Roger Lui

Biological growth and spread modeled by systems of recursions. I. mathematical theory

We study the asymptotic behavior of solutions to a system of recursions u in+1 = Qi[mu n], i = 1, ..., k. The vector operator Q has the origin theta and a positive vector beta as fixed points and is defined for vector-valued functions bounded between theta and gamma where gamma greater than or equal to beta. In addition, Q is order-preserving, commutes with translation, and is continuous in the topology of uniform convergence on compact subsets. Let theta less than or equal to pi much less than beta, and suppose that for all pi much less than alpha much less than beta, Q(n) alpha]----beta as n----infinity. If u0 much greater than pi on a sufficiently large ball and has bounded support, then un propagates with a speed c*(xi) in the direction of the unit vector xi as n----infinity. In certain cases, c*(xi) can be calculated explicitly. The results generalize those of a scalar equation studied by Weinberger.

Exploring the Control Circuit of Cell Migration by Mathematical Modeling

We have developed a top-down, rule-based mathematical model to explore the basic principles that coordinate mechanochemical events during animal cell migration, particularly the local-stimulation-global-inhibition model suggested originally for chemotaxis. Cells were modeled as a shape machine that protrudes or retracts in response to a combination of local protrusion and global retraction signals. Using an optimization algorithm to identify parameters that generate specific shapes and migration patterns, we show that the mechanism of local stimulation global inhibition can readily account for the behavior of Dictyostelium under a large collection of conditions. Within this collection, some parameters showed strong correlation, indicating that a normal phenotype may be maintained by complementation among functional modules. In addition, comparison of parameters for control and nocodazole-treated Dictyostelium identified the most prominent effect of microtubules as regulating the rates of retraction and protrusion signal decay, and the extent of global inhibition. Other changes in parameters can lead to profound transformations from amoeboid cells into cells mimicking keratocytes, neurons, or fibroblasts. Thus, a simple circuit of local stimulation-global inhibition can account for a wide range of cell behaviors. A similar top-down approach may be applied to other complex problems and combined with molecular manipulations to define specific protein functions.

Biological growth and spread modeled by systems of recursions. II. biological theory

The mathematical theory developed in Part I is applied to a selection-migration model in population genetics with sex-linked locus and to the host-vector or venereal disease epidemic model. In both models, a constant c*(xi) is found for each unit vector xi. The mathematical results imply that under certain initial conditions, the frequency of the advantageous gene in the male and female gametic outputs or the epidemic will spread at a speed c*(xi) in the direction xi as time goes to infinity. Time is measured in discrete nonoverlapping generations. In most cases, we can find a formula for c*(xi).

A Nonlinear Integral Operator Arising from a Model in Population Genetics I. Monotone Initial Data

We study the asymptotic behavior of the solutions to the recursion

A Nonlinear Integral Operator Arising from a Model in Population Genetics II. Initial Data with Compact Support

u_{n + 1} (x) = Q[u_n ](x)

A mathematical model for the electrokinetic remediation of contaminated soil

for

Analysis of a continuous crystallizer with agglomeration

n \geq 0

Some remarks about the wave speed and travelling wave solutions of a nonlinear integral operator

. Here

A nonlinear integral operator arising from a model in population genetics IV. clines

Q[u](x) = (K * g \circ u)(x)

Analysis of an electrochemistry model with zero-flux boundary conditions

acts on functions bounded between 0 and 1,