Asymptotic analysis of dissipative waves with applications to their numerical simulation

Various problems involving the interplay of asymptotics and numerics in the analysis of wave propagation in dissipative systems are studied. A general approach to the asymptotic analysis of linear, dissipative waves is developed. It was applied to the derivation of asymptotic boundary conditions for numerical solutions on unbounded domains. Applications include the Navier-Stokes equations. Multidimensional traveling wave solutions to reaction-diffusion equations are also considered. A preliminary numerical investigation of a thermo-diffusive model of flame propagation in a channel with heat loss at the walls is presented

Modeling, analysis, and numerical approximations of the forced Fisher\u27s equation and related control problems

The Fisher equation with inhomogeneous forcing is considered in this work. First, a forced Fisher equation and boundary conditions are derived. Then, the existence of a local and global solution for the forced equation with a homogeneous Dirichlet condition is proved and the results are generalized to the case of less regular forces. Semi-discrete finite element approximations, semi-discrete approximations in the time variable, and fully discrete approximations are studied under certain minimal regularity assumptions. Numerical experiments are carried out and computational results are presented. An optimal distributed control problem related to the forced Fisher equation is also considered, the optimality system is derived, and numerical approximations of the optimality system are discussed

Topics in Evolutionary Ecology

46 pages, 1 article*Topics in Evolutionary Ecology* (Levin, Simon A.; Castillo-Chavez, Carlos) 46 page

Partial Differential Equation Models of Collective Migration During Wound Healing

This dissertation is concerned with the derivation, analysis, and parameter inference of mathematical models of the collective migration of epithelial cells. During the wound healing process, epidermal keratinocytes collectively migrate from the wound edge into the wound area as a means to re-establish the outermost layer of skin. This migration into the wound is stimulated by the presence of epidermal growth factor. Accordingly, this dissertation focuses on the migratory response of epidermal keratinocytes in response to this growth factor. Such studies will suggest suitable clinical treatments to consider for chronic wounds and invasive cancers.We begin with a study into the role of cell-cell adhesions on keratinocyte migration during wound healing. We use an inverse problem methodology in combination with model validation to show that cells use these connections to promote migration by pulling on their follower cells as they migrate into the wound. We next derive a biochemically-structured version of Fisher's Equation that provides a framework to study how patterns of biochemical activation influence migration into the wound. We prove the existence of a self-similar traveling wave solution. In considering a more complicated scenario where cell migration depends on biochemical activity levels, we show numerically that the threshold parameter where all cells in the population become activated yields the simulations that migrate farthest into the wound. Lastly, we consider the role of numerical error on an inverse problem methodology. The numerical approximation of a cost function is dominated by either numerical or experimental error in computations, which leads to different rates of convergence as numerical resolution increases. We use residual analysis to derive an autocorrelative statistical model for cases where numerical error is the main source of error for first order schemes. This autocorrelative statistical model can correct confidence interval computation for these methods and hence improve uncertainty quantification

Spatio-temporal patterns generated by Salmonella typhimurium

We present experimental results on the bacterium Salmonella typhimurium which show that cells of chemotactic strains aggregate in response to gradients of amino acids, attractants that they themselves excrete. Depending on the conditions under which cells are cultured, they form periodic arrays of continuous or perforated rings, which arise sequentially within a spreading bacterial lawn. Based on these experiments, we develop a biologically realistic cell-chemotaxis model to describe the self-organization of bacteria. Numerical and analytical investigations of the model mechanism show how the two types of observed geometric patterns can be generated by the interaction of the cells with chemoattractant they produce

Modeling of Cell Migration Assays Including Electrotaxis

Undergraduate Research ScholarshipCell motility is important in embryonic development, wound-healing, and the metastasis of cancer. There are different stimuli that guide cell motility in biological tissue. One of these stimuli is electrical in nature (the others being chemical and mechanical), and the mobility of cells under the presence of electric fields is called electrotaxis. It is well known that potential differences on the order of 20mV to 50mV exist across epithelial tissue. It is also known that when epithelial tissue is compromised resulting in a wound, a short circuit is created across the basement membrane. This endogenous electric field drives a flow of electrical current and cells towards the more negative site of the wound, driving closure. If bioelectricity plays an important role in moving cells in specific directions, then increasing the strength of the electric field and varying its direction in vivo can either accelerate or decelerate cell movement as desired. Bioelectricity therefore offers the possibility of controlling and potentially accelerating wound-healing or retarding the metastasis of cancer. Conventional methods of studying and inducing electrotaxis have involved the use of metal electrodes placed in contact with the tissue or medium containing cells using agar salt bridges. This approach raises the possibility of contamination as well as unwanted effects of Ohmic heating. In this research, electric fields are induced in vitro in a non-contact manner, thereby eliminating any interfering electrochemical interactions or unwanted heating arising from flow of direct current through the culture medium. The goal of this research is to quantify and simulate cell movement in a standard wound-healing assay using numerical methods to solve the relevant two-dimensional transient governing equations. Preliminary results have been obtained from non-contact electrotaxis experiments, and a time-varying 2-D model has been developed that simulates and can eventually predict the migration of cells in response to an electrical stimulus. This model can be useful for further studies delving into mechanisms driving electrotaxis. Moreover, this work may lead to development of non-invasive means of treating patients with chronic wounds or burns or halting metastasis.College of EngineeringNo embargoAcademic Major: Mechanical Engineerin

Dynamics of Macrosystems; Proceedings of a Workshop, September 3-7, 1984

There is an increasing awareness of the important and persuasive role that instability and random, chaotic motion play in the dynamics of macrosystems. Further research in the field should aim at providing useful tools, and therefore the motivation should come from important questions arising in specific macrosystems. Such systems include biochemical networks, genetic mechanisms, biological communities, neutral networks, cognitive processes and economic structures. This list may seem heterogeneous, but there are similarities between evolution in the different fields. It is not surprising that mathematical methods devised in one field can also be used to describe the dynamics of another. IIASA is attempting to make progress in this direction. With this aim in view this workshop was held at Laxenburg over the period 3-7 September 1984. These Proceedings cover a broad canvas, ranging from specific biological and economic problems to general aspects of dynamical systems and evolutionary theory

Some problems in nonlinear diffusion

In this thesis we investigate mathematical models for a number of topics in the field of nonlinear diffusion, using similarity, asymptotic and numerical methods and focussing on the time-asymptotic behaviour in most cases. Firstly, we consider `fast' diffusion in the vicinity of a mask-edge, with application to dopant diffusion into a semiconductor. A variety of approaches are used to determine concentration contours and aspect ratios. Next we consider flow by curvature. Using group analysis, we determine a number of new symmetries for the governing equations in two and three dimensions. By tracking a moving front numerically, we also construct single and double spiral patterns (reminiscent of those observed in the Belousov-Zhabotinskii chemical reaction), and classify the types of behaviour that can occur. Finally, we analyse travelling wave solutions and the behaviour near to extinction for closed loops. We next consider relaxation waves in a system that can be used to model target patterns, also observed in the Belousov-Zhabotinskii reaction. Numerical and asymptotic results are presented, and a number of new cases of front behaviour are obtained. Finally, we investigate a number of systems using an approach based on the WKB method, analysing the motion of invasive fronts and also the form of the pattern left behind. For Fisher's equation, we demonstrate how modulated travelling waves can be obtained by prescribing an oscillatory initial profile. The method is then extended, firstly to Turing systems and then to oscillatory systems, for which we use an additional periodic plane wave argument to determine the unequal front and pattern speeds, as well as the periodicity. Finally, we illustrate how these methods apply to a recently-used `chaotic' model from ecology

Analysis of complex nonlinear reaction-diffusion equations

A mathematical analysis has been carried out for some nonlinear reaction- diffusion equations on open bounded convex domains Ω C R(^d)(d < 3) with Robin boundary conditions- Existence, uniqueness and continuous dependence on initial data of weak and strong solutions are proved. A numerical analysis has also been undertaken for these nonlinear reaction- diffusion equations on the above domains. A fully practical piecewise linear finite element approximation is proposed for which existence and uniqueness of the numerical solution are proved. Semi-discrete and fully discrete error estimates are given. A practical algorithm for computing the numerical solution is given and its convergence is proved. Finally, some numerical simulations in one-dimensional space are exhibited