This paper is concerned with a predator-prey model in $N$-dimensional spaces ($N=1, 2, 3$), given bybegin{align*}left{begin{aligned}&frac{partial u}{partial t}=Delta u-chiablacdot(uabla v),&frac{partial v}{partial t}=Delta v+xiablacdot(vabla u),end{aligned}ight. end{align*}which describes random movement of both predator and prey species, as well as the spatial dynamics involving predators pursuing prey and prey attempting to evade predators. It is shown that any global strong solutions of the corresponding Cauchy problem converge to zero in the sense of $L^p$-norm for any $1<ple infty$, and also converge to the heat kernel with respect to $L^p$-norm for any $1le ple infty$. In particular, the decay rate thereof is optimal in the sense thatit is consistent with that of the heat equation in $mathbb R^N$ ($N=2, 3$).Undoubtedly, the global existence of solutions appears to be among the most challenging topic in the analysis of this model. Indeed even in the one-dimensional setting, only global weak solutions in a bounded domain have been successfully constructed by far. Nevertheless, to provide a comprehensive understanding of the main results, we append the conclusion on the global existence and asymptotic behavior of strong solutions, although certain smallness conditions on the initial data are required.