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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.

MSC:

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