The purpose of this paper is to give sufficient conditions for the existence and uniqueness of positive solutions to a rather general type of elliptic system of the Dirichlet problem on a bounded domain \(\Omega\) in \(R^{n}\). Also considered are the effects of perturbations on the coexistence state and uniqueness. The techniques used in this paper are super-sub solutions method, eigenvalues of operators, maximum principles, spectrum estimates, inverse function theory, and general elliptic theory. The arguments also rely on some detailed properties for the solution of logistic equations. These results yield an algebraically computable criterion for the positive coexistence of species of animals with predator-prey relation in many biological models.
One of the prominent subjects of study and analysis in mathematical biology concerns the survival of two or more species of animals in the same environment. Especially, pertinent areas of investigation include the conditions under which the species can coexist, as well as the conditions under which any one of the species becomes extinct, that is, one of the species is excluded by the others. In this paper, we focus on the general predator-prey model to better understand the competitive interactions between multiple species. Specifically, we investigate the conditions needed for the coexistence of species when the factors affecting them are fixed or perturbed.
In this paper, we focus on the existence and uniqueness of the positive steady state solution of the general predator-prey model for arbitrary \(N\) species, \[\begin{aligned} \begin{cases} (u_{i})_{t}(x,t) = \Delta u_{i}(x,t) + u_{i}(x,t)g_{i}(u_{1}(x,t),…,u_{i}(x,t),…,u_{N}(x,t))& \mbox{in}\;\;\Omega \times R^{+}, \\ u_{i}(x,t)|_{\partial\Omega} = 0,& i = 1,…,N, \end{cases} \end{aligned}\] or, equivalently, the positive solution to \[\begin{cases} \Delta u_{i}(x) + u_{i}(x)g_{i}(u_{1}(x),…,u_{i}(x),…,u_{N}(x)) = 0& \mbox{in}\;\;\Omega,\\ u_{i}|_{\partial\Omega} = 0,& i = 1,…,N, \end{cases}\label{eq:a} \tag{1}\] where \(\Omega\) is a bounded domain with smooth boundary \(\partial\Omega\), \(g_{i} \in C^{1}\) are relative growth rates such that \((g_{1})_{u_{j}} < 0, j = 1,…,N\) and for each \(i = 2,…,N\), \[\left.\begin{array}{lll} (g_{i})_{u_{1}} > 0,\\ (g_{i})_{u_{j}} < 0, j = 2,…,N. \end{array}\right.\]
Because of its broader applicability, the general predator-prey model has become a more popular subject of research within the mathematical community over the past few years.
The functions \(g_{i}\) describe how species \(u_{i}\) interact among themselves and with each other.
The followings are questions raised in the general model with nonlinear growth rates.
Problem 1. What are the sufficient conditions for existence of positive solutions?
Problem 2. What are the sufficient conditions for uniqueness of positive solutions?
Problem 3. What is the effect of perturbation for existence and uniqueness? In our analysis, we focus on the conditions required for the maintenance of the coexistence state of the model when interaction rates \((g_{1},…,g_{N})\) are slightly perturbed. Biologically, our conclusion implies that the species may slightly relax ecologically and yet continue to coexist at unique densities.
In §3, we establish sufficient conditions for the existence and non-existence of positive solution of the system. We also achieve solution estimates in §4 to prove the uniqueness and the invertibility of linearization in §5, §6 and §7, where we investigate the effect of perturbation for existence and uniqueness. In §8, we study an easy concrete example to illustrate abstract mathematical conditions and the practical relevance, and to discuss of how the mathematical results could inform ecological understanding and management.
An especially significant aspect of the global uniqueness result is the stability of the positive steady state solution, which has become an important subject of mathematical study. Indeed, researchers have obtained several stability results for the Lotka-Volterra model with constant rates (see [1], [2].) The research presented in this paper therefore begins the mathematical community’s discussion on the stability of the steady state solution for the general predator-prey model.
Before entering into our primary arguments and results, we must first present a few preliminary items that we later employ throughout the proofs detailed in this paper. The following definition and lemmas are established and accepted throughout the literature on our topic.
Definition 1. (Super and Sub solutions). The vector functions \((\bar{u}^{1},…,\bar{u}^{N}), (\underline{u}^{1},…,\underline{u}^{N})\) form a super/sub solution pair for the system \[\begin{aligned} \begin{cases} \Delta u^{i} + g^{i}(u^{1},…,u^{N}) = 0& \mbox{in}\;\;\Omega,\\ u^{i} = 0& \mbox{on}\;\;\partial\Omega, \end{cases} \end{aligned}\] if for \(i = 1,…,N\), \[\left\{\begin{array}{r} \left.\begin{array}{l} \Delta \bar{u}^{i} + g^{i}(u^{1},…,u^{i-1},\bar{u}^{i},u^{i+1},…,u^{N}) \leq 0,\\ \Delta \underline{u}^{i} + g^{i}(u^{1},…,u^{i-1},\underline{u}^{i},u^{i+1},…,u^{N}) \geq 0,\\ \end{array}\right.\mbox{in}\;\;\Omega\;\;\mbox{for}\;\;\underline{u}^{j} \leq u^{j} \leq \bar{u}^{j}, j \neq i, \end{array}\right.\] and \[\begin{cases} \underline{u}^{i} \leq \bar{u}^{i}&\mbox{on}\;\;\Omega,\\ \underline{u}^{i} \leq 0 \leq \bar{u}^{i}&\mbox{on}\;\;\partial\Omega. \end{cases}\]
Lemma 1. If \(g^{i}\) in the Definition 1 are in \(C^{1}\) and the system admits a super/sub solution pair \((\underline{u}^{1},…,\underline{u}^{N}), (\bar{u}^{1},…,\bar{u}^{N})\), respectively, then there is a solution \((u_{1},…,u_{N})\) to the system in Definition 1 with \(\underline{u}^{i} \leq u^{i} \leq \bar{u}^{i}\) in \(\bar{\Omega}\). If \[\left.\begin{array}{l} \Delta\bar{u}^{i} + g^{i}(\bar{u}^{1},…,\bar{u}^{N}) \neq 0,\\ \Delta\underline{u}^{i} + g^{i}(\underline{u}^{1},…,\underline{u}^{N}) \neq 0, \end{array}\right.\] in \(\Omega\) for \(i = 1,…,N\), then \(\underline{u}^{i} < u^{i} < \bar{u}^{i}\) in \(\Omega\).
Lemma 2. (The first eigenvalue). Consider \[\left\{ \begin{array}{l} -\Delta u + q(x)u = \lambda u\;\;\mbox{in}\;\;\Omega,\\ u|_{\partial \Omega} = 0, \end{array} \right. \label{eq:2} \tag{2}\] where \(q(x)\) is a smooth function from \(\Omega\) to \(R\) and \(\Omega\) is a bounded domain.
(A) The first eigenvalue \(\lambda_{1}(q)\) of (2), denoted by simply \(\lambda_{1}\) when \(q \equiv 0\), is simple with a positive eigenfunction \(\phi_{q}\).
(B) If \(q_{1}(x) < q_{2}(x)\) for all \(x \in \Omega\), then \(\lambda_{1}(q_{1}) < \lambda_{1}(q_{2})\).
(C) (Variational Characterization of the first eigenvalue)
\[\lambda_{1}(q) = \min_{\phi \in W_{0}^{1}(\Omega),\phi \neq 0}\frac{\int_{\Omega}(|\nabla \phi|^{2}+q\phi^{2})dx}{\int_{\Omega}\phi^{2}dx}.\]
In our proof, we also employ accepted conclusions concerning the solutions of the following logistic equations.
Lemma 3. Consider \[\begin{cases} \Delta u + uf(u) = 0& \mbox{in}\;\; \Omega,\\ u|_{\partial\Omega} = 0,& u > 0, \end{cases}\] where \(f\) is a decreasing \(C^{1}\) function such that there exists \(c_{0} > 0\) such that \(f(u) \leq 0\) for \(u \geq c_{0}\) and \(\Omega\) is a bounded domain.
(A) If \(f(0) > \lambda_{1}\), then the above equation has a unique positive solution. We denote this unique positive solution as \(\theta_{f}\).
(B) If \(f(0) \leq \lambda_{1}\), then \(u \equiv 0\) is the only nonnegative solution to the above equation.
The main property about this positive solution is that \(\theta_{f}\) is increasing as \(f\) is increasing.
Especially, for \(\alpha > \lambda_{1}\), we denote \(\theta_{\alpha}\) as the unique positive solution of \[\begin{cases} \Delta u + u(\alpha – u) = 0& \mbox{in}\;\; \Omega,\\ u|_{\partial\Omega} = 0,& u > 0. \end{cases}\]
Hence, \(\theta_{\alpha}\) is increasing as \(\alpha > 0\) is increasing.
Having established these preliminaries, we now commence our investigation of the general predator-prey model.
We establish the following existence result for (1):
Theorem 1. If for each \(i = 2,…,N\),
(A) \[\sup(g_{1})_{u_{1}}\sup(g_{i})_{u_{i}} + (N – 1)\inf(g_{1})_{u_{i}}\sup(g_{i})_{u_{1}} > 0,\] and
(B) \[g_{1}(0,…,0) > \dfrac{\sup(g_{1})_{u_{1}}\sup(g_{i})_{u_{i}}\left(\lambda_{1} + \frac{(N – 1)\inf(g_{1})_{u_{i}}g_{i}(0,…,0)}{\sup(g_{i})_{u_{i}}}\right)}{\sup(g_{1})_{u_{1}}\sup(g_{i})_{u_{i}} + (N – 1)\inf(g_{1})_{u_{i}}\sup(g_{i})_{u_{1}}},\] and \[g_{i}(0,…,0) > \lambda_{1} – (N – 2)\inf_{j=2,…,N, j\neq i}\{\bar{u_{j}}\inf(g_{i})_{u_{j}}\},\] where \(\bar{u_{j}}\) is defined below, then (1) has a positive solution.
Proof. Let \(\underline{u_{i}} = \gamma_{i}\omega,\; i = 1,…,N,\; \bar{u_{1}} = -\frac{g_{1}(0,…,0)}{\sup(g_{1})_{u_{1}}},\; \bar{u_{i}} = -\frac{1}{\sup(g_{i})_{u_{i}}}\left(g_{i}(0,…,0) – \frac{g_{1}(0,…,0)\sup(g_{i})_{u_{1}}}{\sup(g_{1})_{u_{1}}}\right),\; i = 2,…,N\), where \(\gamma_{i} > 0\) are constants and \(\omega\) is the eigenfunction of (2) with \(q(x) \equiv 0\) corresponding to the first eigenvalue \(\lambda_{1}\).
Then by the Mean Value Theorem, for all \(u_{i}\) such that \(\underline{u_{i}} \leq u_{i} \leq \bar{u_{i}}, i = 2,…,N\), \[\begin{aligned} \Delta\bar{u_{1}} + \bar{u_{1}}g_{1}(\bar{u_{1}},u_{2},…,u_{N}) = & \bar{u_{1}}[g_{1}(0,…,0) + g_{1}(\bar{u_{1}},u_{2},…,u_{N}) – g_{1}(0,u_{2},…,u_{N})\\ & + g_{1}(0,u_{2},…,u_{N}) – g_{1}(0,0,u_{3},…,u_{N})] + g_{1}(0,0,u_{3},…,u_{N}) – g_{1}(0,0,0,u_{4},…,u_{N})\\ & + … + g_{1}(0,…,0,u_{N}) – g_{1}(0,…,0)]\\ \leq & \bar{u_{1}}[g_{1}(0,…,0) + \bar{u_{1}}\sup(g_{1})_{u_{1}} + u_{2}\sup(g_{1})_{u_{2}} + … + u_{N}\sup(g_{1})_{u_{N}}]\\ = & \bar{u_{1}}[u_{2}\sup(g_{1})_{u_{2}} + …, + u_{N}\sup(g_{1})_{u_{N}} < 0, \end{aligned}\] and for all \(u_{1}\) such that \(\underline{u_{1}} \leq u_{1} \leq \bar{u_{1}}\) and for all \(i = 2,…,N\), \[\begin{aligned} \Delta\bar{u_{i}} + \bar{u_{i}}g_{i}(u_{1},…,u_{i-1},\bar{u_{i}},u_{i+1},…,u_{N}) = & \bar{u_{i}}[g_{i}(0,…,0) + g_{i}(u_{1},…,u_{i-1},\bar{u_{i}},u_{i+1},…,u_{N})\\ & – g_{i}(0,u_{2},…,u_{i-1},\bar{u_{i}},u_{i+1},…,u_{N})\\ & + g_{i}(0,u_{2},…,u_{i-1},\bar{u_{i}},u_{i+1},…,u_{N}) – g_{i}(0,0,u_{3},…,u_{i-1},\bar{u_{i}},u_{i+1},…,u_{N})\\ & + g_{i}(0,0,u_{3},…,u_{i-1},\bar{u_{i}},u_{i+1},…,u_{N})\\ & – g_{i}(0,0,0,u_{4},…,u_{i-1},\bar{u_{i}},u_{i+1},…,u_{N}) + … +\\ & + g_{i}(0,…,0,\bar{u_{i}},u_{i+1},…,u_{N}) – g_{i}(0,…,0,u_{i+1},…,u_{N})\\ & + g_{i}(0,…,0,u_{i+1},…,u_{N}) – g_{i}(0,…,0,u_{i+2},…,u_{N})\\ & + … + g_{i}(0,…,0,u_{N}) – g_{i}(0,…,0)]\\ \leq & \bar{u_{i}}[g_{i}(0,…,0) + u_{1}\sup(g_{i})_{u_{1}} + … u_{i-1}\sup(g_{i})_{u_{i-1}} + \bar{u_{i}}\sup(g_{i})_{u_{i}}\\ & + u_{i+1}\sup(g_{i})_{u_{i+1}} + … + u_{N}\sup(g_{i})_{u_{N}}]\\ \leq & \bar{u_{i}}[g_{i}(0,…,0) + \bar{u_{1}}\sup(g_{i})_{u_{1}} + \bar{u_{i}}\sup(g_{i})_{u_{i}}] = 0. \end{aligned}\]
By the condition and the Mean Value Theorem again, for all \(u_{i}\) such that \(\underline{u_{i}} \leq u_{i} \leq \bar{u_{i}}, i = 2,…,N\), \[\begin{aligned} \Delta\underline{u_{1}} + &\underline{u_{1}}g_{1}(\underline{u_{1}},u_{2},…,u_{N})\\ = & \Delta(\gamma_{1}\omega) + \gamma_{1}\omega g_{1}(\gamma_{1}\omega,u_{2},…,u_{N})\\ = & \Delta(\gamma_{1}\omega) + \gamma_{1}\omega[g_{1}(0,…,0) + g_{1}(\gamma_{1}\omega,u_{2},…,u_{N}) – g_{1}(0,u_{2},…,u_{N}) + g_{1}(0,u_{2},…,u_{N}) – g_{1}(0,0,u_{3},…,u_{N})\\ & + g_{1}(0,0,u_{3},…,u_{N}) – g_{1}(0,0,0,u_{4},…,u_{N}) + … + g_{1}(0,…,0,u_{N}) – g_{1}(0,…,0)]\\ \geq & -\gamma_{1}\lambda_{1}\omega + \gamma_{1}\omega\left[g_{1}(0,…,0) + \gamma_{1}\omega\inf(g_{1})_{u_{1}} + \sum\limits_{i=2}^{N}u_{i}\inf(g_{1})_{u_{i}}\right]\\ \geq & -\gamma_{1}\lambda_{1}\omega + \gamma_{1}\omega\left[g_{1}(0,…,0) + (N – 1)\inf_{i=2,…,N}\{\bar{u_{i}}\inf(g_{1})_{u_{i}}\} + \gamma_{1}\omega\inf(g_{1})_{u_{1}}\right]\\ = & -\gamma_{1}\lambda_{1}\omega + \gamma_{1}\omega\left[g_{1}(0,…,0) + (N – 1)\inf_{i=2,…,N}\left\{-\frac{1}{\sup(g_{i})_{u_{i}}}(g_{i}(0,…,0) \right.\right.\\ &\left.\left. – \frac{g_{1}(0,…,0)\sup(g_{i})_{u_{1}}}{\sup(g_{1})_{u_{1}}})\inf(g_{1})_{u_{i}}\right\} + \gamma_{1}\omega\inf(g_{1})_{u_{1}}\right]\\ = & -\gamma_{1}\lambda_{1}\omega + \gamma_{1}\omega\left[g_{1}(0,…,0) + (N – 1)\inf_{i=2,…,N}\left\{\inf(g_{1})_{u_{i}}\left(-\frac{g_{i}(0,…,0)}{\sup(g_{i})_{u_{i}}}\right.\right.\right.\\ &\left.\left.\left. + \frac{g_{1}(0,…,0)\sup(g_{i})_{u_{1}}}{\sup(g_{1})_{u_{1}}\sup(g_{i})_{u_{i}}}\right)\right\} + \gamma_{1}\omega\inf(g_{1})_{u_{1}}\right]\\ = & -\gamma_{1}\lambda_{1}\omega + \gamma_{1}\omega\left[\inf_{i=2,…,N}\left\{g_{1}(0,…,0) – \frac{(N – 1)\inf(g_{1})_{u_{i}}g_{i}(0,…,0)}{\sup(g_{i})_{u_{i}}}\right.\right.\\ &\left.\left. + \frac{(N – 1)g_{1}(0,…,0)\inf(g_{1})_{u_{i}}\sup(g_{i})_{u_{1}}}{\sup(g_{1})_{u_{1}}\sup(g_{i})_{u_{i}}}\right\} + \gamma_{1}\omega\inf(g_{1})_{u_{1}}\right]\\ = & -\gamma_{1}\lambda_{1}\omega + \gamma_{1}\omega\left[\inf_{i=2,…,N}\left\{g_{1}(0,…,0)\left(1 + \frac{(N – 1)\inf(g_{1})_{u_{i}}\sup(g_{i})_{u_{1}}}{\sup(g_{1})_{u_{1}}\sup(g_{i})_{u_{i}}}\right)\right.\right.\\ &\left.\left. – \frac{(N – 1)\inf(g_{1})_{u_{i}}g_{i}(0,…,0)}{\sup(g_{i})_{u_{i}}}\right\} + \gamma_{1}\omega\inf(g_{1})_{u_{1}}\right]\\ = & \gamma_{1}\omega\left[-\lambda_{1} + \inf_{i=2,…,N}\left\{g_{1}(0,…,0)\left(1 + \frac{(N – 1)\inf(g_{1})_{u_{i}}\sup(g_{i})_{u_{1}}}{\sup(g_{1})_{u_{1}}\sup(g_{i})_{u_{i}}}\right)\right.\right.\\ &\left.\left. – \frac{(N – 1)\inf(g_{1})_{u_{i}}g_{i}(0,…,0)}{\sup(g_{i})_{u_{i}}}\right\} + \gamma_{1}\omega\inf(g_{1})_{u_{1}}\right]\\ > & 0 \end{aligned}\] with small enough \(\gamma_{1} > 0\), and for all \(i = 2,…,N\), \(j = 1,…,N, j \neq i\) and \(u_{j}\) such that \(\underline{u_{j}} \leq u_{j} \leq \bar{u_{j}}\), \[\begin{aligned} \Delta\underline{u_{i}} + \underline{u_{i}}g_{i}(u_{1},…,u_{i-1},\underline{u_{i}},u_{i+1},…,u_{N}) = & -\gamma_{i}\lambda_{1}\omega + \gamma_{i}\omega g_{i}(u_{1},…,u_{i-1},\gamma_{i}\omega,u_{i+1},…,u_{N})\\ = & -\gamma_{i}\lambda_{1}\omega + \gamma_{i}\omega[g_{i}(0,…,0) + g_{i}(u_{1},…,u_{i-1},\gamma_{i}\omega,u_{i+1},…,u_{N})\\ & – g_{i}(0,u_{2},…,u_{i-1},\gamma_{i}\omega,u_{i+1},…,u_{N}) + g_{i}(0,u_{2},…,u_{i-1},\gamma_{i}\omega,u_{i+1},…,u_{N})\\ & – g_{i}(0,0,u_{3},…,u_{i-1},\gamma_{i}\omega,u_{i+1},…,u_{N})] + …+\\ & + g_{i}(0,…,0,\gamma_{i}\omega,u_{i+1},…,u_{N}) – g_{i}(0,…,0,u_{i+1},…,u_{N})\\ & + g_{i}(0,…,0,u_{i+1},…,u_{N}) – g_{i}(0,…,0,u_{i+2},…,u_{N})\\ & + … + g_{i}(0,…,0,u_{N}) – g_{i}(0,…,0)]\\ \geq & -\gamma_{i}\lambda_{1}\omega + \gamma_{i}\omega\left[g_{i}(0,…,0) + \sum\limits_{j=1,j\neq i}^{N}u_{j}\inf(g_{i})_{u_{j}} + \gamma_{i}\omega\inf(g_{i})_{u_{i}}\right]\\ \geq & \gamma_{i}\omega\left[g_{i}(0,…,0) – \lambda_{1} + \gamma_{i}\omega\inf(g_{i})_{u_{i}} + \sum\limits_{j=2,j\neq i}^{N}\bar{u_{j}}\inf(g_{i})_{u_{j}}\right] > 0 \end{aligned}\] with small enough \(\gamma_{i} > 0\). Furthermore, \[\underline{u_{i}} = \bar{u_{i}} = 0\;\;\mbox{on}\;\;\partial\Omega,\] and \[\underline{u_{i}} \leq \bar{u_{i}},\] with small enough \(\gamma_{i} > 0\). Hence, by the Lemma 1, there is a solution \((u_{1},…,u_{N})\) to (1) with \[\underline{u_{i}} \leq u_{i} \leq \bar{u_{i}}.\] ◻
We also establish the following nonexistence results.
Theorem 2. Suppose \(g_{i}(0,…,0) \leq \lambda_{1}, i = 1,…,N\). Then \(u_{i} \equiv 0, i = 1,…,N\) is the only nonnegative solution to (1).
Proof. Let \((u_{1},…,u_{N})\) be a nonnegative solution to (8). By the Mean Value Theorem, for \(i = 2,…,N\), there are \(\tilde{u_{i}}\) such that \[\left.\begin{array}{l} g_{1}(u_{1},0,…,0,u_{i},0,…,0) – g_{1}(u_{1},0,…,0) = (g_{1})_{u_{i}}(u_{1},0,…,0,\tilde{u_{i}},0,…,0)u_{i},\\ g_{i}(u_{1},0,…,0,u_{i},0,…,0) – g_{i}(0,…,0,u_{i},0,…,0) = (g_{i})_{u_{1}}(\tilde{u_{1}},0,…,0,u_{i},0,…,0)u_{1}. \end{array}\right.\]
Hence, by the monotonicity, for \(i = 2,…,N\), (1) implies that \[\begin{aligned} \Delta u_{1} + &u_{1}[g_{1}(u_{1},0,…,0) + (g_{1})_{u_{i}}(u_{1},0,…,0,\tilde{u_{i}}0,…,0)u_{i}]\\ = & \Delta u_{1} + u_{1}[g_{1}(u_{1},0,…,0) + g_{1}(u_{1},0,…,0,u_{i},0,…,0) – g_{1}(u_{1},0,…,0)]\\ \geq & \Delta u_{1} + u_{1}[g_{1}(u_{1},0,…,0) + g_{1}(u_{1},…,u_{N}) – g_{1}(u_{1},0,…,0)]\\ = & \Delta u_{1} + u_{1}g_{1}(u_{1},…,u_{N})\\ = & 0\;\;\mbox{in}\;\;\Omega, \end{aligned}\] and \[\begin{aligned} \Delta u_{i} + &u_{i}[g_{i}(0,…,0,u_{i},0,…,0) + (g_{i})_{u_{1}}(\tilde{u_{1}},0,…,0,u_{i},0,…,0)u_{1}]\\ = & \Delta u_{i} + u_{i}[g_{i}(0,…,0,u_{i},0,…,0) + g_{i}(u_{1},0,…,0,u_{i},0,…,0) \\ & – g_{i}(0,…,0,u_{i},0,…,0)]\\ = & \Delta u_{i} + u_{i}g_{i}(u_{1},0,…,0,u_{i},0,…,0)\\ \geq & \Delta u_{i} + u_{i}g_{i}(u_{1},…,u_{N}) = 0\;\;\mbox{in}\;\;\Omega. \end{aligned}\]
Hence, for \(i = 2,…,N\), \[\left.\begin{array}{l} \Delta u_{1} + u_{1}[g_{1}(u_{1},0,…,0) + \sup(g_{1})_{u_{i}}u_{i}] \geq 0\;\;\mbox{in}\;\;\Omega,\\ \Delta u_{i} + u_{i}[g_{i}(0,…,0,u_{i},0,…,0) + \sup(g_{i})_{u_{1}}u_{1}] \geq 0\;\;\mbox{in}\;\;\Omega.\\ \end{array}\right.\]
Therefore, for \(i = 2,…,N\), \[\left.\begin{array}{l} \sup(g_{i})_{u_{1}}\phi_{1}\Delta u_{1} + \sup(g_{i})_{u_{1}}\phi_{1}u_{1}[g_{1}(u_{1},0,…,0) + \sup(g_{1})_{u_{i}}u_{i}] \geq 0\;\;\mbox{in}\;\;\Omega,\\ -\sup(g_{1})_{u_{i}}\phi_{1}\Delta u_{i} – \sup(g_{1})_{u_{i}}\phi_{1} u_{i}[g_{i}(0,…,0,u_{i},0,…,0) + \sup(g_{i})_{u_{1}}u_{1}] \geq 0\;\;\mbox{in}\;\;\Omega.\\ \end{array}\right.\]
So, for \(i = 2,…,N\), \[\begin{aligned} \int_{\Omega}-\sup(g_{i})_{u_{1}}\phi_{1}\Delta u_{1} dx \leq & \int_{\Omega}[g_{1}(u_{1},0,…,0)\sup(g_{i})_{u_{1}}u_{1} + \sup(g_{1})_{u_{i}}\sup(g_{i})_{u_{1}}u_{1}u_{i}]\phi_{1}dx\\ \int_{\Omega}\sup(g_{1})_{u_{i}}\phi_{1}\Delta u_{i} dx \leq & \int_{\Omega}[-g_{i}(0,…,0,u_{i},0,…,0)\sup(g_{1})_{u_{i}}u_{i} – \sup(g_{1})_{u_{i}}\sup(g_{i})_{u_{1}}u_{1}u_{i}]\phi_{1}dx. \end{aligned}\]
Hence, by the Green’s Identity, we have \[\begin{aligned} \int_{\Omega}\sup(g_{i})_{u_{1}}\lambda_{1}\phi_{1}u_{1} dx \leq & \int_{\Omega}[g_{1}(u_{1},0,…,0)\sup(g_{i})_{u_{1}}u_{1} + \sup(g_{1})_{u_{i}}\sup(g_{i})_{u_{1}}u_{1}u_{i}]\phi_{1}dx\\ \int_{\Omega}-\sup(g_{1})_{u_{i}}\lambda_{1}\phi_{1}u_{i} dx \leq & \int_{\Omega}[-g_{i}(0,…,0,u_{i},0,…,0)\sup(g_{1})_{u_{i}}u_{i} – \sup(g_{1})_{u_{i}}\sup(g_{i})_{u_{1}}u_{1}u_{i}]\phi_{1}dx. \end{aligned}\]
Therefore, \[\int_{\Omega}\sup(g_{i})_{u_{1}}([\lambda_{1} – g_{1}(u_{1},0,…,0)]u_{1}\phi_{1} – \sup(g_{1})_{u_{i}}[\lambda_{1} – g_{i}(0,…,0,u_{i},0,…,0)]u_{i}\phi_{1}dx \leq 0.\]
Since the left hand side is nonnegative from \[\left.\begin{array}{l} g_{1}(u_{1},0,…,0) \leq g_{1}(0,…,0) \leq \lambda_{1},\\ g_{i}(0,…,0,u_{i},0,…,0) \leq g_{i}(0,…,0) \leq \lambda_{1}, \end{array}\right.\] we conclude that \(u_{i} \equiv 0, i = 1,…,N\). ◻
Theorem 3. Let \(u_{i} \geq 0, i = 1,…,N\) be a solution to (1). If \(g_{1}(0,…,0) \leq \lambda_{1}\), then \(u_{1} \equiv 0\).
Proof. Proceeding as in the proof of the Theorem 2, we obtain \[0 \leq \int_{\Omega}[\lambda_{1} – g_{1}(u_{1},0,…,0)]u_{1}\phi_{1}dx \leq \int_{\Omega}\sup(g_{1})_{u_{i}}u_{1}u_{i}\phi_{1}dx \leq 0,\] and so, \(u_{1} \equiv 0\). ◻
In order to prove further results, we will need the following solution estimate.
For the rest of this section, we assume the following additional growth condition:
\[\lim_{x_{i}\rightarrow\infty}g_{i}(x_{1},…,x_{i},…,x_{N}) = -\infty, i = 1,…,N.\]
Lemma 4. Let \((u_{1},…,u_{N}), u_{i} \geq 0, i = 1,…,N\) be a solution of the problem \[\begin{cases} -\Delta u_{i} = tu_{i}g_{i}(u_{1},,…,u_{N})&\mbox{in}\;\;\Omega,\\ u_{i}|_{\partial\Omega} = 0,& i = 1,…,N, \end{cases}\label{eq:2.2} \tag{3}\] where \(t \in [0,1]\). Then
(A) \[u_{1} \leq M_{1}, u_{i} \leq M_{i},\] where \[M_{1} = -\frac{g_{1}(0,…,0)}{\sup(g_{1})_{u_{1}}},\qquad \qquad M_{i} = \frac{\sup(g_{i})_{u_{1}}(M_{1}) + g_{i}(0,…,0)}{-\sup(g_{i})_{u_{i}}}, i = 2,…,N.\]
(B) For \(t = 1\), if \(u_{i} > 0\), \[\sup(g_{1})_{u_{1}}\sup(g_{i})_{u_{i}} + \inf(g_{1})_{u_{i}}\sup(g_{i})_{u_{1}} > 0,\] \[g_{1}(0,M_{2},…,M_{i-1},0,M_{i+1},…,M_{N}) + \inf(g_{1})_{u_{i}}\left[\frac{g_{1}(0,…,0)\sup(g_{i})_{u_{1}} – g_{i}(0,…,0)\sup(g_{1})_{u_{1}}}{\sup(g_{1})_{u_{1}}\sup(g_{i})_{u_{i}}}\right] > \lambda_{1},\] and \[g_{i}(0,M_{2},…,M_{i-1},0,M_{i+1},…,M_{N}) > \lambda_{1}, i = 2,…,N,\] then \[\begin{aligned} \theta_{M(g_{1},g_{i})} \leq & u_{1} \leq \theta_{g_{1}(\cdot,0,…,0)},\\ \theta_{g_{i}(0,M_{2},…,M_{i-1},\cdot,M_{i+1},…,M_{N})} \leq& u_{i} \leq \theta_{g_{i}(0,…,0,\cdot,0,…,0) – \frac{g_{1}(0,…,0)\sup(g_{i})_{u_{1}}}{\sup(g_{1})_{u_{1}}}}, i = 2,…,N, \end{aligned}\] where \[\left.\begin{array}{lll} M(g_{1},g_{i}) & = & g_{1}(\cdot,M_{2},…,M_{i-1},0,M_{i+1},…,M_{N})\\ & & + \inf(g_{1})_{u_{i}}\left[\dfrac{g_{1}(0,…,0)\sup(g_{i})_{u_{1}} – g_{i}(0,…,0)\sup(g_{1})_{u_{1}}}{\sup(g_{1})_{u_{1}}\sup(g_{i})_{u_{i}}}\right], i = 2,…,N. \end{array}\right.\]
Proof. (A) Since \((u_{1},…,u_{N})\) is a solution to (3), by the Mean Value Theorem, \[\begin{aligned} \Delta u_{1} + &u_{1}[g_{1}(0,…,0) + \sup(g_{1})_{u_{1}}u_{1} + … + \sup(g_{1})_{u_{N}}u_{N}]\\ \geq & \Delta u_{1} + u_{1}[g_{1}(0,…,0) + g_{1}(u_{1},…,u_{N}) – g_{1}(0,u_{2},…,u_{N})\\ & + g_{1}(0,u_{2},…,u_{N}) – g_{1}(0,0,u_{3},…,u_{N}) + … + g_{1}(0,…,0,u_{N}) – g_{1}(0,…,0)]\\ = & \Delta u_{1} + u_{1}g_{1}(u_{1},…,u_{N})\\ = & 0, \end{aligned}\] and so, \[\Delta u_{1} + u_{1}[g_{1}(0,…,0) + \sup(g_{1})_{u_{1}}u_{1}] \geq -u_{1}\sum\limits_{i=2}^{N}\sup(g_{1})_{u_{i}}u_{i} \geq 0.\]
Hence, by the Maximum Principles, \[g_{1}(0,…,0) + \sup(g_{1})_{u_{1}}u_{1} \geq 0,\] equivalently, \[u_{1} \leq -\frac{g_{1}(0,…,0)}{\sup(g_{1})_{u_{1}}}.\]
Since \((u_{1},…,u_{N})\) is a solution to (3), by the Mean Value Theorem and above estimation, for \(i = 2,…,N\), \[\begin{aligned} \Delta u_{i} + &u_{i}\left[g_{i}(0,…,0) + \sup(g_{i})_{u_{i}}u_{i} + \sup(g_{i})_{u_{1}}\left(-\frac{g_{1}(0,…,0)}{\sup(g_{1})_{u_{1}}}\right)\right]\\ \geq & \Delta u_{i} + u_{i}[g_{i}(0,…,0) + \sup(g_{i})_{u_{i}}u_{i} + \sup(g_{i})_{u_{1}}u_{1}]\\ \geq & \Delta u_{i} + u_{i}[g_{i}(0,…,0) + g_{i}(u_{1},…,u_{N}) – g_{i}(0,u_{2},…,u_{N})]\\ & + g_{i}(0,u_{2},…,u_{N}) – g_{i}(0,0,u_{3},…,u_{N}) + … + g_{i}(0,…,0,u_{N}) – g_{i}(0,…,0)]\\ = & 0. \end{aligned}\]
Hence, by the Maximum Principles again, \[g_{i}(0,…,0) + \sup(g_{i})_{u_{i}}u_{i} + \sup(g_{i})_{u_{1}}\left[-\frac{g_{1}(0,…,0)}{\sup(g_{1})_{u_{1}}}\right] \geq 0,\] equivalently, \[u_{i} \leq \frac{\sup(g_{i})_{u_{1}}\left[-\frac{g_{1}(0,…,0)}{\sup(g_{1})_{u_{1}}}\right] + g_{i}(0,…,0)}{-\sup(g_{i})_{u_{i}}}, i = 2,…,N.\]
(B) By the monotonicity of \(g_{1}\), \[\Delta u_{1} + u_{1}g_{1}(u_{1},0,…,0) = u_{1}[g_{1}(u_{1},0,…,0) – g_{1}(u_{1},…,u_{N})] \geq 0,\] and so, \(u_{1}\) is a subsolution to \[\left.\begin{array}{l} \Delta Z + Zg_{1}(Z,0,…,0) = 0\;\;\mbox{in}\;\;\Omega,\\ Z|_{\partial\Omega} = 0. \end{array}\right.\]
But, since any sufficiently large positive constant is a supersolution to \[\left.\begin{array}{l} \Delta Z + Zg_{1}(Z,0,…,0) = 0\;\;\mbox{in}\;\;\Omega,\\ Z|_{\partial\Omega} = 0, \end{array}\right.\] by the Lemmas 1 and 3, we conclude that \[\left.\begin{array}{l} u_{1} \leq \theta_{g_{1}(\cdot,0,…,0)}. \end{array}\right.\label{eq:11} \tag{4}\]
Since \[\begin{aligned} \Delta u_{1} + u_{1}&\left[g_{1}(u_{1},M_{2},…,M_{i-1},0,M_{i+1},…,M_{N}) + \inf(g_{1})_{u_{i}}\left(\frac{\sup(g_{i})_{u_{1}}\left[-\frac{g_{1}(0,…,0)}{\sup(g_{1})_{u_{1}}}\right] + g_{i}(0,…,0)}{-\sup(g_{i})_{u_{i}}}\right)\right]\\ = & u_{1}\Bigg[-g_{1}(u_{1},…,u_{N}) + g_{1}(u_{1},M_{2},…,M_{i-1},0,M_{i+1},…,M_{N})\\ &\left. + \inf(g_{1})_{u_{i}}\left(\frac{\sup(g_{i})_{u_{1}}\left[-\frac{g_{1}(0,…,0)}{\sup(g_{1})_{u_{1}}}\right] + g_{i}(0,…,0)}{-\sup(g_{i})_{u_{i}}}\right)\right]\\ \leq & u_{1}\Bigg[-g_{1}(u_{1},M_{2},…,M_{i-1},u_{i},M_{i+1},…,M_{N}) + g_{1}(u_{1},M_{2},…,M_{i-1},0,M_{i+1},…,M_{N})\\ &\left. + \inf(g_{1})_{u_{i}}\left(\frac{\sup(g_{i})_{u_{1}}\left[-\frac{g_{1}(0,…,0)}{\sup(g_{1})_{u_{1}}}\right] + g_{i}(0,…,0)}{-\sup(g_{i})_{u_{i}}}\right)\right]\\ \leq & u_{1}\left[-\inf(g_{1})_{u_{i}}u_{i} + \inf(g_{1})_{u_{i}}\left(\frac{\sup(g_{i})_{u_{1}}\left[-\frac{g_{1}(0,…,0)}{\sup(g_{1})_{u_{1}}}\right] + g_{i}(0,…,0)}{-\sup(g_{i})_{u_{i}}}\right)\right]\\ = & -u_{1}\inf(g_{1})_{u_{i}}\left[u_{i} – \frac{\sup(g_{i})_{u_{1}}[-\frac{g_{1}(0,…,0)}{\sup(g_{1})_{u_{1}}}] + g_{i}(0,…,0)}{-\sup(g_{i})_{u_{i}}}\right]\\ \leq & 0,\qquad i = 2,…,N, \end{aligned}\] by the Mean Value Theorem, monotonicity and (A), \(u_{1}\) is a supersolution to \[\begin{aligned} \begin{cases} \Delta Z + Z\left[g_{1}(Z,M_{2},…,M_{i-1},0,M_{i+1},…,M_{N}) +\inf(g_{1})_{u_{i}}\left(\frac{\sup(g_{i})_{u_{1}}\left[-\frac{g_{1}(0,…,0)}{\sup(g_{1})_{u_{1}}}\right] + g_{i}(0,…,0)}{-\sup(g_{i})_{u_{i}}}\right)\right] = 0&\mbox{in}\;\;\Omega,\\ Z|_{\partial\Omega} = 0,& i = 2,…,N.\end{cases} \end{aligned}\]
But, by the continuity of \(g_{1}\) and the condition, for sufficiently small \(\epsilon > 0\), \[\begin{aligned} \Delta\epsilon\phi_{1}& + \epsilon\phi_{1}\left[g_{1}(\epsilon\phi_{1},M_{2},…,M_{i-1},0,M_{i+1},…,M_{N}) + \inf(g_{1})_{u_{i}}\left(\frac{\sup(g_{i})_{u_{1}}\left[-\frac{g_{1}(0,…,0)}{\sup(g_{1})_{u_{1}}}\right] + g_{i}(0,…,0)}{-\sup(g_{i})_{u_{i}}}\right)\right]\\ = & \epsilon\phi_{1}\left[-\lambda_{1} + g_{1}(\epsilon\phi_{1},M_{2},…,M_{i-1},0,M_{i+1},…,M_{N}) + \inf(g_{1})_{u_{i}}\left(\frac{\sup(g_{i})_{u_{1}}\left[-\frac{g_{1}(0,…,0)}{\sup(g_{1})_{u_{1}}}\right] + g_{i}(0,…,0)}{-\sup(g_{i})_{u_{i}}}\right)\right]\\ > & 0,\qquad i = 2,…,N, \end{aligned}\] and so, \(\epsilon\phi_{1}\) is a subsolution to \[\begin{aligned} \begin{cases} \Delta Z + Z\left[g_{1}(Z,M_{2},…,M_{i-1},0,M_{i+1},…,M_{N}) + \inf(g_{1})_{u_{i}}\left(\frac{\sup(g_{i})_{u_{1}}\left[-\frac{g_{1}(0,…,0)}{\sup(g_{1})_{u_{1}}}\right] + g_{i}(0,…,0)}{-\sup(g_{i})_{u_{i}}}\right)\right] = 0&\mbox{in}\;\;\Omega,\\ Z|_{\partial\Omega} = 0,& i = 2,…,N.\end{cases} \end{aligned}\]
Therefore, by the Lemmas 1 and 3, we have \[\begin{aligned} \theta_{g_{1}(\cdot,M_{2},…,M_{i-1},0,M_{i+1},…,M_{N}) + \inf(g_{1})_{u_{i}}\left(\frac{\sup(g_{i})_{u_{1}}\left[-\frac{g_{1}(0,…,0)}{\sup(g_{1})_{u_{1}}}\right] + g_{i}(0,…,0)}{-\sup(g_{i})_{u_{i}}}\right)} \leq u_{1},\qquad i = 2,…,N. \label{eq:12} \end{aligned} \tag{5}\]
By the monotonicity of \(g_{i}\), \[\begin{aligned} \Delta u_{i} + u_{i}g_{i}(0,M_{2},…,M_{i-1},u_{i},M_{i+1},…,M_{N})= & u_{i}[g_{i}(0,M_{2},…,M_{i-1},u_{i},M_{i+1},…,M_{N}) – g_{i}(u_{1},…,u_{N})]\\ < & 0,\qquad i = 2,…,N, \end{aligned}\] and so, \(u_{i}\) is a supersolution to \[\begin{cases} \Delta Z + Zg_{i}(0,M_{2},…,M_{i-1},Z,M_{i+1},…,M_{N}) = 0&\mbox{in}\;\;\Omega,\\ Z|_{\partial\Omega} = 0,& i = 2,…,N. \end{cases}\]
But, by the continuity of \(g_{i}\) and the condition, for sufficiently small \(\epsilon > 0\), \[\begin{aligned} \Delta\epsilon\phi_{1} + \epsilon\phi_{1}g_{i}(0,M_{2},…,M_{i-1},\epsilon\phi_{1},M_{i+1},…,M_{N}) = & \epsilon\phi_{1}[-\lambda_{1} + g_{i}(0,M_{2},…,M_{i-1},\epsilon\phi_{1},M_{i+1},…,M_{N})]\\ > & 0,\qquad i = 2,…,N, \end{aligned}\] and so, \(\epsilon\phi_{1}\) is a subsolution to \[\begin{cases} \Delta Z + Zg_{i}(0,M_{2},…,M_{i-1},Z,M_{i+1},…,M_{N}) = 0&\mbox{in}\;\;\Omega,\\ Z|_{\partial\Omega} = 0, &i = 2,…,N. \end{cases}\]
Hence, by the Lemmas 1 and 3, we have \[\left.\begin{array}{l} \theta_{g_{i}(0,M_{2},…,M_{i-1},\cdot,M_{i+1},…,M_{N})} \leq u_{i},\qquad i = 2,…,N. \end{array}\right.\label{eq:13} \tag{6}\]
Since \((u_{1},…,u_{N})\) is a solution to (3), by the Mean Value Theorem and (A), \[\begin{aligned} \Delta u_{i} + &u_{i}\left[g_{i}(0,…,0,u_{i},0,…,0) – \frac{\sup(g_{i})_{u_{1}}g_{1}(0,…,0)}{\sup(g_{1})_{u_{1}}}\right]\\ = & u_{i}\left[-g_{i}(u_{1},…,u_{N}) + g_{i}(0,…,0,u_{i},0,…,0) – \frac{\sup(g_{i})_{u_{1}}g_{1}(0,…,0)}{\sup(g_{1})_{u_{1}}}\right]\\ = & u_{i}\Bigg[-g_{i}(u_{1},…,u_{N}) + g_{i}(0,u_{2},…,u_{N}) – g_{i}(0,u_{2},…,u_{N}) + g_{i}(0,0,u_{3},…,u_{N})\\ & – g_{i}(0,0,u_{3},…,u_{N}) + g_{i}(0,0,0,u_{4},…,u_{N}) – … – g_{i}(0,…,0,u_{i-1},u_{i},…,u_{N}) + g_{i}(0,…,0,u_{i},…,u_{N})\\ & – g_{i}(0,…,0,u_{i},…,u_{N}) + g_{i}(0,…,0,u_{i},0,u_{i+2},…,u_{N}) – g_{i}(0,…,0,u_{i},0,u_{i+2},…,u_{N})\\ & + g_{i}(0,…,0,u_{i},0,0,u_{i+3},…,u_{N}) – … – g_{i}(0,…,0,u_{i},0,…,0,u_{N}) \\ &+ g_{i}(0,…,0,u_{i},0,…,0) – \frac{\sup(g_{i})_{u_{1}}g_{1}(0,…,0)}{\sup(g_{1})_{u_{1}}}\Bigg]\\ \geq & u_{i}\left[-\sup(g_{i})_{u_{1}}u_{1} – \sum\limits_{j=2,j\neq i}^{N}u_{j}\sup(g_{i})_{u_{j}} – \frac{\sup(g_{i})_{u_{1}}g_{1}(0,…,0)}{\sup(g_{1})_{u_{1}}}\right]\\ \geq & u_{i}\left[-u_{1}\sup(g_{i})_{u_{1}} – \frac{\sup(g_{i})_{u_{1}}g_{1}(0,…,0)}{\sup(g_{1})_{u_{1}}}\right]\\ = & \sup(g_{i})_{u_{1}}u_{i}\left[-u_{1} – \frac{g_{1}(0,…,0)}{\sup(g_{1})_{u_{1}}}\right]\\ \geq & 0,\qquad i = 2,…,N. \end{aligned}\]
Therefore, \(u_{i}\) is a subsolution to \[\begin{cases} \Delta Z + Z\left[g_{i}(0,…,0,Z,0,…,0) – \frac{\sup(g_{i})_{u_{1}}g_{1}(0,…,0)}{\sup(g_{1})_{u_{1}}}\right] = 0&\mbox{in}\;\;\Omega,\\ Z|_{\partial\Omega} = 0,& i = 2,…,N. \end{cases}\]
But, since any sufficiently large constant is a supersolution to \[\begin{cases} \Delta Z + Z[g_{i}(0,…,0,Z,0,…,0) – \frac{\sup(g_{i})_{u_{1}}g_{1}(0,…,0)}{\sup(g_{1})_{u_{1}}}] = 0&\mbox{in}\;\;\Omega,\\ Z|_{\partial\Omega} = 0,& i = 2,…,N, \end{cases}\] by the Lemmas 1 and 3, we have \[\left.\begin{array}{l} u_{i} \leq \theta_{g_{i}(0,…,0,\cdot,0,…,0) – \frac{g_{1}(0,…,0)\sup(g_{i})_{u_{1}}}{\sup(g_{1})_{u_{1}}}},\qquad i = 2,…,N. \end{array}\right.\label{eq:14} \tag{7}\]
By (4), (5), (6) and (7), we establish the desired inequalities. ◻
In this section, we prove the uniqueness of positive solution to (1) with the following additional growth condition: \[\lim_{x_{i}\rightarrow\infty}g_{i}(x_{1},…,x_{i},…,x_{N}) = -\infty,\qquad i = 1,…,N\].
We have the following uniqueness result.
Theorem 4. In addition to the Theorem 1, if
(A) \[\begin{aligned} g_{1}(0,M_{2},…,M_{i-1},0,M_{i+1},…,M_{N})& + \inf(g_{1})_{x_{i}}\left[\frac{g_{1}(0,…,0)\sup(g_{i})_{x_{1}} – g_{i}(0,…,0)\sup(g_{1})_{x_{1}}}{\sup(g_{1})_{x_{1}}\sup(g_{i})_{x_{i}}}\right]\\ >& \lambda_{1}, \,\, g_{i}(0,M_{2},…,M_{i-1},0,M_{i+1},…,M_{N}) > \lambda_{1},\qquad i = 2,…,N, \end{aligned}\] and
(B) \[\begin{aligned} 2\sup(g_{1})_{x_{1}} + &\sum\limits_{i=2}^{N}\sup(g_{i})_{x_{1}}\sup\frac{\theta_{g_{i}(0,…,0,\cdot,0,…,0) – \frac{g_{1}(0,…,0)\sup(g_{i})_{x_{1}}}{\sup(g_{1})_{u_{1}}}}}{\theta_{M(g_{1},g_{i})}}\\ <& \sum\limits_{j=2}^{N}\left[\inf(g_{1})_{x_{j}} + \inf(g_{j})_{x_{1}}\inf\frac{\theta_{g_{j}(0,M_{2},…,M_{j-1},\cdot,M_{j+1},…,M_{N})}}{\theta_{g_{1}(\cdot,0,…,0)}}\right], \end{aligned}\] and \[\begin{aligned} 2\sup(g_{i})_{x_{i}} + \sup(g_{i})_{x_{1}} < \sum\limits_{j=2,j\neq i}^{N}\left[\inf(g_{i})_{x_{j}} + \inf(g_{j})_{x_{i}}\sup\frac{\theta_{g_{j}(0,…,0,\cdot,0,…,0) – \frac{g_{1}(0,…,0)\sup(g_{j})_{x_{1}}}{\sup(g_{1})_{x_{1}}}}}{\theta_{g_{i}(0,M_{2},…,M_{i-1},\cdot,M_{i+1},…,M_{N})}}\right]\qquad \text{ for }i = 2,…,N, \end{aligned}\] then (1) has a unique posotive solution.
The conditions imply that species 1 interacts strongly among themselves and weakly with species 2. Similarly for species 2, they interact more strongly among themselves than they do with species 1.
Proof. The existence was already proved in the last section. We prove the uniqueness. Let \((u_{1},…,u_{N}), (v_{1},…,v_{N})\) be positive solutions to (1), and let \(p_{i} = u_{i} – v_{i}, i = 1,…,N\). We want to show that \(p_{i} \equiv 0, i = 1,…,N\).
Since \((u_{1},…,u_{N}), (v_{1},…,v_{N})\) are solutions to (1), \[\begin{aligned} \Delta p_{i} + p_{i}g_{i}(u_{1},…,u_{N}) = & \Delta u_{i} – \Delta v_{i} + (u_{i} – v_{i})g_{i}(u_{1},…,u_{N})\\ = & -\Delta v_{i} – v_{i}g_{i}(u_{1},…,u_{N})\\ = & -\Delta v_{i} – v_{i}g_{i}(v_{1},…,v_{N}) + v_{i}g_{i}(v_{1},…,v_{N}) – v_{i}g_{i}(u_{1},…,u_{N})\\ = & -v_{i}[g_{i}(u_{1},…,u_{N}) – g_{i}(v_{1},…,v_{N})],\qquad i = 1,…,N. \end{aligned}\]
So, \[\left.\begin{array}{l} \Delta p_{i} + p_{i}g_{i}(u_{1},…,u_{N}) -v_{i}[g_{i}(v_{1},…,v_{N}) – g_{i}(u_{1},…,u_{N})] = 0,\qquad i = 1,…,N.\\ \end{array}\right.\]
So, for \(i = 1,…,N\), \[\left.\begin{array}{l} -p_{i}\Delta p_{i} – (p_{i})^{2}g_{i}(u_{1},…,u_{N}) + v_{i}p_{i}[g_{i}(v_{1},…,v_{N}) – g_{i}(u_{1},…,u_{N})] = 0. \end{array}\right.\]
Since \[\left.\begin{array}{l} \Delta u_{i} + u_{i}g_{i}(u_{1},..,u_{N}) = 0,\qquad i = 1,…,N,\\ \end{array}\right.\] by the Lemma 2, we have \[\left.\begin{array}{l} \int_{\Omega}-p_{i}\Delta p_{i} – g_{i}(u_{1},…,u_{N})(p_{i})^{2}dx \geq 0,\qquad i = 1,…,N,\\ \end{array}\right.\] and so, \[\int_{\Omega}v_{i}p_{i}[g_{i}(v_{1},…,v_{N}) – g_{i}(u_{1},…,u_{N})]dx \leq 0,\qquad i = 1,…,N,\] and so, \[\int_{\Omega}\sum\limits_{i=1}^{N}v_{i}p_{i}[g_{i}(v_{1},…,v_{N}) – g_{i}(u_{1},…,u_{N})]dx \leq 0.\]
But, by the Mean Value Theorem, for \(i = 1,…,N\), there are \(z_{ij}\) such that \(z_{ij}\) is between \(u_{j}\) and \(v_{j}\), \(j = 1,…,N\), and \[\begin{aligned} g_{i}(v_{1},…,v_{N}) – g_{i}(u_{1},…,u_{N}) = & g_{i}(v_{1},v_{2},…,v_{N}) – g_{i}(u_{1},v_{2},…,v_{N}) + g_{i}(u_{1},v_{2},v_{3},…,v_{N}) – g_{i}(u_{1},u_{2},v_{3},…,v_{N})\\ & + … – g_{i}(u_{1},u_{2},…,u_{N-1},v_{N}) – g_{i}(u_{1},u_{2},…,u_{N-1},u_{N})\\ = & \sum\limits_{j=1}^{N}(g_{i})_{x_{j}}(u_{1},u_{2},…,u_{j-1},z_{ij},v_{j+1},…,v_{N})(-p_{j}). \end{aligned}\]
Therefore, \[\begin{aligned} \int_{\Omega}\sum\limits_{i=1}^{N}v_{i}p_{i}\left[\sum\limits_{j=1}^{N}(g_{i})_{x_{j}}(u_{1},u_{2},…,u_{j-1},z_{ij},v_{j+1},…,v_{N})(-p_{j})\right]dx \leq 0. \end{aligned}\]
Hence, \[\begin{aligned} \int_{\Omega}\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}(g_{i})_{x_{j}}(u_{1},…,u_{j-1},z_{ij},v_{j+1},…,v_{N})(-v_{i}p_{i}p_{j})dx \leq 0. \end{aligned}\]
So, \[\begin{aligned} \int_{\Omega}\sum\limits_{i=1}^{N}\left\{(g_{i})_{x_{i}}(u_{1},…,u_{j-1},z_{ii},v_{i+1},…,v_{N})[-v_{i}(p_{i})^{2}] + \sum\limits_{j=1,j\neq i}^{N}(g_{i})_{x_{j}}(u_{1},…,u_{j-1},z_{ij},v_{j+1},…,v_{N})(-v_{i}p_{i}p_{j})\right\}dx\leq 0. \end{aligned}\]
So, \[\begin{aligned} &\int_{\Omega}(g_{1})_{x_{1}}(z_{11},v_{2},…,v_{N})[-v_{1}(p_{1})^{2}] + \sum\limits_{j=2}^{N}(g_{1})_{x_{j}}(u_{1},…,u_{j-1},z_{1j},v_{j+1},…,v_{N})(-v_{1}p_{1}p_{j})\\ &+ \sum\limits_{i=2}^{N}\{(g_{i})_{x_{i}}(u_{1},…,u_{i-1},z_{ii},v_{i+1},…,v_{N})[-v_{i}(p_{i})^{2}] + \sum\limits_{j=1,j\neq i}^{N}(g_{i})_{x_{j}}(u_{1},…,u_{j-1},z_{ij},v_{j+1},…,v_{N})(-v_{i}p_{i}p_{j})\}dx\\ &\qquad\leq 0, \end{aligned}\] so, \[\begin{aligned} &\int_{\Omega}(g_{1})_{x_{1}}(z_{11},v_{2},…,v_{N})[-v_{1}(p_{1})^{2}] + \sum\limits_{j=2}^{N}(g_{1})_{x_{j}}(u_{1},…,u_{j-1},z_{1j},v_{j+1},…,v_{N})(-v_{1}p_{1}p_{j})\\ &+ \sum\limits_{i=2}^{N}\left\{(g_{i})_{x_{i}}(u_{1},…,u_{i-1},z_{ii},v_{i+1},…,v_{N})[-v_{i}(p_{i})^{2}] + \sum\limits_{j=2,j\neq i}^{N}(g_{i})_{x_{j}}(u_{1},…,u_{j-1},z_{ij},v_{j+1},…,v_{N})(-v_{i}p_{i}p_{j})\right.\\ &+ (g_{i})_{x_{1}}(z_{i1},v_{2},…,v_{N})(-v_{i}p_{i}p_{1})\Bigg\}dx \leq 0. \end{aligned}\]
But, since \((g_{i})_{x_{1}}(-v_{i}) < 0\) and \(p_{i}p_{1} \leq \frac{(p_{i})^{2}}{2} + \frac{(p_{1})^{2}}{2}\) for \(i = 2,…,N\), \[\begin{aligned} &\int_{\Omega}(g_{1})_{x_{1}}(z_{11},v_{2},…,v_{N})[-v_{1}(p_{1})^{2}] + \sum\limits_{j=2}^{N}(g_{1})_{x_{j}}(u_{1},…,u_{j-1},z_{1j},v_{j+1},…,v_{N})(-v_{1}p_{1}p_{j})\\ &+ \sum\limits_{i=2}^{N}\Bigg\{(g_{i})_{x_{i}}(u_{1},…,u_{i-1},z_{ii},v_{i+1},…,v_{N})[-v_{i}(p_{i})^{2}] + \sum\limits_{j=2,j\neq i}^{N}(g_{i})_{x_{j}}(u_{1},…,u_{j-1},z_{ij},v_{j+1},…,v_{N})(-v_{i}p_{i}p_{j})\\ &+ (g_{i})_{x_{1}}(z_{i1},v_{2},…,v_{N})(-v_{i})\left(\frac{(p_{i})^{2}}{2} + \frac{(p_{1})^{2}}{2}\right)\Bigg\}dx \leq 0. \end{aligned}\]
Hence, \[\begin{aligned} &\int_{\Omega}\left[(g_{1})_{x_{1}}(z_{11},v_{2},…,v_{N})(-v_{1}) + \sum\limits_{i=2}^{N}\frac{(g_{i})_{x_{1}}(z_{i1},v_{2},…,v_{N})(-v_{i})}{2}\right](p_{1})^{2} \\ &+ \sum\limits_{j=2}^{N}(g_{1})_{x_{j}}(u_{1},…,u_{j-1},z_{1j},v_{j+1},…,v_{N})(-v_{1}p_{1}p_{j})\\ &+ \sum\limits_{i=2}^{N}\Bigg\{\left[(g_{i})_{x_{i}}(u_{1},…,u_{i-1},z_{ii},v_{i+1},…,v_{N})(-v_{i}) + \frac{(g_{i})_{x_{1}}((z_{i1},v_{2},…,v_{N})(-v_{i})}{2}\right][(p_{i})^{2}] \\ &+ \sum\limits_{j=2,j\neq i}^{N}(g_{i})_{x_{j}}(u_{1},…,u_{j-1},z_{ij},v_{j+1},…,v_{N})(-v_{i}p_{i}p_{j})\Bigg\}dx \leq 0. \end{aligned}\]
If the integrand on the left side is positive definite, then \(p_{i} \equiv 0, i = 1,…,N\), which means the uniqueness. But, \[\begin{aligned} (g_{1})_{x_{j}}(-v_{1}p_{1}p_{j}) \leq& (g_{1})_{x_{j}}(-v_{1})\left(\frac{(p_{1})^{2}}{2} + \frac{(p_{j})^{2}}{2}\right),\\ (g_{i})_{x_{j}}(-v_{i}p_{i}p_{j}) \leq & (g_{i})_{x_{j}}(-v_{i})\left(\frac{(p_{i})^{2}}{2} + \frac{(p_{j})^{2}}{2}\right),\\ \end{aligned}\] for \(j = 2,…,N, j \neq i\), and so, the result follows if the condition is satisfied by the solution estimates in the Lemma 4. ◻
Define \(B = \{(g_{1},…,g_{N}) \in [C^{1}]^{N} | \lim_{x_{i}\rightarrow\infty}g_{i}(x_{1},…,x_{i},…,x_{N}) = -\infty, i = 1,…,N \}\) with \(\parallel (g_{1},…,g_{N}) \parallel_{B} = \sum\limits_{i=1}^{N}|g_{i}(0,…,0)| + \sum\limits_{i,j=1}^{N}\sup|(g_{i})_{x_{j}}|\) for all \((g_{1},…,g_{N}) \in B\).
Then by the functional analysis theory, \((B,\parallel\cdot\parallel_{B})\) is a Banach space.
The following theorem is our main result about the perturbation of uniqueness.
Theorem 5. Suppose \((g_{1},…,g_{N}) \in B\) is such that
(A) \[\sup(g_{1})_{u_{1}}\sup(g_{i})_{u_{i}} + \inf(g_{1})_{u_{i}}\sup(g_{i})_{u_{1}} > 0, i = 2,…,N,\]
(B) \[\begin{aligned} g_{1}(0,M_{2},…,M_{i-1},0,M_{i+1},…,M_{N}) + \inf(g_{1})_{u_{i}}\left[\frac{g_{1}(0,…,0)\sup(g_{i})_{u_{1}} – g_{i}(0,…,0)\sup(g_{1})_{u_{1}}}{\sup(g_{1})_{u_{1}}\sup(g_{i})_{u_{i}}}\right]& > \lambda_{1},\\ g_{1}(M_{1},…,M_{i-1},0,M_{i+1},…,M_{N}) &> \lambda_{1},\\ g_{i}(0,M_{2},…,M_{N}) &> \lambda_{1}, i = 2,…,N, \end{aligned}\]
(C) (1) has a unique coexistence state \((u_{1},…,u_{N})\),
(D) the Frechet derivative of (1) at \((u_{1},…,u_{N})\) is invertible.
Then there is a neighborhood \(V\) of \((g_{1},…,g_{N})\) in \(B\) such that if \((\bar{g_{1}},…,\bar{g_{N}}) \in V\), then (1) with \((\bar{g_{1}},…,\bar{g_{N}})\) has a unique positive solution.
Biologically, the first two conditions in Theorem 5 indicates that the rates of reproduction are relatively large, and the birth rate of the prey must be larger than that of predator. Furthermore, comparing the two conditions. Similarly, the fourth condition, which requires the invertibility of the Frechet derivative, signifies that the rates of self-limitation are relatively larger than the rates of competition, a relationship that will be established in Lemma 5. When these conditions are fulfilled, the conclusion of our theorem asserts that small perturbations of the rates do not affect the existence and uniqueness of the positive steady state. That is, the two species implied can continue to coexist even if the factors determining the population densities vary slightly.
Now, at first glance, Theorem 5 may appear to be a consequence of the Implicit Function Theorem. However, the Implicit Function Theorem only guarantees local uniqueness. In contrast, our result in Theorem 5 guarantees global uniqueness. The techniques we will use in the proof of Theorem 5 include the Implicit Function Theorem and a priori estimates on solutions of (1).
Proof. Since the Frechet derivative of (1) at \((u_{1},…,u_{N})\) is invertible, by the Implicit Function Theorem, there is a neighborhood \(V\) of \((g_{1},…,g_{N})\) in \(B\) and a neighborhood \(W\) of \((u_{1},…,u_{N})\) in \([C_{0}^{2,\alpha}(\bar{\Omega})]^{N}\) such that for all \((\bar{g_{1}},…,\bar{g_{N}}) \in V\), there is a unique positive solution \((\bar{u_{1}},…,\bar{u_{N}}) \in W\) of (1) with \((\bar{g_{1}},…,\bar{g_{N}})\). Thus, the local uniqueness of the solution is guaranteed.
To prove global uniqueness, suppose that the conclusion of Theorem 5 is false. Then, there are
sequences \((g_{1n},g_{2n},…,g_{Nn},u_{1n},u_{2n},…,u_{Nn})\)
and \((g_{1n},g_{2n},…,g_{Nn},u_{1n}^{*},u_{2n}^{*},…,u_{Nn}^{*})\)
in \(V \times
[C_{0}^{2,\alpha}(\bar{\Omega})]^{N}\) such that \((u_{1n},…,u_{Nn})\) and \((u_{1n}^{*},…,u_{Nn}^{*})\) are positive
solutions of (1) with \((g_{1n},…,g_{Nn})\),
\((u_{1n},…,u_{Nn})
\neq (u_{1n}^{*},…,u_{Nn}^{*})\) and \((g_{1n},…,g_{Nn}) \rightarrow
(g_{1},…,g_{N})\). By Schauder’s estimate in elliptic theory
and the solution estimate in the Lemma 4, there
are uniformly convergent subsequences of \(\{u_{1n}\},…,\{u_{Nn}\}\), which again
will be denoted by \(\{u_{1n}\},…,\{u_{Nn}\}\).
Thus, let \[\left.\begin{array}{ll} (u_{1n},…,u_{Nn}) \rightarrow (\bar{u_{1}},…,\bar{u_{N}}),\\ (u_{1n}^{*},…,u_{Nn}^{*}) \rightarrow (u_{1}^{*},…,u_{N}^{*}). \end{array}\right.\]
Then \((\bar{u_{1}},…,\bar{u_{N}}), (u_{1}^{*},…,u_{N}^{*}) \in (C^{2,\alpha})^{N}\) are also solutions to (1) with \((g_{1},…,g_{N})\). We claim that \(\bar{u_{1}} > 0, …,\bar{u_{N}} > 0, u_{1}^{*} > 0,…,u_{N}^{*} > 0.\) By the Maximum Principles, it suffices to claim \(\bar{u_{1}},…,\bar{u_{N}}, u_{1}^{*},…,u_{N}^{*}\) are not identically zero.
Suppose that it is not true. Then by the Maximum Principles again, either one of the followings will hold:
(1) \(\bar{u_{1}} \equiv 0\) and \(\bar{u_{i}} \equiv 0\) for all \(i = 2,…,N\).
(2) \(\bar{u_{1}} \equiv 0\) and \(\bar{u_{i}} > 0\) for some \(i = 2,…,N\).
(3) \(\bar{u_{1}} > 0\) and \(\bar{u_{i}} \equiv 0\) for some \(i = 2,…,N\).
First, suppose \(\bar{u_{1}} \equiv 0\).
Let \(\tilde{u_{1n}} = \frac{u_{1n}}{{\parallel u_{1n} \parallel}_{\infty}}\) and \(\tilde{u_{in}} = u_{in}\) for \(i = 2,…,N\).
Then \[\left.\begin{array}{l} \Delta\tilde{u_{1n}} + \tilde{u_{1n}}g_{1n}(u_{1n},\tilde{u_{2n}},…,\tilde{u_{Nn}}) = 0,\\ \Delta\tilde{u_{in}} + \tilde{u_{in}}g_{in}(u_{1n},\tilde{u_{2n}},…,\tilde{u_{Nn}}) = 0, i = 2,…,N. \end{array}\right.\]
By the elliptic theory again, there is \(\tilde{u_{1}}\) such that \(\tilde{u_{1n}} \rightarrow \tilde{u_{1}}\), and so, \[\left.\begin{array}{l} \Delta\tilde{u_{1}} + \tilde{u_{1}}g_{1}(0,\bar{u_{2}},…,\bar{u_{N}}) = 0,\\ \Delta\bar{u_{i}} + \bar{u_{i}}g_{i}(0,\bar{u_{2}},…,\bar{u_{N}}) = 0, i = 2,…,N. \end{array}\right.\]
Hence, \(\lambda_{1}[-g_{1}(0,\bar{u_{2}},…,\bar{u_{N}})] = 0\).
If \(\bar{u_{i}} \equiv 0\) for all \(i = 2,…,N\), then \(\lambda_{1} – g_{1}(0,…,0) = \lambda_{1}[-g_{1}(0,…,0)] = 0\), which is a contradiction to our assumption. If \(\bar{u_{i}} > 0\) for some \(i = 2,…,N\), then by the monotonicity, \[\begin{aligned} \lambda_{1} – g_{i}(0,M_{2},…,M_{N}) = & \lambda_{1}[- g_{i}(0,M_{2},…,M_{N})]\geq \ \lambda_{1}[-g_{i}(0,\bar{u_{2}},…,\bar{u_{N}})] = 0, \end{aligned}\] which is a contradiction.
Suppose \(\bar{u_{1}} > 0\) and \(\bar{u_{i}} \equiv 0\) for some \(i = 2,…,N\).
Then \[\left.\begin{array}{l} \Delta u_{1n} + u_{1n}g_{1n}(u_{1n},u_{2n},…,u_{Nn}) = 0. \end{array}\right.\]
So, \[\left.\begin{array}{l} \Delta\bar{u_{1}} + \bar{u_{1}}g_{1}(\bar{u_{1}},…,\bar{u_{i-1}},0,\bar{u_{i+1}},…,\bar{u_{N}}) = 0. \end{array}\right.\]
Therefore, \[\begin{aligned} \lambda_{1} – g_{1}(M_{1},M_{2},…,M_{i-1},0,M_{i+1},…,M_{N}) = & \lambda_{1}[ – g_{1}(M_{1},M_{2},…,M_{i-1},0,M_{i+1},…,M_{N})]\\ \geq & \lambda_{1}[-g_{1}(\bar{u_{1}},…,\bar{u_{i-1}},0,\bar{u_{i+1}},…,\bar{u_{N}})]\\ = & 0, \end{aligned}\] which is a contradiction.
Consequently, \((\bar{u_{1}},…,\bar{u_{N}})\) and \((u_{1}^{*},…,u_{N}^{*})\) are positive solutions to (1) with \((g_{1},…,g_{N})\), and so \((\bar{u_{1}},…,\bar{u_{N}}) = (u_{1}^{*},…,u_{N}^{*}) = (u_{1},…,u_{N})\) by the uniqueness condition. But, this is a contradiction to the Implicit Function Theorem, since \((u_{1n},…,u_{Nn}) \neq (u_{1n}^{*},…,u_{Nn}^{*})\). ◻
In biological terms, the proof of our theorem indicates that if one of two species living in the same domain becomes extinct, that is, if one species is excluded by the other, then the reproduction rates of both must be small. In other words, the region condition of reproduction rates \((A)\) is reasonable.
Now, the condition \((C)\) in Theorem 5 requiring the invertibility of the Frechet derivative is too artificial to have any direct biological implications. We therefore turn our attention to more applicable conditions that will guarantee the invertibility of the Frechet derivative. We then obtain the following relationship:
Lemma 5. Suppose \((u_{1},…,u_{N})\) is a positive solution to (1). If \[\begin{aligned} 2\sup(g_{1})_{u_{1}} + \sum\limits_{i=2}^{N}\sup(g_{i})_{u_{1}}\frac{u_{i}}{u_{1}} <& \sum\limits_{j=2}^{N}\left[\inf(g_{1})_{u_{j}} + \inf(g_{j})_{u_{1}}\frac{u_{j}}{u_{1}}\right],\\ 2\sup(g_{i})_{u_{i}} + \sup(g_{i})_{u_{1}} <& \sum\limits_{j=2,j\neq i}^{N}\left[\inf(g_{i})_{u_{j}} + \inf(g_{j})_{u_{i}}\frac{u_{j}}{u_{i}}\right], i = 2,…,N, \end{aligned}\] then the Frechet derivative of (1) at \((u_{1},…,u_{N})\) is invertible.
Proof. The Frechet derivative of (1) at \((u_{1},…,u_{N})\) is \(A = (A_{ij})\), where \[\left.\begin{array}{lll} A_{ij} & = & \left\{\begin{array}{cc} -\Delta – g_{i}(u_{1},…,u_{N}) – u_{i}\frac{\partial g_{i}(u_{1},…,u_{N})}{\partial u_{i}}, & i = j\\ -u_{i}\frac{\partial g_{i}(u_{1},…,u_{N})}{\partial u_{j}}, & i \neq j \end{array}\right. \end{array}\right.\] for \(i,j = 1,…,N.\) We need to show that \(N(A) = \{0\}\) by the Fredholm Alternative, where \(N(A)\) is the null space of \(A\). In fact, from the equations \[\begin{aligned} \int_{\Omega} & |\nabla\phi_{1}|^{2} – \left(g_{1}(u_{1},…,u_{N}) + u_{1}\frac{\partial g_{1}(u_{1},…,u_{N})}{\partial u_{1}}\right)\phi_{1}^{2}\\ & – \left(\frac{\partial g_{1}(u_{1},…,u_{N})}{\partial u_{2}}\phi_{2} + … + \frac{\partial g_{1}(u_{1},…,u_{N})}{\partial u_{N}}\phi_{N}\right)u_{1}\phi_{1}dx = 0, \\ \int_{\Omega} & |\nabla\phi_{2}|^{2} – \left(g_{2}(u_{1},…,u_{N}) + u_{2}\frac{\partial g_{2}(u_{1},…,u_{N})}{\partial u_{2}}\right)\phi_{2}^{2}\\ & – \left(\frac{\partial g_{2}(u_{1},…,u_{N})}{\partial u_{1}}\phi_{1} + \frac{\partial g_{2}(u_{1},…,u_{N})}{\partial u_{3}}\phi_{3} + … + \frac{\partial g_{2}(u_{1},…,u_{N})}{\partial u_{N}}\phi_{N}\right)u_{2}\phi_{2}dx = 0 ,\\ &\qquad\qquad\qquad\vdots\,\,\, ,\\ \int_{\Omega} & |\nabla\phi_{N}|^{2} – \left(g_{N}(u_{1},…,u_{N}) + u_{N}\frac{\partial g_{N}(u_{1},…,u_{N})}{\partial u_{N}}\right)\phi_{N}^{2}\\ & – \left(\frac{\partial g_{N}(u_{1},…,u_{N})}{\partial u_{1}}\phi_{1} + … + \frac{\partial g_{N}(u_{1},…,u_{N})}{\partial u_{N-1}}\phi_{N-1}\right)u_{N}\phi_{N}dx = 0. \end{aligned}\]
Since \(\lambda_{1}(-g_{i}(u_{1},…,u_{N})) = 0\) for \(i = 1,…,N\), we see that \[\begin{aligned} \int_{\Omega}|\nabla\phi_{i}|^{2} – g_{i}(u_{1},…,u_{N})\phi_{i}^{2}dx \geq 0, i = 1,…,N. \end{aligned}\]
Hence, \[\begin{aligned} \int_{\Omega} – u_{1}\frac{\partial g_{1}(u_{1},…,u_{N})}{\partial u_{1}}\phi_{1}^{2} -\left(\frac{\partial g_{1}(u_{1},…,u_{N})}{\partial u_{2}}\phi_{2} +… + \frac{\partial g_{1}(u_{1},…,u_{N})}{\partial u_{N}}\phi_{N}\right)u_{1}\phi_{1}dx &\leq 0 ,\\ \int_{\Omega} – u_{2}\frac{\partial g_{2}(u_{1},…,u_{N})}{\partial u_{2}}\phi_{2}^{2} -\left(\frac{\partial g_{2}(u_{1},…,u_{N})}{\partial u_{1}}\phi_{1} + \frac{\partial g_{2}(u_{1},…,u_{N})}{\partial u_{3}}\phi_{3} +… + \frac{\partial g_{2}(u_{1},…,u_{N})}{\partial u_{N}}\phi_{N}\right)u_{2}\phi_{2}dx & \leq 0,\\ &\vdots\,\, ,\\ \int_{\Omega} – u_{N}\frac{\partial g_{N}(u_{1},…,u_{N})}{\partial u_{N}}\phi_{N}^{2} – \left(\frac{\partial g_{N}(u_{1},…,u_{N})}{\partial u_{1}}\phi_{1} + … + \frac{\partial g_{N}(u_{1},…,u_{N})}{\partial u_{N-1}}\phi_{N-1}\right)u_{N}\phi_{N}dx &\leq 0. \end{aligned}\]
Therefore, \[\int_{\Omega}-\sum\limits_{i=1}^{N}u_{i}\frac{\partial g_{i}(u_{1},…,u_{N})}{\partial u_{i}}\phi_{i}^{2} – \sum\limits_{i=1}^{N}u_{i}\phi_{i}\sum\limits_{j=1,j\neq i}^{N}\frac{\partial g_{i}(u_{1},…,u_{N})}{\partial u_{j}}\phi_{j}dx \leq 0.\]
It implies that \[\int_{\Omega}\sum\limits_{i=1}^{N}(-u_{i}\frac{\partial g_{i}(u_{1},…,u_{N})}{\partial u_{i}}\phi_{i}^{2} – \sum\limits_{j=1,j\neq i}^{N}\frac{\partial g_{i}(u_{1},…,u_{N})}{\partial u_{j}}u_{i}\phi_{j}\phi_{i})dx \leq 0.\]
Hence, \[\begin{aligned} &\int_{\Omega}-u_{1}(g_{1})_{u_{1}}(u_{1},…,u_{N})(\phi_{1})^{2} – \sum\limits_{j=2}^{N}(g_{1})_{u_{j}}(u_{1},…,u_{N})u_{1}\phi_{j}\phi_{1}+ \sum\limits_{i=2}^{N}[-u_{i}(g_{i})_{u_{i}}(u_{1},…,u_{N})(\phi_{i})^{2}\\ &\qquad – (g_{i})_{u_{1}}(u_{1},…,u_{N})u_{i}\phi_{1}\phi_{i}- \sum\limits_{j=2,j\neq i}^{N}(g_{i})_{u_{j}}(u_{1},…,u_{N})u_{i}\phi_{j}\phi_{i}]dx \leq 0. \end{aligned}\]
But, since \(-(g_{i})_{u_{1}}(u_{1},…,u_{N})u_{i}\phi_{1}\phi_{i} \geq -(g_{i})_{u_{1}}(u_{1},…,u_{N})u_{i}(\frac{(\phi_{1})^{2}}{2} + \frac{(\phi_{i})^{2}}{2})\), for all \(i = 2,…,N\), \[\begin{aligned} &\int_{\Omega}-u_{1}(g_{1})_{u_{1}}(u_{1},…,u_{N})(\phi_{1})^{2} – \sum\limits_{j=2}^{N}(g_{1})_{u_{j}}(u_{1},…,u_{N})u_{1}\phi_{j}\phi_{1}+ \sum\limits_{i=2}^{N}\Bigg[-u_{i}(g_{i})_{u_{i}}(u_{1},…,u_{N})(\phi_{i})^{2}\\ &\quad – (g_{i})_{u_{1}}(u_{1},…,u_{N})u_{i}(\frac{(\phi_{1})^{2}}{2} + \frac{(\phi_{i})^{2}}{2})- \sum\limits_{j=2,j\neq i}^{N}(g_{i})_{u_{j}}(u_{1},…,u_{N})u_{i}\phi_{j}\phi_{i}\Bigg]dx \leq 0. \end{aligned}\]
Hence, \[\begin{aligned} &\int_{\Omega}\left[-u_{1}(g_{1})_{u_{1}}(u_{1},…,u_{N}) – \sum\limits_{i=2}^{N}\frac{(g_{i})_{u_{1}}(u_{1},…,u_{N})}{2}u_{i}\right](\phi_{1})^{2}- \sum\limits_{j=2}^{N}(g_{1})_{u_{j}}(u_{1},…,u_{N})u_{1}\phi_{j}\phi_{1}\\ & + \sum\limits_{i=2}^{N}\left[(-u_{i}(g_{i})_{u_{i}}(u_{1},…,u_{N}) – \frac{(g_{i})_{u_{1}}(u_{1},…,u_{N})}{2})(\phi_{i})^{2} – \sum\limits_{j=2,j\neq i}^{N}(g_{i})_{u_{j}}(u_{1},…,u_{N})u_{i}\phi_{j}\phi_{i}\right]dx \leq 0. \end{aligned}\]
Since \[\begin{aligned} -(g_{1})_{u_{j}}(u_{1},…,u_{N})u_{1}\phi_{j}\phi_{1} \leq& -(g_{1})_{u_{j}}(u_{1},…,u_{N})u_{1}[\frac{(\phi_{j})^{2}}{2} + \frac{(\phi_{1})^{2}}{2}], j = 2,…,N,\\ -(g_{i})_{u_{j}}(u_{1},…,u_{N})u_{i}\phi_{j}\phi_{i} \leq& -(g_{i})_{u_{j}}(u_{1},…,u_{N})u_{i}[\frac{(\phi_{j})^{2}}{2} + \frac{(\phi_{i})^{2}}{2}], i,j = 2,…,N, i \neq j, \end{aligned}\] if \[\begin{aligned} -u_{1}(g_{1})_{u_{1}}(u_{1},…,u_{N}) – \frac{1}{2}\sum\limits_{i=2}^{N}(g_{i})_{u_{1}}(u_{1},…,u_{N})u_{i} >& -\frac{1}{2}\sum\limits_{j=2}^{N}\left[(g_{1})_{u_{j}}(u_{1},…,u_{N})u_{1} + (g_{j})_{u_{1}}(u_{1},…,u_{N})u_{j}\right],\\ -u_{i}(g_{i})_{u_{i}}(u_{1},…,u_{N}) – \frac{(g_{i})_{u_{1}}(u_{1},…,u_{N})}{2}u_{i} >& -\frac{1}{2}\sum\limits_{j=2,j\neq i}^{N}\Big[(g_{i})_{u_{j}}(u_{1},…,u_{N})u_{i}\\& + (g_{j})_{u_{i}}(u_{1},…,u_{N})u_{j}\Big], i = 2,…,N, \end{aligned}\] then the integrand in above inequality is positive definite, which means \((\phi_{1},…,\phi_{N})\) is trivial. But, it holds if the conditions are satisfied. ◻
Combining Lemma 4, Theorem 4, Theorem 5, and Lemma 5, we obtain the following corollary.
Corollary 1. If \((g_{1},…,g_{N}) \in B\) is such that
(A) \[\sup(g_{1})_{u_{1}}\sup(g_{i})_{u_{i}} + (N – 1)\inf(g_{1})_{u_{i}}\sup(g_{i})_{u_{1}} > 0, \,\,\,i = 2,…,N,\]
(B) \[g_{1}(0,…,0) > \frac{\sup(g_{1})_{u_{1}}\sup(g_{i})_{u_{i}}\left(\lambda_{1} + \frac{(N – 1)\inf(g_{1})_{u_{i}}g_{i}(0,…,0)}{\sup(g_{i})_{u_{i}}}\right)}{\sup(g_{1})_{u_{1}}\sup(g_{i})_{u_{i}} + (N – 1)\inf(g_{1})_{u_{i}}\sup(g_{i})_{u_{1}}},\] and
\[g_{i}(0,…,0) > \lambda_{1} – (N – 2)\inf_{j=2,…,N, j\neq i}\{\bar{u_{j}}\inf(g_{i})_{u_{j}}\}, \,\,\,i = 2,…,N,\]
(C) \[\begin{aligned} g_{1}(0,M_{2},…,M_{i-1},0,M_{i+1},…,M_{N}) + \inf(g_{1})_{x_{i}}\left[\frac{g_{1}(0,…,0)\sup(g_{i})_{x_{1}} – g_{i}(0,…,0)\sup(g_{1})_{x_{1}}}{\sup(g_{1})_{x_{1}}\sup(g_{i})_{x_{i}}}\right] >& \lambda_{1},\\ g_{1}(M_{1},…,M_{i-1},0,M_{i+1},…,M_{N}) >& \lambda_{1}, \qquad \text{and}\\ g_{i}(0,M_{2},…,M_{i-1},0,M_{i+1},…,M_{N}) >& \lambda_{1}, i = 2,…,N, \end{aligned}\] and
(D) \[\begin{aligned} &2\sup(g_{1})_{x_{1}} + \sum\limits_{i=2}^{N}\sup(g_{i})_{x_{1}}\sup\frac{\theta_{g_{i}(0,…,0,\cdot,0,…,0) – \frac{g_{1}(0,…,0)\sup(g_{i})_{x_{1}}}{\sup(g_{1})_{u_{1}}}}}{\theta_{M(g_{1},g_{i})}}\\ &\qquad< \sum\limits_{j=2}^{N}\left[\inf(g_{1})_{x_{j}} + \inf(g_{j})_{x_{1}}\inf\frac{\theta_{g_{j}(0,M_{2},…,M_{j-1},\cdot,M_{j+1},…,M_{N})}}{\theta_{g_{1}(\cdot,0,…,0)}}\right], \end{aligned}\] and \[\begin{aligned} &2\sup(g_{i})_{x_{i}} + \sup(g_{i})_{x_{1}}\\ &\quad < \sum\limits_{j=2,j\neq i}^{N}\left[\inf(g_{i})_{x_{j}} + \inf(g_{j})_{x_{i}}\sup\frac{\theta_{g_{j}(0,…,0,\cdot,0,…,0) – \frac{g_{1}(0,…,0)\sup(g_{j})_{x_{1}}}{\sup(g_{1})_{x_{1}}}}}{\theta_{g_{i}(0,M_{2},…,M_{i-1},\cdot,M_{i+1},…,M_{N})}}\right]\text{ for }i = 2,…,N, \end{aligned}\] then there is a neighborhood \(V\) of \((g_{1},…,g_{N})\) such that if \((\bar{g_{1}},…,\bar{g_{N}}) \in V\), then (1) with \((\bar{g_{1}},…,\bar{g_{N}})\) has a unique positive solution.
In biological terms, the result obtained in Corollary 1 confirms that under certain conditions, two species who relax ecologically can continue to coexist at fixed rates. The requirements given in \((A)\) and \((B)\) simply state that each species must interact strongly with itself and weakly with the other species.
The following Theorem is the main result.
Theorem 6. Suppose \(\Gamma \subseteq B\) be a closed, bounded, convex region in \(B\) such that
(A) for all \((g_{1},…,g_{N}) \in \Gamma\), \[\sup(g_{1})_{u_{1}}\sup(g_{i})_{u_{i}} + \inf(g_{1})_{u_{i}}\sup(g_{i})_{u_{1}} > 0, i = 2,…,N,\]
(B) for all \((g_{1},…,g_{N}) \in \Gamma\), \[\begin{aligned} g_{1}(0,M_{2},…,M_{i-1},0,M_{i+1},…,M_{N}) + \inf(g_{1})_{u_{i}}\left[\frac{g_{1}(0,…,0)\sup(g_{i})_{u_{1}} – g_{i}(0,…,0)\sup(g_{1})_{u_{1}}}{\sup(g_{1})_{u_{1}}\sup(g_{i})_{u_{i}}}\right] >& \lambda_{1},\\ g_{1}(M_{1},M_{2},…,M_{i-1},0,M_{i+1},…,M_{N}) >& \lambda_{1},\\ g_{i}(0,M_{2},…,M_{i-1},0,M_{i+1},…,M_{N}) > &\lambda_{1}, i = 2,…,N, \end{aligned}\]
(C) for all \((g_{1},…,g_{N}) \in \partial_{L}\Gamma\), (1) has a unique positive solution, where \(\partial_{L}\Gamma = \{(\lambda_{g_{2},…,g_{N}},g_{2},…,g_{N}) \in \Gamma |\;\;\mbox{for any fixed}\;\;g_{2},…,g_{N}, \lambda_{g_{2},…,g_{N}} = \inf\{\parallel g_{1} \parallel | (g_{1},g_{2},…,g_{N}) \in \Gamma\}.\}\),
(D) for all \((g_{1},…,g_{N}) \in \Gamma\), the Frechet derivative of (1) at every positive solution \((u_{1},…,u_{N})\) is invertible.
Then for all \((g_{1},…,g_{N}) \in \Gamma\), (1) has a unique positive solution. Furthermore, there is an open set \(W\) in \(B\) such that \(\Gamma \subseteq W\) and for every \((g_{1},…,g_{N}) \in W\), (1) has a unique positive solution.
Theorem 6 goes even further than Theorem 5 which states uniqueness in the whole region whenever we have uniqueness on the left boundary and invertibility of the linearized operator at any particular solution inside the domain.
Proof. For each fixed \(g_{2},…,g_{N}\), consider \((g_{1},g_{2},…,g_{N}) \in \partial_{L}\Gamma\) and \((\bar{g_{1}},g_{2},…,g_{N}) \in \Gamma\). We need to show that for all \(0 \leq t \leq 1\), (1) with \((1 – t)(g_{1},…,g_{N}) + t(\bar{g_{1}},g_{2},…,g_{N})\) has a unique positive solution. Since (1) with \((g_{1},…,g_{N})\) has a unique positive solution \((u_{1},…,u_{N})\) and the Frechet derivative of (1) at \((u_{1},…,u_{N})\) is invertible, Theorem 5 implies that there is an open neighborhood \(V\) of \((g_{1},…,g_{N})\) in \(B\) such that if \((g_{10},…,g_{N0}) \in V\), then (1) with \((g_{10},…,g_{N0})\) has a unique positive solution.
Let \(\lambda_{s} = \sup\{0 \leq \lambda \leq 1 |\;\;\mbox{(\ref{eq:a}) with}\;\;(1 – t)(g_{1},…,g_{N}) + t(\bar{g_{1}},g_{2},…,g_{N})\) has a unique coexistence state for \(0 \leq t \leq \lambda.\}\). We need to show that \(\lambda_{s} = 1\). Suppose \(\lambda_{s} < 1\). From the definition of \(\lambda_{s}\), there is a sequence \(\{\lambda_{n}\}\) such that \(\lambda_{n} \rightarrow \lambda_{s}^{-}\) and there is a sequence \((u_{1n},…,u_{Nn})\) of the unique positive solutions of ([eq:a]) with \((1 – \lambda_{n})(g_{1},…,g_{N}) + \lambda_{n}(\bar{g_{1}},g_{2},…,g_{N})\). Then by elliptic theory, there is \((u_{10}…,,u_{N0})\) such that \((u_{1n},…,u_{Nn})\) converges to \((u_{10},…,u_{N0})\) uniformly and \((u_{10},…,u_{N0})\) is a solution of (1) with \((1 – \lambda_{s})(g_{1},…,g_{N}) + \lambda_{s}(\bar{g_{1}},g_{2},…,g_{N})\).
But, by the same proof as in the §7, \(u_{10} > 0,…,u_{N0} > 0\).
We claim that (1) has a unique coexistence state with \((1 – \lambda_{s})(g_{1},…,g_{N}) + \lambda_{s}(\bar{g_{1}},g_{2},…,g_{N})\). In fact, if not, assume that \((\bar{u}_{10},…,\bar{u}_{N0}) \neq (u_{10},…,u_{N0})\) is another coexistence state. By the Implicit Function Theorem, there exists \(0 < \tilde{a} < \lambda_{s}\) and very close to \(\lambda_{s}\) such that (1) with \((1 – \tilde{a})(g_{1},…,g_{N}) + \tilde{a}(\bar{g_{1}},g_{2},…,g_{N})\) has a coexistence state very close to \((\bar{u}_{10},…,\bar{u}_{N0})\), which means that (1) with \((1 – \tilde{a})(g_{1},…,g_{N}) + \tilde{a}(\bar{g_{1}},g_{2},…,g_{N})\) has more than one coexistence state. This is a contradiction to the definition of \(\lambda_{s}\). But, since (1) with \((1 – \lambda_{s})(g_{1},…,g_{N}) + \lambda_{s}(\bar{g_{1}},g_{2},…,g_{N})\) has a unique coexistence state and the Frechet derivative is invertible, Theorem 5 implies that \(\lambda_{s}\) can not be as defined. Therefore, for each \((g_{1},..,g_{N}) \in \Gamma\), (1) with \((g_{1},…,g_{N})\) has a unique coexistence state \((u_{1},…,u_{N})\). Furthermore, by the assumption, for each \((g_{1},…,g_{N}) \in \Gamma\), the Frechet derivative of (1) with \((g_{1},…,g_{N})\) at the unique solution \((u_{1},…,u_{N})\) is invertible. Hence, Theorem 5 concluded that for each \((g_{1},…,g_{N}) \in \Gamma\), there is an open neighborhood \(V_{(g_{1},…,g_{N})}\) of \((g_{1},…,g_{N})\) in \(B\) such that if \((\bar{g_{1}},…,\bar{g_{N}}) \in V_{(g_{1},…,g_{N})}\), then (1) with \((\bar{g_{1}},…,\bar{g_{N}})\) has a unique coexistence state. Let \(W = \bigcup_{(g_{1},…,g_{N})\in\Gamma}V_{(g_{1},…,g_{N})}\). Then \(W\) is an open set in \(B\) such that \(\Gamma \subseteq W\) and for each \((\bar{g_{1}},…,\bar{g_{N}}) \in W\), (1) with \((\bar{g_{1}},…,\bar{g_{N}})\) has a unique coexistence state.
Apparently, Theorem 6 generalizes Theorem 5. ◻
Within the academia of mathematical biology, extensive academic work has been devoted to investigation of the simple predator-prey model, commonly known as the Lotka-Volterra predator-prey model. This system describes the predator-prey interaction of two species residing in the same environment in the following manner:
Suppose \(N\) species of animals with predator-prey interaction, rabbits and tigers for instance, are residing in a bounded domain \(\Omega\). Let \(u_{i}(x,t)\) be densities of prey(\(i = 1\)) and predators(\(i = 2,…,N\)) in the place \(x\) of \(\Omega\) at time \(t\), respectively. Then we have the dynamic predator-prey model \[\left\{ \begin{array}{l} \left.\begin{array}{l} (u_{i})_{t}(x,t) = \Delta (u_{i})(x,t) + u_{i}(x,t)(a_{i} + \sum\limits_{j=1}^{N}b_{ij}u_{j}(x,t))\\ \end{array} \right.\;\;\mbox{in}\;\;\Omega \times [0,\infty),\\ (u_{i})(x,t) = 0, i = 1,…,N\;\;\mbox{for}\;\;x \in \partial\Omega, \end{array} \right.\] where \(a_{i} > 0, i = 1,…,N\) are reproduction rates and \(b_{ij}\) are self-limitation and competition rates such that \[\left.\begin{array}{lll} b_{ij} & \left\{\begin{array}{l} < 0, i = 1, j = 1,…,N,\\ > 0, i = 2,…,N, j=1,\\ < 0, i = 2,…,N, j = 2,…,N. \end{array}\right. \end{array}\right.\]
Here we are interested in the time independent, positive solutions, i.e. the positive solutions \(u_{i}(x), i = 1,…,N\) of the elliptic interacting system of \(N\) functions with homogeneous boundary conditions \[\left\{\begin{array}{l} \left.\begin{array}{rcl} \Delta u_{i} + u_{i}(a_{i} + \sum\limits_{j=1}^{N}b_{ij}u_{j}) & = & 0\\ \end{array}\right.\;\;\mbox{in}\;\;\Omega,\\ u_{i}|_{\partial\Omega} = 0, i = 1,…,N, \end{array}\right.\label{eq:1} \tag{8}\] which are called the coexistence state or the steady state. The coexistence state is the positive density solution depending only on the spatial variable \(x\), not on the time variable \(t\), and so, its existence means that the two species of animals can live peacefully and forever.
We establish the following existence result from the Theorem 1:
Corollary 2. If for each \(i = 2,…,N\),
(A) \[b_{11}b_{ii} + (N – 1)b_{1i}b_{i1} > 0, i = 2,…,N,\]
(B) \[a_{1} > \frac{b_{11}b_{ii}(\lambda_{1} + \frac{(N – 1)b_{1i}a_{i}}{b_{ii}})}{b_{11}b_{ii} + (N – 1)b_{1i}b_{i1}}\] and
(C) \[a_{i} > \lambda_{1} – (N – 2)\inf_{j=2,…,N, j\neq i}\{\bar{u_{j}}b_{ij}\}, i = 2,…,N,\] where \(\bar{u_{j}}\) is defined below, then (8) has a positive solution.
Biologically, the conditions in Corollary 2 implies that if the self-reproduction and self-limitation rates are relatively large, and the competition rates are relatively small, in other words, if members of each species interact strongly among themselves and weakly with members of the other species, then there is a positive steady state solution to (8), that is, the two species within the same domain will coexist indefinitely at population densities.
We also establish the following nonexistence results from the Theorem 2, which means that they can not peacefully coexist, in other words, they are extinct, if they don’t have sufficiently large reproduction capacities.
Corollary 3. Suppose \(a_{i} \leq \lambda_{1}, i = 1,…,N\). Then \(u_{i} \equiv 0, i = 1,…,N\) is the only nonnegative solution to (8).
The Theorem 4 derives the following uniqueness result:
Corollary 4. In addition to the Corollary 2, if
(A) \[a_{1} + \sum\limits_{j=2,j\neq i}^{N}b_{1j}M_{j} + b_{1i}\left[\frac{a_{1}b_{i1} – a_{i}b_{11}}{b_{11}b_{ii}}\right] > \lambda_{1},\,\, a_{i} + \sum\limits_{j=2,j\neq i}^{N}b_{ij}M_{j} > \lambda_{1}, i = 2,…,N,\] and
(B) \[2b_{11} + \sum\limits_{i=2}^{N}b_{i1}\sup\frac{-\frac{1}{b_{ii}}\theta_{a_{i} – \frac{a_{1}b_{i1}}{b_{11}}}}{-\frac{1}{b_{11}}\theta_{a_{1}+\sum\limits_{j=2,j\neq i}^{N}b_{1j}M_{j}+b_{1i}\frac{a_{1}b_{i1}-a_{i}b_{11}}{b_{11}b_{ii}}}} < \sum\limits_{j=2}^{N}\left(b_{1j} \\+ b_{j1}\inf\frac{-\frac{1}{b_{jj}}\theta_{a_{j} + \sum\limits_{k=2,k\neq j}^{N}b_{jk}M_{k}}}{-\frac{1}{b_{11}}\theta_{a_{1}}}\right),\] and \[2b_{ii} + b_{i1} < \sum\limits_{j=2,j\neq i}^{N}\left(b_{ij} + b_{ji}\sup\frac{-\frac{1}{b_{jj}}\theta_{a_{j} – \frac{a_{1}b_{j1}}{b_{11}}}}{-\frac{1}{b_{ii}}\theta_{a_{i}+\sum\limits_{k=2,k\neq i}^{N}b_{ik}M_{k}}}\right), \text{ for }i = 2,…,N,\] then (8) has a unique posotive solution.
The conditions imply that species interacts strongly among themselves and weakly with others. The following theorem is our main result about the perturbation of uniqueness derived from the Corollary 1:
Corollary 5. If
(A) \[b_{11}b_{ii} + (N – 1)b_{1i}b_{i1} > 0,\text{ for }i = 2,…,N,\]
(B) \[\begin{aligned} a_{1} >& \frac{b_{11}b_{ii}(\lambda_{1} + \frac{(N – 1)b_{1i}a_{i}}{b_{ii}})}{b_{11}b_{ii} + (N – 1)b_{1i}b_{i1}},\\ a_{i} >& \lambda_{1} – (N – 2)\inf_{j=2,…,N, j\neq i}\{b_{ij}\bar{u_{j}}\}, \end{aligned}\]
(C) \[\begin{aligned} a_{1} + \sum\limits_{j=2,j\neq i}^{N}b_{1j}M_{j} + b_{1i}\left(\frac{a_{1}b_{i1} – a_{i}b_{11}}{b_{11}b_{ii}}\right) >& \lambda_{1},\\ a_{1} + \sum\limits_{j=1,j\neq i}^{N}b_{1j}M_{j} >& \lambda_{1},\\ a_{i} + \sum\limits_{j=2,j\neq i}^{N}b_{ij}M_{j} >& \lambda_{1}, i = 2,…,N, \end{aligned}\] and
(D) \[2b_{11} + \sum\limits_{i=2}^{N}b_{i1}\sup\frac{-\frac{1}{b_{ii}}\theta_{a_{i} – \frac{a_{1}b_{i1}}{b_{11}}}}{-\frac{1}{b_{11}}\theta_{a_{1}+\sum\limits_{j=2,j\neq i}^{N}b_{1j}M_{j}+b_{1i}\frac{a_{1}b_{i1}-a_{i}b_{11}}{b_{11}b_{ii}}}}< \sum\limits_{j=2}^{N}\left(b_{1j} + b_{j1}\inf\frac{-\frac{1}{b_{jj}}\theta_{a_{j}+\sum\limits_{k=2,k\neq j}^{N}b_{jk}M_{k}}}{-\frac{1}{b_{11}}\theta_{a_{1}}}\right),\] and
\[2b_{ii} + b_{i1} < \sum\limits_{j=2,j\neq i}^{N}\left(b_{ij} + b_{ji}\sup\frac{-\frac{1}{b_{jj}}\theta_{a_{j} – \frac{a_{1}b_{j1}}{b_{11}}}}{-\frac{1}{b_{ii}}\theta_{a_{i}+\sum\limits_{k=2,k\neq i}^{N}b_{ik}M_{k}}}\right)\text{ for }i = 2,…,N,\] then there is a neighborhood \(V\) of \((a_{1},…,a_{N})\) such that if \((\bar{a_{1}},…,\bar{a_{N}}) \in V\), then (8) with \((\bar{a_{1}},…,\bar{a_{N}})\) has a unique positive solution.
In biological terms, the result obtained in Corollary 5 confirms that under certain conditions, the species who relax ecologically can continue to coexist at fixed rates. The requirements given in \((A)\) and \((B)\) simply state that each species must interact strongly with itself and weakly with the other species.
These results provide insight into the predator-prey interactions of species operating under the conditions described in the Lotka-Volterra model.
In this paper, our investigation of the effects of perturbations on the general predator-prey model resulted in the development and proof of Theorem 5, Lemma 5, and Corollary 1 as detailed above. The three together assert that given the existence of a unique solution \((u_{1},…,u_{N})\) to the system ([eq:5]), perturbations of the birth rates \((g_{1},…,g_{N})\), within a specified neighborhood, will maintain the existence and uniqueness of the positive steady state. Indeed, our results specifically outline conditions sufficient to maintain the positive, steady state solution when the general predator-prey model is perturbed within some region.
Applying this mathematical result to real world situations, our results establish that the species residing in the same environment can vary their interactions, within certain bounds, and continue to survive together indefinitely at unique densities. The conditions necessary for coexistence, as described in the theorem, simply require that members of each species interact strongly with themselves and weakly with members of the other species.
The research presented in this paper has a number of strengths, which confirm both the validity and the applicability of the project. First, the mathematical conditions required in Corollary 1 are identical to those required in Theorem 4. However, in the Theorem 4, we used these conditions to prove the existence and uniqueness of the positive steady state solution for the general predator-prey model. In contrast, the Corollary 1 employs the same conditions to establish that the existence and uniqueness of this solution is maintained when the model is perturbed within some neighborhood. Thus, our findings extend and improve established mathematical theory.
Secondly, perturbations of the general model render its implications more applicable both mathematically and biologically. Because our theorem extends the steady state to any value within some neighborhood of \((g_{1},…,g_{N})\), results for the general model pertain to a far wider variety of values. Biologically, perturbations extend the model’s description to species affected by factors that vary slightly yet erratically. Thus, the description of competitive interactions given by the model becomes a closer approximation of real-world population dynamics.
While our research therefore represents a progression in the field, the results obtained have an important limitation. Theorem 5, Lemma 5, and Corollary 1 establish that a region of perturbation exists within which the coexistence state is maintained for the general predator-prey model. However, the exact extent of that region remains unknown. Therefore, the results presented in this paper may serve as a platform for research of the question given above. Mathematicians should now attempt to establish the exact extent of the perturbation region in which coexistence is maintained for the general model. Such information would prove very useful not only mathematically but also biologically. Specifically, knowledge of the extent of the region would imply exactly how far the species can relax and yet continue to coexist. Thus, the results achieved through our research will enable the field to continue the development of theory on predator-prey interaction of populations.
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Cosner, C., & Lazer, A. C. (1984). Stable coexistence states in the Volterra–Lotka competition model with diffusion. SIAM Journal on Applied Mathematics, 44(6), 1112-1132.
