基于植物疾病控制的非光滑动力学模型的研究

Plant disease is a key constraint in yield and quality of cultivated crops. Classical plant disase mathematical models with ordinary differential equations are proposed to describe the development and control of plant diseases, however, periodicity, non-instantaneousness and state-dependency of the control behavior are usually ignored. On the basis of the economic threshold, and the integrated strategies including cultural and chemical tactics and so on, we establish non-smooth plant disease models to do theoretical and applied research. Firstly, the impulsive model with nonlinear cultural control strategy is proposed due to the availability of resource. We extend the linear impulse to the nonlinear impulse, then explore the threshold concerning disease eradication and the key factor that can affect the development of diseases. Secondly, according to transmission approaches of plant diseases, the hybrid model with viral dynamics and population dynamics is established. By setting the cost for the implementation of various strategies as the target function, the optimal control period and economic threshold with the least expensive are investigated. Meanwhile, the optimal integrated control scheme will be obtained. Thirdly, the numbers of infected plants and susceptible plants can be controlled as the criterion, then the Filippov model with double swithching surfaces is proposed. We analyze the complex dynamic behavior of the piecewise smooth system to seek the effective control strategies that can be used to maintain the number of infected plants below the economic threshold. According to the statistical data of plant diseases in our country, such as sweet potato virus disease and so on, we estimate the parameters, forecast the development trend of disease and study the effectiveness of prevention and control measures. The results obtained are potential to provide the assessment and optimal suggestion on the control methods for the agricultural sector.

植物疾病严重影响农作物生产。经典的植物疾病模型通常利用常微分方程刻画疾病的发展与控制，忽略了人为控制的周期性、持续性和状态依赖性。本项目引入经济阈值，考虑种植和化学等综合策略，提出非光滑模型进行理论和应用研究。基于资源的有限性建立具有非线性种植策略的脉冲微分模型，将线性脉冲推广至非线性脉冲，寻求根除疾病的阈值，揭示影响疾病发展的关键因素。根据疾病的传播机理建立具有综合控制的病毒动力学与种群动力学的复合模型，以实施策略的费用为目标函数，探求费用最少的最优控制周期和经济阈值，探寻最佳防治方案。以染病植物和易感植物的数量为准则，建立具有双切换面的Filippov模型，探究该分段光滑系统的复杂动力学，寻求有效的防治策略使得染病植物数量最终不超过经济阈值。根据我国甘薯病毒病等疾病的统计数据估计模型参数，预测疾病发展趋势，探讨防治方案的有效性，为农业部门实施的防治措施提供定量评估和优化建议。

植物疾病严重影响农作物生产。经典的植物疾病模型通常利用常微分方程刻画疾病的发展与控制，忽略了人为控制的周期性、持续性和状态依赖性。本项目考虑病毒、经济阈值、传播媒介、种植和生物策略等因素，提出非光滑模型、反应扩散模型以及随机模型进行理论分析和应用研究。(1)建立具有非线性移除和非线性重植策略的植物疾病模型，将周期脉冲或状态依赖脉冲微分模型中关于线性脉冲函数的研究理论推广至非线性脉冲函数的研究理论中，寻求周期脉冲模型疾病根除的阈值，探讨各参数对该阈值的影响程度，揭示影响疾病发展的关键因素。分析状态脉冲模型全局稳定的周期解，最终实现疾病的周期控制。(2)以易感植物和染病植物的数量为控制准则，建立具有双切换面的Filippov模型，根据首次积分和Lyapunov函数定性地分析滑动模式解、滑动平衡态、盆吸引子和全局吸引子的性态，结果表明：合理有效地进行有限次切换控制后，能够将易感植物和染病植物的数量维持在各自的阈值范围内。(3)立足于植物病虫害和有益天敌之间的食饵与捕食者的关系，研究具有双状态脉冲控制的食饵—捕食者模型，通过对病虫害数量进行双重阈值控制，使得平衡点左侧出现阶1周期解，据此可达到周期控制。(4)考虑疾病的传播媒介昆虫，建立不具拟单调性的非局部时滞反应扩散模型，系统深入地对传播速度和行波解进行理论研究，得出渐近传播速度等于波前解的最小波速。(5)基于疾病在病毒水平的传播机制，建立具有潜伏期的HIV随机模型，得到病毒灭绝的充分条件，同时发现大强度的白噪音有利于病毒的灭绝。本项目为进一步进行基于综合控制的病毒动力学与种群动力学复合模型的研究提供了强有力的理论依据，运用的分析方法为研究具有非线性策略的脉冲模型、具有Lotka-Volterra型的Filippov切换模型提供了一般的理论框架，所得结论将为预测疾病的发展趋势和制定合理的防治措施提供定量评估和优化建议。