Multiple periodic solutions for a discrete time model of plankton allelopathy

Abstract

We study a discrete time model of the growth of two species of plankton with competitive and allelopathic effects on each other N 1 ( k +1) = N 1 ( k )exp{ r 1 ( k )- a 11 ( k ) N 1 ( k )- a 12 ( k ) N 2 ( k )- b 1 ( k ) N 1 ( k ) N 2 ( k )}, N 2 ( k +1) = N 2 ( k )exp{ r 2 ( k )- a 21 ( k ) N 2 ( k )- b 2 ( k ) N 1 ( k ) N 1 ( k ) N 2 ( k )}. A set of sufficient conditions is obtained for the existence of multiple positive periodic solutions for this model. The approach is based on Mawhin's continuation theorem of coincidence degree theory as well as some a priori estimates. Some new results are obtained.