| Abstract: |
| In this talk we will analyze the existence, uniqueness and multiplicity of coexistence states for the generalized spatially heterogeneous predator-prey model
\begin{equation}
\label{1}
\left\{
\begin{array}{lll}
\mathfrak{L}_1 u=\lambda u - a(x)u^2 -b(x)\dfrac{uv}{1+\gamma m(x)u} &\quad \hbox{in}\;\;\Omega,\[10pt]
\mathfrak{L}_2 v=\mu v + c(x)\dfrac{uv}{1+\gamma m(x)u} - d(x)v^2 &\quad \hbox{in}\;\;\Omega,\[10pt]
\mathfrak{B}_1 u=\mathfrak{B}_2 v=0 &\quad\hbox{on}\;\;\partial\Omega,
\end{array}
\right.
\end{equation}
where $\mathfrak{L}_1$ and $\mathfrak{L}_2$ are second order uniformly elliptic operators, and $\mathfrak{B}_1$ and $\mathfrak{B}_2$ are general boundary operators of mixed type. In \eqref{1},
\[
a>0,\; d>0,\; b \gneq 0,\;
c \gneq 0,\;\gamma>0\;\;\hbox{and}\;\; m\geq0\quad\hbox{in}\;\,\bar{\Omega},
\]
while $\lambda,\mu\in\mathbb{R}$ are regarded as bifurcation parameters. The term $m(x)$ measures the level of saturation of the predator at any particular location $x\in\Omega$ where $m(x)>0$, while saturation effects do not play any role if $m(x)=0$. Thus, \eqref{1} is an homotopy in $m$ which combines, within the same territory, the classical interactions of Lotka--Volterra type in the region $m^{-1}(0)$ with the Holling-Tanner functional responses in $\{x\in\Omega: m(x)>0\}$.
\par
During the talk, they will be ascertained the regions in the plane $(\lambda,\mu)$ in which coexistence states exist or could exist and the conditions for which model \eqref{1} in its one-dimensional counterpart has uniqueness of coexistence states. Then, after a comprehensive analysis of a shadow system appearing when $\gamma\uparrow+\infty$, it will be provided a generic multiplicity result ensuring the existence of, at least, two coexistence states of \eqref{1} for $\gamma$ large enough. |
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