Grant: Discovery Grants Program - Individual

Most real physical systems (transmission dynamics of the COVID-19 outbreaks, fish population growth) are inherently nonlinear in nature, even simple nonlinear dynamical systems (DS) can exhibit predictable/unpredictable behaviour. Time delay plays a fundamental role in almost all fields. Therefore, time delay in nonlinear dynamics is not just of crucial importance in applications but still poses a fundamental challenge for theoretical and mathematical studies. The longterm objectives of my proposed research program include: introducing time delay appropriately in applied DS, particularly in biology/ecology/epidemiology (B/E/EP); providing more realistic mathematical modelling; gaining a more in-depth understanding of nonlinear dynamical behaviour, and the factors that govern such behaviour; thus contributing to new and improved analytical theories and applications to problems in B/P/EP. In the literature, there has been considerable effort devoted to the problem of stabilization and bifurcation in delay differential equations and continuous DS with discrete time delay. By contrast, when time delay appears in the boundary forces in a nonlinear way, or the real application system is non-smooth, much less is known, at least not for spatial models with advection or symmetry. In this research proposal, the short term objectives include: investigate the spatiotemporal patterns for the systems with local or non-local delays and different boundary conditions; seek more harmonious ways to study the discontinuity-induced bifurcations with delay and apply the theoretical results to real world applications, such as fish harvesting and biological pest control in fishery and disease transmission dynamics. We will use mathematical techniques based on nonlinear analysis, functional differential equations, mathematical modelling and continuous/discontinuous DS. While pursuing our objectives, we will remain flexible enough to explore promising avenues of research as they emerge. This proposed research is expected to advance and enrich the theory of continuous/discontinuous DS with time delay. By combining theoretical and practical perspectives, we expect to gain new insights for the development of mathematical theory with predictive power for real world applications. The theories and methodologies in this proposal will have a substantial impact on the nonlinear dynamics community. We expect our work will not only strengthen the foundation for theoretical development in nonlinear dynamics with time delay, but also provide the evidence and methods to support decision making (e.g., COVID masking policies, fishing quotas, etc.) in B/E/EP.  The proposed program of research will provide students at all levels with training in mathematical theory and applications. They will acquire highly logical thought ability and skills in applied mathematics that will prepare them to study and solve real-world problems in academia, government and industry, in Canada and beyond.