SOME IMPULSIVE MATHEMATICAL MODELS IN PEST MANAGEMENT

Abstract

In this thesis, attempts have been made to investigate the dynamics of pest control models considering various pest management tactics with birth pulses. The mathematical models in pest management are the system of differential equation with impulsive conditions. These impulsive effects may occur due to the instantaneous killing of pest using pesticides, releasing natural enemies and harvesting pest. Dynamical behaviors of some impulsive models with time-dependent strategies as well as state-dependent strategies have been explored. Emphasis is given to exploring the factors that are responsible for pest eradication. Due to impulses in pest control systems underlying equations have complex dynamical behavior, including periodic solutions, quasiperiodic, chaotic behavior etc.. The numerical simulations are carried out to explore the dynamic complexity in impulsive models. The efforts have been made to interpret mathematical results and to explore the biological relevance of these results. Chapter 1 includes brief introduction including basic concepts for pest control, mathematical tools, literature survey and summary of the thesis. The chapters 2, 3 and 4 are devoted to the control of pest using single pest management strategy. The next three chapters incorporate the Integrated Pest Management approach to control the pest population. The effect of pesticides on the environment is considered in chapter 8. A state dependent control is discussed in chapter 9. The conclusions and future scope are presented in the last chapter. In particular, the second chapter deals with the dynamics of an impulsive stagestructured pest control model. In this model, birth pulses occur at regular intervals to release immature pest. The Ricker type birth function is assumed. Pest population is controlled by periodic spray killing mature as well as immature pest instantaneously. This is synchronized with birth pulses. The discrete dynamical system determined i ii by the stroboscopic map is analyzed. The threshold condition for pest eradication is established. It is found that if the birth rate parameter is below the critical level, then the pest can be effectively controlled. Finally, numerical simulations depict the complex dynamics of the model. In chapter 3, birth pulse and chemical spray are no more synchronous. The pesticide is sprayed periodically before the birth at the fixed time. The effect of pesticide spray timing on threshold condition for pest eradication is studied. The pest will extinct when the time of pesticide spray exceeds the critical value. The maximum reduction in immature and mature densities will occur near the birth pulse when the basic reproduction number exceeds one. Further, asynchronous pulses reduce the complexity of the system. A pest control model using pesticides having residual effects is discussed in chapter 4. The effect of residual pesticide can be described by kill function. Birth pulse and chemical spray are assumed to be asynchronous. The basic reproduction number for pest eradication has been computed. The effects of various model parameters on the threshold condition are investigated. It is found that the killing efficiency rate reduces the threshold below unity which is required for effective pest control. Further, the decay rate of the pesticide enhances the threshold and pest outbreak may occur. Finally, numerical simulations depict the complex dynamical behaviors. In chapter 5, a model is developed considering the continuous mechanical effort (harvesting) to control the immature pest while mature pest is controlled impulsively by the pesticide. It is found that the pest cannot be controlled successfully in the absence of harvesting effort. The combination of harvesting effort and pesticide is needed for successful pest control. The use of pesticides can be reduced by incorporating mechanical control. It is found that when mature pest density goes beyond a critical level, then the pest-free state will be stable. Further, the rate reduces the complexity of the system with an increase in immature pest mortality rate. Due to asynchronous pulses, the harvesting reduces the complexity of the system. The chances of pesteradication also increase with less toxic pesticides. In chapter 6, an impulsive model with three pulses is considered where the mechanical, chemical control and birth pulse occur at three different fixed times. The increase iii in time delay of chemical control reduces the mature pest density as well as the threshold for pest eradication. The threshold value of harvesting effort has been obtained for the stable pest-free state. The critical level of pulse period is obtained to control the pest population. Numerical simulations have been performed to show the complex dynamical behavior, including period-doubling bifurcation and chaotic dynamics. The Lyapunov exponent and Lyapunov dimension are computed to establish the pest outbreak in the form of chaotic attractor. In chapter 7, integrated pest management approach comprising of the impulsive chemical as well as mechanical control is considered. The residual effect of pesticide with the delayed response is assumed. The harvesting effort and birth pulse occur asynchronously. The basic reproduction number has been computed. The bifurcation diagram with respect to birth rate has been plotted to show the stability regions of the pest-free state and interior fixed point. The harvesting effort of immature and mature pest reduces the threshold condition and thereby enhancing the stability of pest-free state. Numerical simulations reveal that increasing the delayed response may stabilize the pest-free state. It is found that the shorter delayed response rate is not sufficient for pest eradication. The combination of time delay in harvesting and the delayed response rate is required to control the pest population. Chapter 8 investigates the toxic effects of pesticides on the environment. The sufficiently small toxicant removal from the environment may eradicate the pest successfully. Otherwise, the pest will occur in regular/irregular periodic manner. For the lower birth rate pest can be eradicated completely. Similarly, the pest outbreak will occur when toxin input into the environment is sufficiently small. A state-dependent combined strategy for biological and chemical control is discussed in chapter 9. The Poincare map is used to explore the system dynamics. Sufficient conditions for the existence and stability of natural enemy-free and positive period-1 solutions are obtained. The positive period-1 solution bifurcates from the semi-trivial solution through a fold bifurcation. Complex dynamical behavior, including chaos is obtained. It is also observed that if more natural enemies are released, the complexity of the system increase, but the pest population will remain below the threshold level.