Global dynamic behavior of a plant disease model with ratio dependent impulsive control strategy

Abstract

In this paper, we consider the dynamics of a plant disease model with a ratio-dependent state impulsive control strategy. It is shown that the boundary equilibrium point of the controlled system is globally asymptotically stable. By combining LaSalle’s invariant theorem, Brouwer’s fixed point theorem and some analysis techniques, we are able to determine the basic reproduction number, confirm the well-posedness of the model, describe the structure of possible equilibria as well as establish the stability of the equilibria. Most interestingly, we find that in the case that the basic reproduction number is more than unity and the endemic equilibrium locates above the impulsive control strategy, we can obtain a unique k-order periodic solution and the critical values between 1-order and 2-order periodic solutions. Furthermore, it is found that the endemic equilibrium point is also globally asymptotically stable under the control strategy. Finally, we present a numerical example to substantiate the effectiveness of the theoretical results.