Non-Local Cell Adhesion Models: Symmetries and Bifurcations in 1-D (CMS/CAIMS Books in Mathematics, 1)

Main subject categories: • Differential equations • Mathematical biology • Mathematical modelling of biologic processes • Cell adhesionMathematics Subject Classification: • 35R09 Integro-partial differential equations • 45K05 Integro-partial differential equations • 35Q92 Integro-partial differential equations • 92C15 Developmental biology, pattern formation • 47G20 Integro-differential operatorsThis monograph considers the mathematical modeling of cellular adhesion, a key interaction force in cell biology. While deeply grounded in the biological application of cell adhesion and tissue formation, this monograph focuses on the mathematical analysis of non-local adhesion models. The novel aspect is the non-local term (an integral operator), which accounts for forces generated by long ranged cell interactions. The analysis of non-local models has started only recently, and it has become a vibrant area of applied mathematics. This monograph contributes a systematic analysis of steady states and their bifurcation structure, combining global bifurcation results pioneered by Rabinowitz, equivariant bifurcation theory, and the symmetries of the non-local term. These methods allow readers to analyze and understand cell adhesion on a deep level. Preface Contents Part I Introduction 1 Introduction 1.1 The Effect of Cellular Adhesions in Tissues 1.2 Prior Modelling of Cellular Adhesions 1.3 Non-local Partial Differential Equation Models 1.4 Outline of the Main Results 2 Preliminaries 2.1 Biological Derivation of the Non-local Adhesion Model 2.2 Introduction to Nonlinear Analysis 2.3 Abstract Bifurcation Theory 2.4 The Averaging Operator in Periodic Domains 2.5 Local and Global Existence 2.6 Adhesion Potential Part II The Periodic Problem 3 Basic Properties 3.1 Non-dimensionalization and Mass Conservation 3.2 The Non-local Operator in 1-D 3.3 Spectral Properties 3.4 The Behavior of `3́9`42`"̇613A``45`47`"603AKR for R→0 3.5 Properties of Steady-State Solutions 3.6 Summary 4 Local Bifurcation 4.1 The Abstract Bifurcation Problem 4.2 Symmetries and Equivariant Flows 4.3 Singular Points of `3́9`42`"̇613A``45`47`"603AF 4.4 Local Bifurcation Result 4.5 Summary 5 Global Bifurcation 5.1 An Area Function 5.1.1 A Non-local Maximum Principle 5.2 Global Bifurcation Branches for Linear Adhesion Function 5.3 Bifurcation Type for Linear Adhesion Function 5.4 Stability of Solutions 5.5 Numerical Verification 5.5.1 Numerical Implementation 5.5.2 Numerical Test Cases 5.6 Summary of the Analytical Challenges 5.7 Further Reading Part III Non-local Equations with Boundary Conditions 6 No-Flux Boundary Conditions for Non-local Operators 6.1 Non-local No-Flux Boundary Conditions 6.1.1 Independent Fluxes 6.1.2 Dependent Flux 6.2 Naive Boundary Conditions 6.3 No-Flux Boundary Conditions 6.3.1 Approximate Steady States for No-Flux Non-local Term 6.4 General Sampling Domain 6.4.1 Set Convergence 6.4.2 Continuity near the Boundary 6.4.3 Differentiation near the Boundary 6.5 Local and Global Existence 6.6 Neutral Boundary Conditions 6.7 Weakly Adhesive and Repulsive Boundary Conditions 6.8 Conclusion 7 Discussion and Future Directions 7.1 Further Thoughts 7.2 Systems and Higher Dimensions References Index "This monograph considers the mathematical modeling of cellular adhesion, a key interaction force in cell biology. While deeply grounded in the biological application of cell adhesion and tissue formation, this monograph focuses on the mathematical analysis of non-local adhesion models. The novel aspect is the non-local term (an integral operator), which accounts for forces generated by long ranged cell interactions. The analysis of non-local models has started only recently, and it has become a vibrant area of applied mathematics. This monograph contributes a systematic analysis of steady states and their bifurcation structure, combining global bifurcation results pioneered by Rabinowitz, equivariant bifurcation theory, and the symmetries of the non-local term. These methods allow readers to analyze and understand cell adhesion on a deep level."-- Back cover