Dynamical Analysis of Delayed Plant Disease Models with Continuous or Impulsive Cultural Control Strategies

Tongqian Zhang,Yi Song,Xinzhu Meng,Zhenqing Li

doi:10.1155/2012/428453

Abstract

Delayed plant disease mathematical models including continuous cultural control strategy and impulsive cultural control strategy are presented and investigated. Firstly, we consider continuous cultural control strategy in which continuous replanting of healthy plants is taken. The existence and local stability of disease‐free equilibrium and positive equilibrium are studied by analyzing the associated characteristic transcendental equation. And then, plant disease model with impulsive replanting of healthy plants is also considered; the sufficient condition under which the infected plant‐free periodic solution is globally attritive is obtained. Moreover, permanence of the system is studied. Some numerical simulations are also given to illustrate our results.

Highlights

Plant viruses or pathogens are an important constraint to crop production worldwide and cause major production and economic losses in agriculture and forestry
Under the continuous control strategy, we consider the hypothetical set of parameter values as ρ 0.3, μ 0.2, β 1.5, α 1, r 0.5, d 0.2, and ω 0.1, with S 0 1, and I 0 1
Its epidemiological implication is that if time delay τ is greater than some key value τ1, diseased plants will disappear in local scope

Summary

Introduction

Plant viruses or pathogens are an important constraint to crop production worldwide and cause major production and economic losses in agriculture and forestry. Several plant diseases caused by plant viruses in cassava Manihot esculenta and sweet potato Ipomoea batatas are among the main constraints to sustainable production of these vegetatively propagated staple food crops in lesser-developed countries 2–4. The integrated disease management IDM which combines biological, cultural, and chemical tactics and so on to reduce the numbers of infected individuals to a tolerable level and aims to minimize losses and maximize returns 6, 7 has been developed gradually. IDM includes four main control strategies for vegetatively propagated plant diseases, which are containing transmission vectors, improving the production of planting material, controlling the crop sanitation through removal of infected plants, and breeding plants for resistance to the virus. Meng and Li have investigated vegetatively propagated plant diseases and developed a mathematical model with continuous control strategies and impulsive cultural control strategies 17 , which leads to

St ρ

St αS t

SΔ αSΔ

Numerical Simulation and Conclusion