Dynamics of positive steady-state solutions of a nonlocal dispersal logistic model with nonlocal terms

In this paper, we investigate a class of nonlocal dispersal logistic equations with nonlocal terms \begin{document}$\left\{ {\begin{array}{*{20}{l}}{{u_t} = Du + {u^q}\left( {\lambda + a(x)\int_\Omega b (x){u^p}} \right),}&{{\rm{\quad\quad in\quad\quad}}\Omega \times (0, + \infty ),}\\{u(x,0) = {u_0}(x) \ge 0}&{{\rm{\quad\quad\quad\quad\quad\quad\quad in\quad\quad}}\Omega ,}\\{u = 0,}&{{\rm{ on\quad\quad}}{{\mathbb{R}}^N}\backslash \Omega \times (0, + \infty ),}\end{array}} \right.$\end{document}where $ \Omega\subset \mathbb{R}^N(N\geq1) $ is a bounded domain, $ \lambda\in \mathbb{R} $, $ 0<q\leq1 $, $ p>0 $, $ a,b\in C(\overline{\Omega}) $, $ b\geq0 $, $ b\neq0 $ and $ a $ verifies either $ a>0 $ or $ a<0 $. $ Du = \int_\Omega J(x-y)u(y,t){\rm{d}}y-u(x,t) $ represents the nonlocal dispersal operators, which is continuous and nonpositive. Under some suitable assumptions we establish the existence, uniqueness or multiplicity and stability of positive stationary solution with nonlocal reaction term by using sub-supersolution methods, Lerray-Schauder degree theory and Lyapunov-Schmidt reduction and so on.