SPATIAL ASPECTS OF SOME TUMOR–IMMUNE AND ECOLOGICAL MODELS

Abstract

This thesis investigates the role of spatial pattern formation in tumor{immune and predator{prey systems. In the rst half of the thesis, we have proposed spatiotemporal mathematical models using a system of non-linear partial di erential equations (reaction-di usion equations), to study the qualitative and quantitative analysis of tumor{immune interaction, and predator{prey interaction, considering the role of di usion and other system parameters. The tumor{immune interaction consists of two separate problems, namely, (i) interaction of solid tumor and e ector cells (ii) interaction of malignant gliomas and four immune components with the administration of immunotherapeutic agent T11 target structure (T11TS). Using the combination of analytical and numerical techniques, we investigate spatiotemporal dynamics due to the e ect of e ector cells and T11 target structure on the growth and spread of solid tumor and malignant gliomas respectively. However, Turing zone is absent in both the problems. The second half of the thesis consists of di usive predator{prey system (i) with hunting cooperation in predators and (ii) exhibiting herd behavior for prey with linear and quadratic mortality. Using extensive numerical simulations, we obtain complex patterns, namely, spotted pattern, stripe pattern and mixed pattern in the Turing domain by varying (a) hunting cooperation parameter and (b) linear and quadratic mortality's rates respectively. We also have a non-Turing pattern that exhibits spatiotemporal chaos. Thus, the study of the predator{prey system focusses in many pattern dynamics and help in better understanding of their interaction in real environment.