  The global boundedness and asymptotic behavior are investigated for the
solutions of a nonlocal time fractional reaction-diffusion equation (NTFRDE)
  $$ \frac{\partial^{\alpha }u}{\partial t^{\alpha }}=\Delta u+\mu
u^{2}(1-kJ*u)-\gamma u, \qquad(x,t)\in\mathbb{R}^{N}\times(0,+\infty)$$ with
$0<\alpha <1,\beta, \mu ,k>0,N\leq 2$ and $u(x,0)=u_{0}(x)$. Under appropriate
assumptions on $J$ and the property of time fractional derivative, it is proved
that for any nonnegative and bounded initial conditions, the problem has a
global bounded classical solution if $k^{*}=0$ for $N=1$ or $k^{*}=(\mu
C^{2}_{GN}+1)\eta^{-1}$ for $N=2$, where $C_{GN}$ is the constant in
Gagliardo-Nirenberg inequality. With further assumptions on the initial datum,
for small $\mu$ values, the solution is shown to converge to $0$ exponentially
or locally uniformly as $t \rightarrow \infty$, which is referred as the Allee
effect in sense of Caputo derivative. Moreover, under the condition of $J
\equiv 1$, it is proved that the nonlinear NTFRDE has a global bounded solution
in any dimensional space with the nonlinear diffusion terms $\Delta u^{m}\,
(2-\frac{2}{N}< m\leq 3)$.
