^{n}$: $$ \left\{ \alignedat2 &\partial _{t}u=\Delta u+\frac{1}{\varepsilon }u(1-u)-hv\frac{u-q}{u+q}\qquad &\text{ in }\Bbb{R}^{n}\times \left( 0,\infty \right),\\ &\partial _{t}v=d\Delta v-v+u\ &\text{in }\Bbb{R}^{n}\times \left( 0,\infty \right),\\ &u\mid _{t=0}=u_{0},\ v\mid _{t=0}=v_{0}&\text{ in }\Bbb{R}^{n}. \endalignedat \right. $$ They prove the existence and properties of time-global unique non-negative classical solutions in the $L^{\infty }$ setting. \par For $n\in \Bbb{N}$, $\varepsilon ,h,d>0$ and $q\in ( 0,1) $ let $\overline{u}\in ( q,1) $ be a root of the function $ g(u)=u(1-u)(u+q)-\varepsilon hq(u-q)$. Denote $S=( q,\overline{u} ) ^{2}$. For $u_{0},v_{0}\in BUC(\Bbb{R}^{n})$, where $BUC(\Bbb{R }^{n})$ is the space of bounded uniformly continuous functions, the main result of the paper states that: \roster \item"$\bullet$" If $u_{0},v_{0}\geq 0$, then there exists a unique classical positive globally defined solution $( u,v) $ in $ C([0,\infty );BUC(\Bbb{R}^{n}))$. \item"$\bullet$" If $(u_{0}( x) ,\ v_{0}( x) )\in S$ for all $ x\in \Bbb{R}^{n}$, then $( u( x,t) ,v( x,t) ) \in S$ for all $x\in \Bbb{R}^{n},\ t\geq 0$. \item"$\bullet$" If $u(x,t_{\ast })\geq c_{\ast },\ v(x,t_{\ast })\geq c_{\ast }$ for all $x\in \Bbb{R}^{n}$ and some $t_{\ast }\geq 0,\ c_{\ast }>0$, then there exists $T_{1}\geq t_{\ast }$ such that $( u( x,t) ,v( x,t) ) \in S$ for all $x\in \Bbb{R}^{n},\ t\geq T_{1}$. \endroster