On almost periodic solutions of Lotka-Volterra almost periodic competition systems. (English) Zbl 0781.34037

The author considers the system of Lotka-Volterra equations \(\dot x_ i= x_ i(b_ i(t)- \sum a_{ij}(t)x_ j)\), \(1\leq i\leq n\) with \(a_{ij}\), \(b_ i\) continuous and almost periodic. He proves the existence and uniqueness of a positive bounded and almost periodic solution \(x_ i\) for \(1\leq i\leq n\), if the following conditions are met: \(a_{ij}(t)>0\), \(a_{ii}(t)> \sum_{j+i} a_{ij}(t)+\mu\) for some \(\mu>0\), \(\overline {b}_ i=\lim {1\over T} \int^ T_ 0 b_ i(t)dt>0\) and \(\overline {B}_ i>0\) with \(B_ i(t)= b_ i(t)- \sum_{j\neq i} a_{ij}(t) \widetilde {x}_ j(t)\) and \(\widetilde {x}_ j\) the well known solution of the logistic equation \(\dot x_ i=x_ i\cdot (b_ i(t)- a_{ii}(t)x_ i)\). This result generalizes previous results of Gopalsamy (1985, 1986), Zhao Xiaoqiang (1991) and Ahmad (1988).
Reviewer: R.Repges (Aachen)


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