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Basic mathematical techniques for partial differential equations (PDE) with applications to the life sciences form an integral part of the core curriculum for programs in mathematical biology. Yet, students in such a program with an undergraduate training in biology are typically deficient in any exposure to PDE. This volume starts with simple first order PDE and progresses through higher order equations and systems but with interesting applications, even at the level of a single first order PDE with constant coefficients.

Similar to the two previous volumes by the author, another unique feature of the book is highlighting the scientific theme(s) of interest for the biological phenomena being modelled and analysed. In addition to temporal evolution of a biological phenomenon, its limiting equilibrium states and their stability, the possibility of locational variations leads to a study of additional themes such as (signal and wave) propagation, spatial patterning and robustness. The requirement that biological developments are relatively insensitive to sustained environmental changes provides an opportunity to examine the issue of feedback and robustness not encountered in the previous two volumes of this series.

Contents:

• Evolution:
  • Renewal and Propagation
  • Method of Characteristics
• Diffusion:
  • Brownian Motion and Heat Conduction
  • Eigenfunction Expansions
• Stability:
  • Well-Posedness
  • Offshore Territorial Sovereignty
  • Reaction-Diffusion Systems
• Propagation:
  • Traveling Waves
  • Traveling Waves in Higher Order Nonlinear Models
• Robustness:
  • An Extracellular Model with One Diffusive Morphogen
  • Nonreceptors
  • Robust Development
• Feedback:
  • Early Feedback Models toward Robust Signaling
  • Spatially Uniform Concurrent Feedback
  • Spatially Nonuniform Concurrent Feedback

Readership: Beginning graduate students and upper division undergraduates in mathematical biology. Also accessible to researchers in the life sciences interested in spatially nonuniform models but with little or no background in partial differential equations.

Key Features:

• Provides a working knowledge on basic mathematical techniques for solving partial differential equations for graduate students and researchers in mathematical biology with no prior knowledge in this area
• Formulates and analyzes a number of mathematical models in the life sciences (including one on COVID-19) to illustrate the importance of these techniques for application in that area
• Introduces the models and the relevant methods of analysis through the underlying scientific themes such as evolution, equilibrium, stability, diffusion, propagation, etc., along with two new themes, feedback and robustness, not previously encountered in the two previous volumes
• Describes an important current area of research on biological tissue patterning ab initio leading to a recent breakthrough on the role of feedback for ensuring robust biological development amidst a changing ambient environment, to equip graduate student and researchers interested in spatial dynamics models in the life sciences not having previous exposure to partial differential equations

Spatial Dynamics Models in the Life Sciences and the Role of Feedback in Robust Developments