A numerical study of modelling in a predator-prey system

The study in this thesis concerns the interactions of the two-spotted spider mite Tetranychus urticae and its predator Phytoseiulus persimilis. The logistic Lotka–Volterra predator–prey-taxis equations with diffusion and advection have been solved numerically to re-estimate parameters using three different response functions with two data sets. In order to observe the effect of prey-taxis on the periodicity of predator–prey dynamics, limits of prey-taxis parameter have been obtained using Routh-Hurwitz’s conditions for stability. It has been shown that for different values of prey-taxis it is possible to achieve periodic, quasi-periodic and chaotic nature of solutions in a predator–prey system. The formation of spatial patterns in the predator–prey system has been studied under various environmental conditions. First spatial patterns are generated with the inclusion of prey-taxis in the predator–prey system. Next spatial patterns are generated with the introduction of diffusion-driven instability and without prey-taxis in the predator–prey system. Among all parameters involved in predator–prey equations, only the predator interference parameter is varied to generate diffusion-driven instability leading to spatial patterns of population density. It has been shown that it is possible to generate spatial patterns with zero flux boundary conditions even in a smaller domain with a suitable value of the predator interference or prey–taxis. Next spatial patterns are generated with the presence of a habitat edge inside the domain. Here predators are sensitive to the presence of an internal edge while prey are free to forage in the entire domain. It has been shown that the sensivity of predators to remain in a favourable or in an unfavourable patch as well as the position of the internal edge have major impacts on the development of spatial patterns. Finally biological control of two-spotted spider mite has been achieved with the introduction of prey-taxis in the predator–prey system. It has been shown that both response functions and initial conditions have major contributions in biological control of the prey population. It is possible to achieve successful biological control in space over a longer time-scale with prey population density below economic threshold provided the predator population density is at least twice that of the prey during most of the time. i Acknowledgements I am deeply grateful to my coordinating supervisor Dr. Manmohan Singh for his guidance and continual encouragement for completion of this thesis. Over the years, I really appreciated Manmohan’s invaluable comments and broad knowledge that inspired me to think beyond the squares. I owe sincere thanks to him for supervising my work, stimulating me intellectually and all the time and effort he has put on my PhD project. I also thank my associate supervisor Dr. David Lucy for his constant assistance and invaluable suggestions. I am also grateful to my external supervisor Dr. Peter Ridland for providing me the field data for this research. I appreciate his support, constructive criticism and helpful suggestions during this research work. My special gratitude to Professor Geoffrey Brooks, Head of Mathematics Discipline and Associate Dean (Research), Faculty of Engineering and Industrial Sciences, for his kindly help and warm encouragement during the preparation and completion of the thesis. I am thankful to Mr. Christopher Barling for his help from time to time during my stay at Swinburne University of Technology. I am also indebted to Professor John Beynon, Dean, Faculty of Engineering and Industrial Sciences for providing necessary facilities to continue my research during the last few years. My special thanks to Mr. Ashley Chester, Manager, Faculty of Engineering and Industrial Sciences, for his warm-hearted help whenever I needed it. I am grateful to Swinburne University of Technology for providing me an opportunity to carry out PhD. It has been challenging, enjoyable and worthwhile time. I have enjoyed becoming absorbed by a topic that really interests me. I am deeply indebted to the university for providing me a Swinburne University Postgraduate Research Award (SUPRA). I extend gratitude to my parents who are responsible for the course my life has taken. Without doubt, their unwavering love and support have been fundamental throughout all stages leading up to, and during research towards this thesis.