Singularly perturbed elliptic system modeling the competitive interactions for a single resource (Q2845071)

Let \(\Omega\) be a smooth bounded domain in \(\mathbb{R}^n\) and let \(\varepsilon, d_1,\dots,d_N, m_0,m_1,..,m_N\) be positive parameters. Moreover, let \(I\in C^{0,1}(\overline{\Omega})\) be a nonzero and nonnegative Lipschitz continuous function and let \(f_i:\mathbb{R}_+\times\overline{\Omega}\rightarrow \mathbb{R}_+\) \ be \(N\)-functions such that for each \(i=1,\dots,N\):NEWLINENEWLINE(i) \ \(f_i(\cdot,x)\) is increasing and \(f_i(0,x)=0\) for each \(x\in \overline{\Omega}\),NEWLINENEWLINE(ii) \ \(f_i(r,\cdot)\in C^1(\overline{\Omega})\) for each \(r\in \mathbb{R}_+\).NEWLINENEWLINE\noindent In this paper, the existence of positive solutions for the following elliptic systemNEWLINENEWLINE\(\varepsilon \Delta r(x)+I(x)-\sum_{i=1}^Nf_i(r(x),x)u_i(x)-m_0r(x)=0\) in \(\Omega\),NEWLINENEWLINE\(\varepsilon d_i\Delta u_i(x)+f_i(r(x),x)u_i(x)-m_iu_i(x)=0\), \ \(i=1,\dots,N\), in \(\Omega\),NEWLINENEWLINE\(\partial_\nu u_1(x)=\dots=\partial_\nu u_N(x)=\partial_\nu r(x)=0\) in \(\Omega\),NEWLINENEWLINE\noindent is investigated. This system models the spacial heterogeneous interactions of \(N\) different species competing for a single resource. In particular, \(u_1,\dots,u_N\) denote the densities of the competing species, \(r\) represents the density of the single resource, \(f_i(r,x)\) is the consumption rate at \(x\in\Omega\) of the \(i\)th-species, and \(I\) represents a supply of resource. Finally, for each \(i=1,\dots,N\), the parameter \(m_i\) denotes the natural death rate of the \(i\)th-species and the parameter \(m_0\) denotes the natural decay of \(I\).NEWLINENEWLINE\noindent For each \(i=1,..,N\) and \(x\in \overline{\Omega}\), let \(R_i(x)\) be the solution of the equation \(f_i(r_i(x),x)=m_i\) if \(f_i(r_i(x),x)>m_i\) and put \(R_i(x)=+\infty\) otherwise.NEWLINENEWLINE\noindent In the case \(N=1\) and under assumptions \((i)\) and \((ii)\), the authors prove the existence of a positive solution \((r^\varepsilon,u_1^\varepsilon)\in (C^2(\overline{\Omega}))^2\) for small diffusion approximations (i.e., for small values of \(\varepsilon\)) provided that \(R_1(x)<S(x):=\frac{I(x)}{m_0}\) at some point \(x\in \overline{\Omega}\). Moreover, the authors show that, as \(\varepsilon \rightarrow 0\), \((r^\varepsilon,u_1^\varepsilon)\rightarrow (\min(S(x),R_1(x)),\frac{m_0}{m_1}(S(x)-R_1(x))^+)\) uniformly on \(\overline{\Omega}\).NEWLINENEWLINE\noindent The above results are then generalized to the case \(N>1\) under the assumption that each species is the best competitor (expressed in terms of \(R_i(x)\)) at least at some point \(x\in \Omega\), and under some further structural assumptions on the functions \(f_i\). In particular, for each \(x\in \overline{\Omega}\) and \(j=1,\dots,N\), let \(R^*(x)\), \(\Theta_j\) and \(U_j^*(x)\) be defined by:NEWLINENEWLINE\(R^*(x)=\min (S(x),R_1(x),\dots,R_N(x))\),NEWLINENEWLINE\(\Theta_j=\{x\in \overline{\Omega}: R_j(x)<S(x), \;R_j(x)<R_i(x), \;\forall i\neq j\}\),NEWLINENEWLINE\(U_j^*(x)=\frac{m_0}{m_j}(S(x)-R^*(x))\) if \(x\in \Theta_j\), and \(U_j^*(x)=0\) otherwise.NEWLINENEWLINEThen, the authors prove that, for small diffusion approximations, the above system has a positive solution \((r^\varepsilon,u_1^\varepsilon,\dots,u_1^\varepsilon)\in (C^2(\overline{\Omega}))^{N+1}\) such that, as \(\varepsilon \rightarrow 0\),NEWLINENEWLINE\((r^\varepsilon,u_1^\varepsilon,\dots,u_1^\varepsilon)\rightarrow\) \((R^*(x),U_1^*(x),\dots,U_N^*(x)),\) uniformly on every compact subset of \(\overline{\Omega}\setminus \Gamma\), where \(\Gamma=\{x\in \overline{\Omega}: R_i(x)=R_j(x)<S(s)\), for some \(i\neq j \}\) is the interface set.NEWLINENEWLINE\noindent The proofs of the main results are based on the degree theory in positive cones of Banach spaces.