Mathematical Sciences: Nonlinear Mathematical Transport Theory and Population Models

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This work will focus on the investigation of nonlinear partial integro-differential and integro-difference equations which arise from mathematical transport theory and population models. In the former, studies of the general transport process describing the effects of collision, absorbtion and reflection of molecules or particles in gases whose kinetics is governed by the nonlinear Boltzman equation are carried out. Problems of existence, uniqueness continuous dependence on initial data, smoothness, approximation and asymptotic behavior for large time will be undertaken. Techniques will employ the theory of abstract differential equations in ordered Banach spaces and lattices, strongly continuous operator semigroups and path integrals. Specific models associated with the kinetic behavior of gases where collisions are governed by general interaction potentials will be considered. Work on population dynamics will utilize the discrete time nature of a growth-dispersal process whose evolution is described by a dynamical system modeling competition among species living in the same habitat. Interaction assumptions have two qualitative forms: peaceful competition or predator-prey competition. Fundamental concerns in this work are the establishment of conditions for coexistence of species and obtaining optimal dispersion strategies for the individual species. The mathematical context for this work is the spectral theory of positive operators applied to Frechet derivatives, in studying solutions of Hammerstein equations which relate growth and dispersion.