Qualitative analysis of solutions of some partial differential equations and equations with delay.

This thesis is devoted to the qualitative analysis of solutions of partial differential N equations and delay partial differential equations with applications to population biology. The first part deals with the problem of finding the length scales for the Navier-Stokes system on a rotating sphere and for a class of generalized reaction-diffusion system on a planar domain. Since the reaction-diffusion system under investigation has many biological and physical applications, it is crucial to be able to prove a positivity preserving property for solutions of this system. Motivated by its applications, the question of asymptotic positivity of solutions, as well as positivity for all time for a reaction-diffusion model is investigated. The presence of the fourth-order derivative in the equation makes the application of the maximum principle impossible. It will be shown that with the help of the ladder method, a positivity preserving property for this type of system can be proved. In all calculations, the application of interpolation inequalities of the Gagliardo-Nirenberg type with explicit and sharp constants gives the best possible results, and all calculations contain only known constants. Next, nonlinear analysis of the Extended Burgers-Huxley equation on a planar domain with periodic boundary conditions is performed. The geometric singular perturbation theory is then used to prove persistence of the travelling wave solutions in the case when a small perturbation parameter multiplies the fourth- order derivative. The second part of this thesis considers partial differential equations with time delay. We propose and study two mathematical models of stage-structured population. First, we study a nonlocal time-delayed reaction-diffusion population model on an infinite one-dimensional spatial domain. Depending on the model parameters, a non-trivial uniform equilibrium state may exist. We prove a comparison theorem for our equation for the case when the birth function is monotone, and then we use this result to establish nonlinear stability of the non-trivial uniform equilibrium state when it exists. A certain class of non-monotone birth functions relevant to certain species is also considered, namely, birth functions that are increasing at low densities but decreasing at high densities. In this case we prove that solutions still converge to the non-trivial equilibrium, provided the birth function is increasing at the equilibrium level. Then we derive a stage-structured model for a single species on a finite one-dimensional lattice. There is no migration into or from the lattice. The resulting system of equations, to be solved for the total adult population on each patch, is a system of delay equations involving the maturation delay for the species, and the delay term is nonlocal involving the population on all patches. We prove that the model has a positivity preserving property. The main theorem of the paper is a comparison principle for the case when the birth function is increasing. Using this theorem we prove that, when the model admits a positive equilibrium, the positive equilibrium is a global attractor. Then we establish a comparison principle that works for very general birth functions, and then we use this theorem to prove convergence theorems in the case when the birth function qualitatively resembles one used in the Nicholson's blowflies equation. We conclude by solving system numerically, using DDE tool in MATLAB. The thesis is concluded by a discussion of some open problems