Modeling and analysis of plant disease with delay and logistic growth of insect vector

Abstract

Plants are essential for the survival of human beings. Plants can be subjected to diseases. Plant diseases are caused by pathogens such as fungi, bacteria and viruses. Most of these pathogens are transmitted by insect vectors. In this paper we formulate and analyze a delay differential equation model for plant disease by incorporating the incubation delay which is the time taken for a plant to become infected. The mathematical model is formulated by considering both the plant and the insect vector populations. The total plant population is taken as a constant and the insect vector population is taken as variable. It is assumed that the insect vector population is growing logistically in the environment. The existence and stability of equilibria of the model are discussed in detail. The basic reproduction number $R_0$ of the model is computed and it is observed that the disease-free equilibrium point is stable for all delay whenever $R_0<1$. When $R_0>1$ the endemic equilibrium point is stable in the absence of delay. We have estimated the length of delay which preserves the stability of endemic equilibrium point. So when the delay is less than a threshold value, the endemic equilibrium point is stable. At that threshold value we get Hopf bifurcation and system shows oscillatory behaviour. Here numerical simulation is also performed to support the analytical results.