Plant Disease Models With Economic Thresholds And Cultural Control Strategies

Crop production is always subject to the constraints of plant diseases which can cause severe losses in economy. For instance, fungal,viral and bacterial diseases give rise to losses in many orchards.Therefore, agro-ecological sector,domestic and foreign experts, and farmers are very concerned about the issue of controlling plant diseases. We should control plant diseases actively and effectively if we want to develop agriculture,which means that controlling plant disease plays a key role in agricultural production.However, controlling plant diseases, referring to dynamic processes such as the growth of plants and the spread of plant diseases and so on,is so complex that if you implement a decision,you must be committed to the long term. Hence lots of researchers have been evolving practice for combating various plagues suffered by crops and have established the optimal treatment program gradually.Generally speaking,the methods of controlling plant diseases involve biologi-cal, chemical,and cultural tactics and so on.Such experiences have led to develop the concept of integrated disease management(IDM),which uses a combination of tactics mentioning above to minimize losses and maximize returns.Contributing to sustainable development,IDM is more effective than the classical one proved by experiments. In recent years, on the basis of IDM,several mathematical models have been formulated by many researchers and lots of meaningful results have been obtained. Meanwhile,it has been recognized that cultural control strategies, such as roguing infected plants and replanting healthy plants are useful approaches for the control of plant diseases. And most research work mainly have focused on models of continuous strategies,as well as the existence and stability of disease eradication periodic solutions.However, continuous control doesn't meet the requirements of the production practice.In addition,complete eradication of the infected plants is generally not possible, and it doesn't measure up to the standards of IDM.Conse-quently, the models with continuous control methods are improved in this paper.Firstly, in chapter 2,the classical S-I model of plant disease with continuous cultural control strategies is extended in a system with periodic impulsive cultural treatment of plant diseases. And we expand the model by taking into account the proportional and constant replanting, different death rates of healthy plants and infected plants. Sufficient conditions are derived for the globally asymptotical sta-bility of the disease-eradication periodic solution by using the comparison theory and the theory of impulsive differential equations about stability. But this improve-ment is not complete. With the guiding principle of prevention and the principles of cost-effectivity, we should reduce the infected plants to tolerable levels.This reveals that the plants are rogued and replanted when the number of infected ones reaches a presetting value, to be called the Economic Threshold(ET).Hence, in chapter 3,the plant disease model including impulsive cultural control strategies and ET on the infected plants is analyzed.We examine the existence of the unique positive periodic solution in terms of three cases of a certain parameter space. Then the stability of the periodic solution in three situations is investigated. By using qualitative methods,we prove that the periodic solution in one situation is globally asymptotically stable under some conditions, and the one in other cases is unstable. If there exists a unstable periodic solution, the number of healthy plants either goes to infinity or tends to zero,though the density of infected plants is always no large than the given ET.In order to solve the problem of the instability of the periodic solutions, we fur-ther consider a bi-threshold-value model of plant diseases. We also set a Economic Threshold for healthy plants.The healthy plants will be replanted when their num-ber reaches the Economic Threshold. By analyzing the complex dynamic behavior of the model,we obtain various types of periodic solutions including order 1-k or order k-1 periodic solutions withκ≥1.There doesn't exist order nS-nI periodic solutions with nS,nI≥2.Besides, under some conditions,the infected plant free periodic solution is globally stable.The main results obtained in this paper extend and generalize the classical results for the systems of plant diseases in the integrated disease management.So we can make use of the corresponding conclusions to guide for the process of production and practice,and control plant diseases effectively.