Robert Gardner

The principal focus of my research has been on the development and application of methods pertaining to the existence and stability analysis of traveling wave solutions to nonlinear PDE. In early joint work with C. Conley, a variant of the Conley index was introduced that detects the presence of co-dimension 1 traveling waves, (i.e. waves connecting pairs of stable rest points in reaction-diffusion systems); the methods were applied to the existence of waves arising in a variety of systems arising in mathematical biology, phase transitions, combustion. It also led to one of the earliest existential theories of multidimensional waves in channel domains. In later joint work with C. K. R. T. Jones, J. Alexander, a topological invariant was introduced that counts the multiplicity of eigenvalues of the linearized operator about a traveling wave solution interior to a Jordan curve in the spectral plane lying outside the continuous spectrum of the wave. In subsequent individual and joint projects, similarinvariants were introduced for periodic waves and steady state solutions of boundary value problems, and attendant analytical methods were developed in connection with the establishment of stability, instability, and bifurcation results for a variety of waves arising in specific areas of application. In joint work with Doelman and Kaper, a formal limit problem predicting the stability and bifurcation of singular limits of 1-D patterns in a class of reaction-diffusion systems that includes the Gray-Scott model was derived. The validity of the limiting problem was established through an analysis of the problem's stability index, which in certain regimes must occur with the eigenvalue parameter lying on a portion of a Riemann surface. My current work is concerned with the formulation of theoretical and computational models of self-annihilative relationships in predator-herbivore-host plant interactions in population biology. Explosive population growth during the early phase of the predator-prey cycle inevitably terminates with the annihilation of the predator in confined arenas of interaction. In unconfined1-D environments, the self-annihilating character of the interaction is also reflected in the character of a collection of stable and unstable traveling waves. Wave instabilities, and wave interactions with other waves with boundaries and spatial inhomogeneities often results in spatio-temporal chaos. Numerical simulations suggest that the chaotic invariant set is in some situations hyperbolic; after an unpredictably long evolution near the set, the PDE solutions exit along its unstable manifold and self-annihilate. In other regimes, it appears that the chaotic set may be an attractor. Current efforts are focused on development of a theoretical method of parameter selection that fits the dynamics of a class of nonlinear PDE models in large but confined 2-D arenas to experimental data sets. While it is unrealistic to expect that such models can provide accurate predictions of behavior of long-term experiments, the intent is to find mathematical models that can describe the behavior of transient traveling waves and other spatial patterns that offen occur within persistent experimental interactions.