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Asymptotic behavior of solutions of a periodic reaction-diffusion system of a competitor-competitor-mutualist model. (English) Zbl 0806.35095

From the introduction: We consider a reaction-diffusion model of mutualism, Lotka-Volterra type, which involves interactions among a competitor, a mutualist-competitor, and a mutualist, in a periodic environment. More precisely, we study the asymptotic behaviour of positive solutions of the periodic reaction-diffusion system \[ \begin{aligned} u_{1t} = \alpha_ 1 (t,x) \Delta u_ 1 + \beta_ 1 (t,x)u_ 1 & \left[ 1 - {u_ 1 \over a_ 1(t,x)} - {a_ 2(t,x)u_ 2 \over 1 + a_ 3(t,x)u_ 3} \right] \\ u_{2t} = \alpha_ 2 (t,x) \Delta u_ 2 + \beta_ 2 (t,x) u_ 2 & \left[ 1 - b_ 1(t,x)u_ 1 - {u_ 2 \over b_ 2(t,x)} \right] \\ u_{3t} = \alpha_ 3 (t,x) \Delta u_ 3 + \beta_ 3 (t,x)u_ t & \left[ 1 - {u_ 3 \over c_ 0 (t,x) + c_ 1(t,x)u_ 1} \right] \end{aligned} \] in \([0,\infty) \times \Omega\), with the initial condition \(u_ i (0,x) = \varphi_ i(x)\) and homogeneous Neumann boundary conditions. Here, \(\Omega\) is a bounded domain of \(\mathbb{R}^ 3\), and \(\varphi_ 1, \varphi_ 2, \varphi_ 3 : \overline \Omega \to \mathbb{R} \) are smooth nonnegative functions such that \(\partial \varphi_ i/ \partial\nu = 0\) in \(\partial \Omega\), and \(\alpha_ i\), \(\beta_ i\), \(a_ i\), \(b_ i : \mathbb{R} \times \overline \Omega \to \mathbb{R}\) are positive continuous functions, periodic in the time variable \(t\), with period \(T>0\).
We prove theorems about the extinction of a competitor \((u_ 1\) or \(u_ 2)\) and on the existence of a \(T\)-periodic solution \((u_ 1, u_ 2, u_ 3)\) defined on \(\mathbb{R} \times \overline \Omega\), such that \(u_ i > 0\) \((i = 1,2,3)\).

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
92D25 Population dynamics (general)
35B40 Asymptotic behavior of solutions to PDEs
35B10 Periodic solutions to PDEs
35K57 Reaction-diffusion equations
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