Computation, analysis and control of biological pattern formation

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Complex systems with interacting components frequently give rise to `emergent properties', or pattern formation phenomena. Models for pattern formation are ubiquitous in biology and are an intensive area of research. An important class of models are reaction-diffusion equations (RDEs). RDEs have been used to model many biological pattern phenomena arising in, plankton dynamics, nerve impulses, chemotaxis, cardiac arrhythmias, epidemiology, and morphogenesis, to name a few (see J.D. Murray's classic text 'Mathematical Biology'). I interpret patterning in a broad sense to include both spatio-temporal dynamics and stationary solutions arising due to diffusion induced instability (Turing patterns). Researchers are becoming more aware of the need to incorporate spatial structure in biological and epidemiological models, and to identify spatial scales. Spatial structures correspond to physical features of the environment, or to intrinsic characteristics of ecological processes and phenomena. Most realistic models are nonlinear, and thus numerical methods have a vital role to play in investigating the key mechanisms that regulate self organization in complex systems. However, the numerical analysis of RDEs with real applications lags behind advances in modeling. Therefore, the development of efficient and accurate numerical methods for the approximation of spatially explicit biological models is a fertile and growing research field. Another area that is underdeveloped is the control of pattern formation using the mathematical discipline of optimal control theory. In most situations, the need to control the evolution of a system comes naturally after establishing the mathematical model. Optimal control theory can also be used to identify key parameters in a biological system. The methodology I use in my proposal entails the cross-fertilization between numerical analysis, applied mathematical analysis, and scientific computing to study biological pattern formation modeled by RDEs. My proposal focuses on the numerical approximation and control of nonlinear RDEs modeling biological pattern formation, with applications in epidemiology, natural resource management, ecology, and various biomedical areas.