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 Lotkaâ-Volterra predatorâ-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 predatorâ-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 predatorâ-prey system. The formation of spatial patterns in the predatorâ-prey system has been studied under various environmental conditions. First spatial patterns are generated with the inclusion of prey-taxis in the predatorâ-prey system. Next spatial patterns are generated with the introduction of diffusion-driven instability and without prey-taxis in the predatorâ-prey system. Among all parameters involved in predatorâ-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 preyâ-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 predatorâ-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.