Georg Hetzer

UNIFORM PERSISTENCE, COEXISTENCE, AND EXTINCTION IN ALMOST PERIODIC/NONAUTONOMOUS COMPETITION DIFFUSION SYSTEMS ∗

A two species competition model with diffusion is considered. The parameters describing the growth, interaction, and self-limitation of the species are spatially inhomogeneous and temporally almost periodic. The boundary conditions are homogeneous and of Neumann or Dirichlet type. First, a convergence theorem is derived in the single species case. Roughly speaking, it states that one of the following alternatives will occur: either every positive solution converges to a unique strictly positive almost periodic solution, every positive solution converges to the trivial solution, or every positive solution is neither bounded away from the trivial solution nor converges to it. Then appropriate conditions for uniform persistence of both species as well as for extinction of one of the species are established. Moreover, it is shown that uniform persistence implies coexistence in the sense that there is a strictly positive solution whose hull is almost automorphic. The above results generalize earlier work in the time independent and time periodic cases for both single species population models and two species competition models. The approach developed in this paper for dealing with almost periodic equations can be applied to more general nonautonomous equations, as we will indicate by briefly discussing applications where merely time recurrence is supposed.

Effects of spatial variations and dispersal strategies on principal eigenvalues of dispersal operators and spreading speeds of monostable equations

The current paper is concerned with the following two separate, but related dynamical problems, the effects of spatial variations on the principal eigenvalues of dispersal operators with random or discrete or nonlocal dispersal and periodic boundary condition, and the effects of spatial variations and dispersal strategies on the spreading speeds of monostable equations in periodic environments. It first shows that spatial variation cannot reduce the principal eigenvalue (if exists) of a dispersal operator with random or discrete or nonlocal dispersal and periodic boundary condition, and indeed it is increased except for degenerate cases. It then shows that spatial variation enhances the spatial spreading in a non-degenerate spatially periodic monostable equation with random or discrete or nonlocal dispersal. It also shows that, for two monostable equations with the same population dynamics, but different dispersal strategies, one of which is random and the other is nonlocal, the spatial spreading speed of the equation with random dispersal is greater (resp. smaller) than the spreading speed of the equation with nonlocal dispersal if the dispersal distance is small (resp. large). The results obtained in the paper reveal the importance of spatial variation and dispersal strategies in population dynamics.

A multiparameter sensitivity analysis of a 2D diffusive climate model

Abstract A simple approach in long-range climate modeling relies on the energy budget solely. In this paper we set up a 2-dimensional equilibrium energy-balance model which incorporates several features of earlier 0- and 1-dimensional models. We examine via a sensitivity analysis to what extent one should put trust into the outcomes of such models. Informally speaking we conclude that it is the uncertainty about the rate of albedo decrease per degree Kelvin which restricts their use in climate diagnosis. On the other hand we have some new information which may help to clarify regional effects of the interaction of the two basic feedback mechanisms taken into account: the ice-albedo and the greenhouse feedback. From a mathematical point of view we are dealing with a parameter-dependent semilinear elliptic equation on the sphere. We report about an analytical approach to identify the qualitative structure of the solution branch and an efficient method for its numerical exploitation.

Asymptotic Behavior of Positive Solutions of Random and Stochastic Parabolic Equations of Fisher and Kolmogorov Types

We study the asymptotic behavior as t→∞ of positive solutions for random and stochastic parabolic equations of Fisher and Kolmogorov type. The following alternatives are established. Either (i) all positive solutions converge to one and the same trivial equilibrium, or (ii) every positive solution is neither bounded away from the trivial equilibria nor converges to them, or (iii) every positive solution is bounded away from the trivial equilibria. Moreover, for the random equation, we provide in case of alternative (iii) a fairly general condition under which every positive solution converges to uniformly positive equilibria. In the stochastic case, it is proved that there is no uniformly positive equilibrium, and under an appropriate condition, (iii) never occurs.

S-shapedness for energy balance climate models of Sellers-Type

Climate models are distinguished by the relative importance they attach to the different components and processes of the climate system. One finds the so-called energy balance climate models at the bottom on a scale of models of increasing complexity, and coupled general circulation models of atmosphere and oceans at the top on that scale.

Global existence and asymptotic behavior for a quasilinear reaction-diffusion system from climate modeling☆☆☆

Abstract This paper is concerned with a weakly coupled system of quasilinear autonomous strongly parabolic equations on a compact two-dimensional manifold without boundary; the system arises from an energy balance climate model. We establish L ∞ , Hoelder, and Sobolev estimates, and apply general results on quasilinear evolution equations in order to guarantee the existence of classical nonnegative solutions. More precisely, it is shown that the system generates a global solution semiflow in the positive cone of some fractional order Sobolev space. Employing elements of the theory of infinite-dimensional dissipative systems, we prove the existence of a connected global attractor. Finally, we present some results about stationary solutions and forced periodic oscillations. The present paper extends earlier work of the authors on a semilinear problem.

Preface: Special issue on dissipative systems and applications with emphasis on nonlocal or nonlinear diffusion problems

The strong interest in infinite dimensional dissipative systems originated from the observation that the dynamics of large classes of partial differential equations and systems resembles the behavior known from the modern theory of finite-dimensional dynamical systems. Reaction-diffusion problems are typical examples in this context. In biological applications linear diffusion represents random dispersal of a species, but in many cases other dispersal strategies occur, which has led to models with cross diffusion and nonlocal dispersal. For more information please click the “Full Text” above.

Branches of Stationary Solutions for Parameter-dependent Reaction-Diffusion Systems from Climate Modeling

We are concerned with a parameter-dependent reaction-diffusion system on a two-dimensional compact connected oriented Riemannian manifold M without boundary (e.g., the two-sphere):

An energy balance climate model with hysteresis

{c_j}{\partial _t}{u_j} - div\left( {{k_j}\;grad\;{u_j}} \right) = {f_j}\left( {\mu ;x,{u_1},{u_2},{u_3}} \right){\mkern 1mu} \quad \left( {j = 1,2.3} \right).