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Zbl 0930.34038
On a property of nonautonomous Lotka-Volterra competition model.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 37, No. 5, B, 603-611 (1999). ISSN 0362-546X

The authors consider the Lotka-Volterra computation model $$u_i'(t)= u_i(t) \Biggl[a_i(t)- \sum^n_{j= 1} b_{ij}(t) u_j(t)\Biggr].\tag LV$$ Under the following assumptions that $$(1)\quad b_{ii}(t)>0\text{ for all }t\in\bbfR\text{ and }1\le i\le n,\qquad (2)\quad \int^\infty_0 b_{ii}(t) dt= \infty,\ 1\le i\le n,$$ $$(3)\quad \sup_{t\in\bbfR} {a_i(t)\over b_{ii}(t)}= \Biggl({a_i\over b_{ii}}\Biggr)_M< \infty,\qquad (4)\quad \inf_{t\in\bbfR} {a_i(t)- \sum^n_{j=1} b_{ij}(t)({a_j\over b_{jj}})_M\over b_{ii}(t)}> 0,$$ the authors prove that $$\mu(u(t, t_0, A))\to 0\quad\text{as }t\to\infty,$$ where $u(t, t_0,A)$ denotes the set of all points in $\bbfR^n$ of $u(t)$, $u(t)$ is a solution to (LV) with $u(t_0)\in A$, $\mu(u(t, t_0, A))$ denotes the $n$-dimensional Lebesgue measure of $u(t, t_0,A)$.
[Chen Lan Sun (Beijing)]
MSC 2000:
*34D40 Ultimate boundedness
92D25 Population dynamics
34D35 Stability of manifolds of solutions of ODE

Keywords: nonautonomous systems; Lotka-Volterra systems; ultimate boundedness

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