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In the current paper, we consider the following parabolic-elliptic semilinearKeller-Segel model on $\mathbb{R}^{N}$, \begin{equation*} \begin{cases}u_{t}=\nabla\cdot (\nabla u-\chi u\nabla v)+a u -b u^2, \quadx\in\mathbb{R}^N,\,\, t>0\cr 0=(\Delta- I)v+ u, \quad x\in\mathbb{R}^N,\,\,t>0, \end{cases} \end{equation*} where $ \chi >0, \ a\ge 0,\ b> 0$ are constantreal numbers and $N$ is a positive integer. We first prove the local existenceand uniqueness of classical solutions $(u(x,t;u_0),v(x,t;u_0))$ with$u(x,0;u_0)=u_0(x)$ for various initial functions $u_0(x)$. Next, under someconditions on the constants $a, b, \chi$ and the dimension $N$, we prove theglobal existence and boundedness of classical solution$(u(x,t;u_0),v(x,t;u_0))$ for given initial functions $u_0(x)$. Finally, weinvestigate the asymptotic behavior of the global solutions with strictlypositive initial functions or nonnegative compactly supported initialfunctions. Under some conditions on the constants $a, b, \chi$ and thedimension $N$, we show that for every strictly positive initial function$u_0(\cdot)$, $$\lim_{t\to\infty} \sup_{x\in\mathbb{R}^N}\big[|u(x,t;u_0)-\frac{a}{b}|+|v(x,t;u_0)-\frac{a}{b}|\big]=0,$$ and that forevery nonnegative initial function $u_0(\cdot)$ with non-empty and compactsupport ${\rm supp}(u_0)$, there are $0<c_{\rm low}^*(u_0)\leq c_{\rmup}^*(u_0)<\infty$ such that $$\lim_{t\to\infty} \sup_{|x|\leq ct}\big[|u(x,t;u_0)-\frac{a}{b}|+|v(x,t;u_0)-\frac{a}{b}|\big]=0\quad \forall\,\,0<c<c_{\rm low}^*(u_0)$$ and $$\lim_{t\to\infty}\sup_{|x|\geq ct}\big[u(x,t;u_0)+v(x,t;u_0)\big]=0\quad \forall\,\, c>c_{\rm up}^*(u_0).$$

Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^{N}$