## On the positive almost periodic solutions of a class of Lotka-Volterra type systems with delays.(English)Zbl 0967.34064

The following almost-periodic Lotka-Volterra type systems with delays are investigated ${dx_i(t)\over dt}= x_i(t)(a_i(t)- b_i(t) x_i(t)- f_i(t, x_t)),\quad i= 1,2,\dots, n,\tag{1}$ with $$t\in\mathbb{R}$$, $$x(t)= (x_1(t),x_2(t),\dots, x_n(t))\in \mathbb{R}^n$$, $$x_t= (x_{1t}, x_{2t},\dots, x_{nt})\in C^n[-\tau, 0]$$, and $$x_{it}(s)= x_i(t+ s)$$, $$i= 1,2,\dots, n$$, for all $$s\in [-\tau,0]$$; $$a_i(t)$$ and $$b_i(t)$$ are continuous almost-periodic functions with respect to $$t\in \mathbb{R}$$ and $$b_i(t)\geq 0$$ for all $$t\in\mathbb{R}$$.
The author establishes a general criterion for the existence of positive almost-periodic solutions to system (1). The conditions required in the criterion are quite weak, and can be easily checked and reduced to some well-known results.

### MSC:

 34K14 Almost and pseudo-almost periodic solutions to functional-differential equations 92D25 Population dynamics (general) 34K20 Stability theory of functional-differential equations
Full Text:

### References:

 [1] Ahmad, S.; Lazer, A.C., On the nonautonomous N-competing species problems, Appl. anal., 57, 309-323, (1995) · Zbl 0859.34033 [2] Ahmad, S.; Rao, M.R.Mohana, Asymptotically periodic solutions of N-competing species problem with time delays, J. math. anal. appl., 186, 559-571, (1994) · Zbl 0818.45004 [3] Bereketoglu, H.; Gyori, I., Global asymptotic stability in a nonautonomous lotka – volterra type system with infinite delay, J. math. anal. appl., 210, 279-291, (1997) · Zbl 0880.34072 [4] Deimling, K., Nonlinear functional analysis, (1988), Springer-Verlag/World Beijing [5] Fink, A.M., Almost periodic differential equations, Lecture notes in mathematics, 377, (1974), Springer-Verlag Berlin · Zbl 0325.34039 [6] He, C.Y., Almost periodic differential equations, (1992), Higher Education Beijing [7] Rong, Y.; Hong, J., The existence of almost periodic solution of a population equation with delay, Appl. anal., 61, 45-52, (1996) · Zbl 0879.34044 [8] Seifert, G., Almost periodic solutions for delay logistic equation with almost periodic time dependence, Differential integral equations, 9, 335-342, (1996) · Zbl 0838.34083 [9] Tineo, A., On the asymptotic behaviour of some population models, J. math. anal. appl., 167, 516-529, (1992) · Zbl 0778.92018
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.