Analysis and Control of Polynomial Dynamic Models with Biological Applications synthesizes three mathematical background areas (graphs, matrices and optimization) to solve problems in the biological sciences (in particular, dynamic analysis and controller design of QP and polynomial systems arising from predator-prey and biochemical models). The book puts a significant emphasis on applications, focusing on quasi-polynomial (QP, or generalized Lotka-Volterra) and kinetic systems (also called biochemical reaction networks or simply CRNs) since they are universal descriptors for smooth nonlinear systems and can represent all important dynamical phenomena that are present in biological (and also in general) dynamical systems.
1. Introduction 1. Dynamic models for describing biological phenomena 2. Kinetic systems 3. QP models 2. Basic Notions 1. General nonlinear system representation in the form of ODEs 2. Formal introduction of the QP model form 3. Introduction of kinetic models with mass action and rational reaction rates 4. Basic relations between kinetic and QP models 3. Model Transformations and Equivalence Classes 1. Alfine and linear positive diagonal transformations 2. Nonlinear diagonal transformations 3. Quasi-monomial transformation and the corresponding equivalence classes of QP systems 4. Embedding transformations and the relationship between classes of positive polynomial systems 4. Model analysis 1. Stability analysis of QP models 2. Stability of kinetic systems 3. Invariants (first integrals) for QP and kinetic systems 4. Relations between the Lyapunov functions of QP and kinetic models 5. Computational analysis of the structure of kinetic systems 5. Stabilizing feedback control design 1. Stabilizing control of QP systems by using optimization 2. Stabilizing state feedback control of nonnegative polynomial systems using special CRN realizations 3. Robustness issues and robust design for the stabilizing control of polynomial systems 6. Case studies 1. Optimization-based structural analysis and design of reaction networks 2. Computational distinguishability analysis of an uncertain kinetic model 3. Stability analysis and stabilizing control of fermentation processes in QP form A. Notations and abbreviations B. Mathematical tools B.1. Directed graphs B.2. Matrices of key importance B.3. Basics of the applied computational tools B.4. Basic notions from systems and control theory