Title:

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 NavierStokes system on a rotating sphere and for a class of generalized reactiondiffusion system on a planar domain. Since the reactiondiffusion 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 reactiondiffusion model is investigated. The presence of the fourthorder 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 GagliardoNirenberg type with explicit and sharp constants gives the best possible results, and all calculations contain only known constants. Next, nonlinear analysis of the Extended BurgersHuxley 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 stagestructured population. First, we study a nonlocal timedelayed reactiondiffusion population model on an infinite onedimensional spatial domain. Depending on the model parameters, a nontrivial 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 nontrivial uniform equilibrium state when it exists. A certain class of nonmonotone 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 nontrivial equilibrium, provided the birth function is increasing at the equilibrium level. Then we derive a stagestructured model for a single species on a finite onedimensional 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.
