## 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)