On a model of competition in periodic environments (Q1354134)

The authors study the following discrete-time analogue of the Lotka-Volterra competition model: \[ x_i(n+1)= b_i(n)x_i(n)\Biggl/ \Biggl[1+ \sum_{j=1}^m c_{ij}(n)x_j(n)\Biggr], \qquad i=1,\dots, m;\quad n\in\mathbb{N}, \] where \(\{b_i(n)\}\), \(\{c_{ij}(n)\}\) are positive periodic sequences with a common period. The sufficient conditions for the existence and global attractivity of a positive periodic solution are given. Asymptotic bounds for arbitrary positive solutions of the system in terms of related periodic solutions are also presented.