On the Lotka-Volterra competition model in symmetric domains. (English) Zbl 0924.35057

The authors study the existence and the stability of multiple co-existence solutions and of periodic solutions of the Lotka-Volterra competition systems \[ \begin{cases} \Delta u+u(a-u-bv)= 0,\quad x\in\Omega,\\ \Delta v+ v(a-bu-v)=0,\\ Bu|_{\partial \Omega}= Bv|_{\partial \Omega}=0, \end{cases} \tag{1} \] or \[ \begin{cases} u_t=\Delta u+u\bigl(a(t) -u-bv\bigr),\;x\in \Omega, \;t\geq 0,\\ v_t=\Delta v+v\bigl(a(t)- bu-v\bigr),\\ Bu|_{\partial \Omega}= Bv |_{\partial \Omega}=0,\\ u(x,t+T)= u(x,t),\;v(x,t+T)= v(x,t), \end{cases} \tag{2} \] where \(\Omega\) is a smooth domain in \(\mathbb{R}^n\) with certain symmetry, \(T>0\), \(b>0\) are constants, \(a\) is a positive constant in (1) while it is a periodic function of \(t\) with period \(T\) in (2). These systems arise in mathematical biology and ecology, \(u\) and \(v\) may represent the population densities of two coexisting species which compete in \(\Omega\) and therefore are supposed to be positive \(\Omega\). The parameters \(a\) and \(b\) may denote the birth rate and competition rate, respectively. The symmetry between \(u\) and \(v\) in (1) or (2) implies that neither of these two species has advantages upon the other in the competition. The boundary condition posed for \(u\) and \(v\) is either Dirichlet or Neumann, depending on the relation between the environment and the outside world.


35K50 Systems of parabolic equations, boundary value problems (MSC2000)
92B05 General biology and biomathematics
35B10 Periodic solutions to PDEs
35B32 Bifurcations in context of PDEs
35B35 Stability in context of PDEs
35K57 Reaction-diffusion equations