In this paper, we consider the system \arraycolsep0.14em\begin{array}{rcl {\hskip2em}rcl {\hskip2em}c}u_t&=&\Delta u+v^p,&v_t&=&\Delta v&x\in{\Bbb R}_{+}^N,t>0,\\ \displaystyle-{\partial u\over\partial x_t}&=&0,&\displaystyle-{\partial v\over\partial x_t}&=&u^q&x_1=0,t>0,\\ u(x,0)&=&u_0(x),&v(x,0)&=&v_0(x)&x\in{\Bbb R}_{+}^N, whereR" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: normal; font-size: 14.4px; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">RN+={(x1, x′)|x′∈R" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: normal; font-size: 14.4px; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">RN−1, x1>0}, p, q>0, andu0, v0are nonnegative and bounded. We prove that ifpq≤1 every nonnegative solution is global. Whenpq>1 we let α=(p+2)/2(pq−1), β=(2q+1)/2(pq−1). We show that if max(α, β)>N/2 or max(α, β)=N/2 andp, q≥1, then all nontrivial nonnegative solutions are nonglobal; whereas if max(α, β)<N/2 there exist both global and nonglobal nonnegative solutions.