The Principal Eigenvalue Theory For Some Non-compact Positive Linear Operators And Its Applications

The thesis is focusing on two related topics about the principal eigenvalue theory of non-compact positive operators.The first one is the principal eigenvalue of the partially degenerate periodic parabolic system and its applications.The other one is mainly con-cerned with the basic reproduction ratios(R0)of the abstract delay differential equations and its applications.In preliminaries,we discuss some basic properties about positive operators and prove a strong version of the Krein-Rutman theorem.The celebrated Krein-Rutman theorem established the principal eigenvalue theory for a compact and positive opera-tor.Edmunds,Potter and Stuart and Nussbaum developed the principal eigenvalue the-ory for a positive operator whose spectrum radius is greater than its essential spectrum radius.We call this result a weak version of the generalized Krein-Rutman theorem.Moreover,some important properties hold when the operator is compact and strongly positive.In this thesis,we obtain the same properties for a strongly positive operator whose spectral radius is greater than its essential spectral radius,which is called a strong version of the generalized Krein-Rutman theorem.In the first part,we study the theory of the principal eigenvalue for a linear time-periodic parabolic cooperative system with some zero diffusion coefficients.The main difficulty is that the Poincare map of the corresponding system loses compactness.To prove the existence of the principal eigenvalue,we use the generalized Krein-Rutman theorem and proceed on two steps.In step one,we analyze delicately the essential spec-trum of the Poincare map of the corresponding system.In step two,we give sufficient conditions such that there is a gap between spectral radius and essential spectral radius for the Poincare map of the corresponding system.In this thesis,we also apply the above consequences into a Benthic-Drift model.In the second part,we establish the relationship between R0 and the stability of null solution for the corresponding linear abstract delay differential system.To obtain this result,we employ the properties of the principal eigenvalue of a positive operator under a series of suitable assumptions for a noncompact system.It is notable that the above assumptions hold automatically when the solution map of the corresponding system is eventually compact.Moreover,we also give a numerical method to compute R0.This method seems to be effective for the infinite dimensional time-periodic system.At last,we apply it to a time-periodic Lyme disease model with time-delay and obtain a threshold type result on its global dynamics in terms of R0.