James F. Reineck

Travelling waves in predator-prey systems

The existence of travelling wave solutions to reaction-diffusion equations which model predator-prey systems is proven. Bistable waves, Fisher waves, and higher-dimensional analogues of Fisher waves are found. Some of the systems investigated have bistable homoclinic waves. The proofs use the Conley index, continuation the connection matrix, and bifurcation theory in the Conley index setting.

The Conley index over a base

We construct a generalization of the Conley index for flows. The new index preserves information which in the classical case is lost in the process of collapsing the exit set to a point. The new index has most of the properties of the classical index. As examples, we study a flow with a knotted orbit in R3, and the problem of continuing two periodic orbits which are not homotopic as loops.

The Conley Index for Fast-Slow Systems I. One-Dimensional Slow Variable

We develop a qualitative theory for fast-slow systems with a one-dimensional slow variable. Using Conley index theory for singularity perturbed systems, conditions are given which imply that if one can construct heteroclinic connections and periodic orbits in systems with the derivative of the slow variable set to 0, these orbits persist when the derivative of the slow variable is small and nonzero.

Singular Index Pairs

Using Conleys idea of slow exit points, we construct index pairs for singularity perturbed families of lows. Some applications are also presented.

The Conley Index Over the Circle

We study the Conley index over a base in the case when the base is the circle. Such an index arises in a natural way when the considered flow admits a Poincaré section. In that case the fiberwise pointed spaces over the circle generated by index pairs are semibundles, i.e., admit a special structure similar to locally trivial bundles. We define a homotopy invariant of semibundles, the monodromy class. We use the monodromy class to prove that the Conley index of the Poincaré map may be expressed in terms of the Conley index over the circle.

Connection matrices and transition matrices

This paper is an introduction to connection and transition matrices in the Conley index theory for flows. Basic definitions and simple examples are discussed.