Periodic-Parabolic Boundary Value Problems and Positivity

Part 1 Discrete-time order-preserving dynamical systems: monotone iteration schemes connecting orbits stabilization existence of stable fixed points an example to stabilization - sublinear mappings existence of unstable fixed points prerequisites - the theorems of Krein-Rutman instability for differentiable maps continuous-time order-preserving dynamical systems. Part 2 The linear periodic-parabolic eigenvalue problem: prerequisites - linear evolution equations of parabolic type in a Banach space prerequisites - linear parabolic differential equations the maximum principle the periodic-parabolic eigenvalue problem a one-parameter family of eigenvalue problems the periodic-parabolic eigenvalue problem with respect to an indefinite weight function estimates for the principal eigenvalue. Part 3 The nonlinear periodic-parabolic problem: prerequisites - semilinear evolution equations of parabolic type in a Banach space prerequisites - semilinear parabolic differential equations the semilinear periodic-parabolic initial value problem existence of stable periodic solutions stability of periodic solutions a general convergence result convergence of solutions in a periodic-parabolic Neumann problem bifurcation of periodic solutions bifurcation of periodic solutions bifurcation for concave and convex nonlinearities the periodic-parabolic logistic equation the periodic Fisher equation. Part 4 The periodic Volterra-Lotka competition system with diffusion: a priori estimates for fixed points of S the Volterra-Lotka system in the compressive case extinction in the competition system an unstable coexistence state. Part 5 the periodic Volterra-Lotka predator-prey system with diffusion: nonlocal perturbation of the logistic equation sufficiency for coexistence extinction in the predator-prey system.