Research

I use models to explore the mechanisms that give rise to spatio-temporal patterns in biology. A significant part of my work is mathematical and primarily involves nonlinear partial differential equations (PDEs). Areas of research include: spatial patterns in ecological relationships (such as wolf territories, predator-prey, and plant-herbivore) and the generation of pattern and form in cells and tissues during morphogenesis and cell differentiation. The models in these two areas share striking similarities; the underlying mathematics provides a basis for new understanding in a wide variety of situations. My analysis of these complex nonlinear PDE models entails a variety of methods from bifurcation theory and singular and regular perturbation theory; exact solutions are almost always impossible to find. I undertake numerical solution of the full nonlinear system of equations.

Some previous research includes

1. Geometrical analysis of 2- and 3-D waves in excitable systems via singular perturbation theory. I have made applications to both physiology (analyzing geometrical conditions leading to one-way blocks in cardiac conductive tissue) and ecology (developing models for the 2--D spread of invading organisms).

2. Analysis of patterns arising in the contractile cytogel cortex of cells (such as microvilli and filopodia) via a mechanochemical model. This included application of regular perturbation theory and weakly nonlinear analysis to explain the evolution of long-term dynamic and stationary patterns.

3. A mathematical model explaining experimentally observed patterns of aphid aggregation using a system of coupled PDEs with complex spatial movement terms. Also, analysis of a joint control strategy for insect pests entailing the release of sterile insects coupled with the spraying of a small area to induce a `traveling wave' of extinction.

Current research includes:

1. Mathematically investigating the nature of wolf territoriality and wolf-deer interactions. Striking spatial patterns of territory arise between adjacent wolf packs and many deer are found in the `buffer zones' between territories. Using models with behavior-dependent spatial flux terms I am characterizing the fundamental mechanisms for territorial pattern formation and the effect of wolf territoriality on deer populations. Mathematical analysis indicates that simple behavioral rules regarding behavior and movement give rise to complex territorial patterns. These patterns are also observed from a 2-D numerical solution of the model.

2. Understanding biogical invasions with mathematical models. Classical models for biological invasions are based on the mathematical idea of a traveling wave of organisms spreading into virgin territory. With coworkers, I am extending this approach to understand the effect of rare, long-distance dispersal events. These can not only regulate the patchiness of the invasion process, but also the confidence intervals on predicted invasion speed. The models are nonlocal in space and involve measures such as the spatial covariance in the density of organisms, as well the expected density of organisms.

3. Using mathematical models to understand the potential spread of genetically engineered microbes released in agricultural settings. To date there have been very few models regarding the spread of engineered microbes and thus little quantitative basis for decision making regarding releases. The modelling approach employs coupled reaction-diffusion-convection equations for different classes of stationary and mobile microbes.

4. Understanding the swarming behavior of locusts based on behavioral movement rules for the individual insects.

For more information contact Mark A. Lewis, 1-6195

E-mail: mlewis@math.utah.edu