Biological Growth and Spread: Mathematical Theories and Applications, Proceedings of a Conference Held at Heidelberg, July 16 - 21, 1979

ISBN-13: 9783540102571 / Angielski / Miękka / 1980 / 511 str.

W. Jager; H. Rost; P. Tautu

Opis

These Proceedings have been assembled from papers presented at the Conference on Models of Biological Growth and Spread, held at the German Cancer Research Centre Heidelberg and at the Institute of Applied Mathematics of the University of Heidelberg, July 16-21, 1979. The main theme of the conference was the mathematical representation of biolog- ical populations with an underlying spatial structure. An important feature of such populations is that they and/or their individual com- ponents may interact with each other. Such interactions may be due to external disturbances, internal regulatory factors or a combination of both. Many biological phenomena and processes including embryogenesis, cell growth, chemotaxis, cell adhesion, carcinogenesis, and the spread of an epidemic or of an advantageous gene can be studied in this con- text. Thus, problems of particular importance in medicine (human and veterinary), agriculture, ecology, etc. may be taken into consideration and a deeper insight gained by utilizing (more) realistic mathematical models. Since the intrinsic biological mechanisms may differ considerably from each other, a great variety of mathematical approaches, theories and techniques is required. The aims of the conference were (i) To provide an overview of the most important biological aspects. (ii) To survey and analyse possible stochastic and deterministic approaches. (iii) To encourage new research by bringing together mathematicians interested in problems of a biological nature and scientists actively engaged in developing mathematical models in biology.

Spis treści

Topic I. Proliferation, Spread and Reaction-Dispersal Processes.- Discussion by the Editors.- Ia. Biological Aspects.- Genetic basis of biological growth.- Some observations on the kinetics of haemopoietic stem cells and their relationship to the spatial cellular organisation of the tissue.- Proliferative cell populations in surface epithelia: biological models for cell replacement.- Ib. Mathematical Aspects and Applications.- Density dependent Markov population processes.- A model of development of a cell.- Spatial spread in branching processes.- Pattern formation by bacteria.- Random processes in random environments.- Segregation model and its applications.- Positive recurrence of multi-dimensional population-dependent branching processes.- Population growth with large random fluctuations.- Niche overlap and invasion of competitors in random environments.II. The effects of demographic stochasticity.- Behavior of limiting diffusions for density-dependent branching processes.- Topic II. Random Systems with Locally Interacting Objects.- Discussion by the Editors.- IIa. Theory.- Local interactions with a global signal: a voter model.- Interacting Markov processes.- On a class of branching processes on a lattice with interactions.- IIb. Models.- The asymptotic behavior of a probabilistic model for tumor growth.- An Ising model of a neural network.- Embryogenesis through cellular interactions.- Biological interpretation of a random configuration model for carcinogenesis.- Topic III. Spatial Models in Epidemiology and Genetics.- Discussion by the Editors.- On the spatial spread of a pattern.- Spatial models in the epidemiology of infectious diseases.- Population controlled spread and intensity of diseases in wildlife.- Clines in a discrete time model in population genetics.- Clines induced by variable migration.- Asymptotic behavior of the solutions of the Fisher equation.- Random walks and probabilities of genetic identity in a tree.- Travelling-front solutions for integrodifferential equations II.- Some mathematical considerations of how to stop the spatial spread of a rabies epidemic.- Some deterministic models for the spread of genetic and other alterations.- Topic IV. Models for Cell Motility.- Discussion by the Editors.- Orientation of cells migrating in a chemotactic gradient.- Theoretical models for the specific adhesion of cells to cells or to surfaces.- Chemotaxis in bacteria: a beginner’s guide to the literature.- Assessing the Keller-Segel model: how has it fared?.- Modeling chemosensory responses of swimming eukaryotes.- Mathematical model for tissue inflammation: effects of spatial distribution, cell motility, and Chemotaxis.- Mathematical theories of topotaxis.- Topic V. Stochastic Versus Deterministic Approaches.- Discussion by the Editors.- Connections between stochastic and ordinary integral equations.- Relationships between stochastic and deterministic population models.- Remarks on limit theorems for partial differential equations with random coefficients.- Topic VI. Further Mathematical Approaches and Techniques.- Discussion by the Editors.- Topological techniques in reaction-diffusion equations.- Percolation.- Some ergodic theorems for spatial processes.- Percolation models in two and three dimensions.