## 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)

### Keywords:

nonautonomous, competitive Lotka-Volterra system
Full Text:

### References:

 [1] Shair Ahmad, Convergence and ultimate bounds of solutions of the nonautonomous Volterra-Lotka competition equations, J. Math. Anal. Appl. 127 (1987), no. 2, 377 – 387. · Zbl 0648.34037 [2] Shair Ahmad, On almost periodic solutions of the competing species problems, Proc. Amer. Math. Soc. 102 (1988), no. 4, 855 – 861. · Zbl 0668.34042 [3] Shair Ahmad and Alan C. Lazer, Asymptotic behaviour of solutions of periodic competition diffusion system, Nonlinear Anal. 13 (1989), no. 3, 263 – 284. · Zbl 0686.35060 [4] Carlos Alvarez and Alan C. Lazer, An application of topological degree to the periodic competing species problem, J. Austral. Math. Soc. Ser. B 28 (1986), no. 2, 202 – 219. · Zbl 0625.92018 [5] Bernard D. Coleman, Nonautonomous logistic equations as models of the adjustment of populations to environmental change, Math. Biosci. 45 (1979), no. 3-4, 159 – 173. · Zbl 0425.92013 [6] K. Gopalsamy, Exchange of equilibria in two-species Lotka-Volterra competition models, J. Austral. Math. Soc. Ser. B 24 (1982/83), no. 2, 160 – 170. · Zbl 0498.92016 [7] J. M. Smith, Mathematical ideas in biology, Cambridge Univ. Press, London, 1968. [8] L. Zhou and C. V. Pao, Asymptotic behavior of a competition-diffusion system in population dynamics, Nonlinear Anal. 6 (1982), no. 11, 1163 – 1184. · Zbl 0522.92017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.