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On the nonautonomous Volterra-Lotka competition equations. (English) Zbl 0848.34033

The nonautonomous, competitive Lotka-Volterra system \(u'(t) = u(t) [a(t) - b(t) u(t) - c(t) \nu (t)]\), \(\nu'(t) = \nu (t) [d(t) - e(t) u(t) - f(t) \nu (t)]\) is considered, in which \(a,b,c, d,e,f\) are continuous functions defined on the real line and bounded above and below by positive constants. Under the assumption that \(a_L f_L > c_M d_M\) and \(b_M d_M \leq a_L e_L\), where subscript \(L\) [resp. \(M]\) denotes the infimum [resp. supremum] of the given function, it is shown that if \((u(t), \nu (t))\) is a solution for which \(u(t_0) > 0\) and \(\nu (t_0) > 0\) for some \( t_0\), then \(\lim_{t \to + \infty} \nu (t) = 0\) and \(\lim_{t \to + \infty} [u(t) - u^* (t)] = 0\), where \(u^* (t)\) is the unique solution of the logistic equation \(u'(t) = u(t) [a(t) - b(t) u(t)]\) satisfying \(0 < \delta \leq u^* (t) \leq \Delta < \infty\) for all real \(t\) and some positive numbers \(\delta\) and \(\Delta\).

MSC:

34D05 Asymptotic properties of solutions to ordinary differential equations
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
92D25 Population dynamics (general)
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