A Class Of Plankton Allelopathy Model Of Multiple Positive Periodic Solution

In this paper, we study a class of delay differential equations models and difference equations models which describes the growth of two species of plankton with competitive and allelopathic effects on each other. A set of easily verifiable sufficient conditions are obtained for the existence of multiple positive periodic solutions for these two models. The approach is based on Mawhin's continuation theorem of coincidence degree theory as well as some priori estimates. This is the first time that these models have been studied by using this method.The whole paper consists of three chapters. In Chapter 1, we introduce some elementary concepts and results from coincidence degree theory. In Chapter 2, we consider a delay differential equations model of plankton allelopathy proposed by Maynard-Smith, Chattopadhyay, Mukhopathyay, Jin Zhen and Ma Zhien et al. By applying Mawhin's continuation theorem of coincidence degree theory, we establish some easily verifiable sufficient conditions on the existence of multiple positive periodic solutions. In Chapter 3, with the help of differential equations with piecewise constant arguments, we propose a discrete analogue of a differential equations model of plankton allelopathy modified by Maynard-Smith and Chattopadhyay, which is governed by nonautonomous difference equations. A set of easily verifiable sufficient conditions are obtained for this discrete time model by Mawhin's continuation theorem.