Average growth and extinction in a two dimensional Lotka-Volterra system. (English) Zbl 1081.34513

The authors consider the following nonautonomous two-species Lotka-Volterra competitive system \[ u_1'(t)=u_1(t)\bigl[a_1(t)-b_{11} (t)u_1(t)-b_{12}(t)u_2(t) \bigr], \]
\[ u_2'(t)=u_2(t)\bigl[a_2(t)-b_{21} (t)u_1(t)-b_{22}(t)u_2(t)\bigr], \] A new criterion on the extinction of species is obtained in terms of an average growth rate. That is, if \(0< a_{kL}\leq a_{kM}<\infty\), \(0<b_{kkL}\leq b_{kkM}< \infty\), \(0\leq b_{kjL}\leq b_{kjM}<\infty\), \(k,j=1,2,k\neq j\), and \[ m[a_1]> b_{12M} \frac{M[a_2]}{b_{22L}},\quad M[a_2]<b_{21L}\frac{m[a_1]} {b_{11M}}, \] then \(u_2(t)\to 0\) and \(u_1(t)\to u^*(t)\) as \(t\to\infty\) for any positive solution \((u_1(t),u_2(t))\), where \(a_{kL},b_{kjL}\) and \(a_{kM},b_{kjM}\) express the infimum and supremum of the functions, respectively, \[ M[a_k]=\limsup_{r\to \infty}\left\{\frac{1}{t_2-t_1} \int^{t_2}_{t_1}a_k(s)ds,\;t_2-t_1\geq r\right \}, \]
\[ m[a_k]= \liminf_{r\to\infty}\left\{\frac{1}{t_2-t_1}\int^{t_2}_{t_1} a_k(s)ds,\;t_2-t_1\geq r\right\}, \] and \(u^*(t)\) is the unique solution of the equation \(u'(t)=u(t)[a_1(t)-b_{11}(t)u(t)]\) such that \(0<\delta\leq u^*(t)\leq \Delta\), where \(\delta\) and \(\Delta\) are constant.


34C11 Growth and boundedness of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
92D25 Population dynamics (general)