Essential Mathematical Biology

This monograph presents the timed input/output automaton (TIOA) modeling framework, a basic mathematical framework to support description and analysis of timed (computing) systems. Timed systems are systems in which desirable correctness or performance properties of the system depend on the timing of events, not just on the order of their occurrence. Timed systems are employed in a wide range of domains including communications, embedded systems, real-time operating systems, and automated control. Many applications involving timed systems have strong safety, reliability, and predictability requirements, which makes it important to have methods for systematic design of systems and rigorous analysis of timing-dependent behavior. An important feature of the TIOA framework is its support for decomposing timed system descriptions. In particular, the framework includes a notion of external behavior for a TIOA, which captures its discrete interactions with its environment. The framework also defines what it means for one TIOA to implement another, based on an inclusion relationship between their external behavior sets, and defines notions of simulations, which provide sufficient conditions for demonstrating implementation relationships. The framework includes a composition operation for TIOAs, which respects external behavior, and a notion of receptiveness, which implies that a TIOA does not block the passage of time "Essential Mathematical Biology is a self-contained introduction to the fast-growing field of mathematical biology. Written for students with a mathematical background, it sets the subject in its historical context and then guides the reader towards questions of current research interest, providing a comprehensive overview of the field and a solid foundation for interdisciplinary research in the biological sciences."--BOOK JACKET. Read more... 1. Single Species Population Dynamics -- 2. Population Dynamics of Interacting Species -- 3. Infectious Diseases -- 4. Population Genetics and Evolution -- 5. Biological Motion -- 6. Molecular and Cellular Biology -- 7. Pattern Formation -- 8. Tumour Modelling -- A. Some Techniques for Difference Equations -- B. Some Techniques for Ordinary Differential Equations -- C. Some Techniques for Partial Differential Equations -- D. Non-negative Matrices Cover......Page 1 Title Page......Page 2 Copyright Page......Page 3 Preface......Page 4 Contents......Page 6 List of Figures......Page 12 1.1 Introduction......Page 16 1.2 Linear and Nonlinear First Order Discrete Time Models......Page 17 1.2.1 The Biology of Insect Population Dynamics......Page 18 1.2.2 A Model for Insect Population Dynamics with Competition......Page 19 1.3 Differential Equation Models......Page 26 1.4 Evolutionary Aspects......Page 31 1.5 Harvesting and Fisheries......Page 32 1.6 Metapopulations......Page 36 1.7 Delay Effects......Page 39 1.8 Fibonacci's Rabbits......Page 42 1.9 Leslie Matrices: Age-structured Populations in Discrete Time......Page 45 1.10.1 Discrete Time......Page 49 1.10.2 Continuous Time......Page 53 1.11 The McKendrick Approach to Age Structure......Page 56 1.12 Conclusions......Page 59 2. Population Dynamics of Interacting Species......Page 62 2.2 Host-parasitoid Interactions......Page 63 2.3 The Lotka-Volterra Prey-predator Equations......Page 69 2.4 Modelling the Predator Functional Response......Page 75 2.5 Competition......Page 81 2.6 Ecosystems Modelling......Page 85 2.7 Interacting Metapopulations......Page 89 2.7.1 Competition......Page 90 2.7.2 Predation......Page 92 2.7.3 Predator-mediated Coexistence of Competitors......Page 93 2.7.4 Effects of Habitat Destruction......Page 94 2.8 Conclusions......Page 96 3.1 Introduction......Page 98 3.2 The Simple Epidemic and SIS Diseases......Page 101 3.3 SIR Epidemics......Page 105 3.4 SIR Endemics......Page 111 3.4.1 No Disease-related Death......Page 112 3.4.2 Including Disease-related Death......Page 114 3.5 Eradication and Control......Page 115 3.6.1 The Equations......Page 118 3.6.2 Steady State......Page 120 3.7 Vector-borne Diseases......Page 122 3.8 Basic Model for Macroparasitic Diseases......Page 124 3.9 Evolutionary Aspects......Page 128 3.10 Conclusions......Page 130 4.1 Introduction......Page 132 4.2 Mendelian Genetics in Populations with Non-overlapping Generations......Page 134 4.3 Selection Pressure......Page 138 4.4.2 Selection for a Recessive Allele......Page 142 4.4.4 The Additive Case......Page 144 4.5 Analytical Approach for Weak Selection......Page 145 4.6 The Balance Between Selection and Mutation......Page 146 4.7 Wright's Adaptive Topography......Page 148 4.8 Evolution of the Genetic System......Page 149 4.9 Game Theory......Page 151 4.10 Replicator Dynamics......Page 157 4.11 Conclusions......Page 160 5.1 Introduction......Page 162 5.2.1 General Derivation......Page 163 5.2.2 Some Particular Cases......Page 166 5.3 Directed Motion, or Taxis......Page 169 5.4.1 Steady State Equations in One Spatial Variable......Page 171 5.4.2 Transit Times......Page 172 5.5 Biological Invasions: A Model for Muskrat Dispersal......Page 175 5.6 Travelling Wave Solutions of General Reaction-diffusion Equations......Page 179 5.6.1 Node-saddle Orbits (the Monostable Equation)......Page 181 5.6.2 Saddle-saddle Orbits (the Bistable Equation)......Page 182 5.7 Travelling Wave Solutions of Systems of Reaction-diffusion Equations: Spatial Spread of Epidemics......Page 183 5.8 Conclusions......Page 187 6.1 Introduction......Page 190 6.2 Biochemical kinetics......Page 191 6.3 Metabolic Pathways......Page 198 6.3.1 Activation and Inhibition......Page 199 6.3.2 Cooperative Phenomena......Page 201 6.4 Neural Modelling......Page 206 6.5 Immunology and AIDS......Page 212 6.6 Conclusions......Page 217 7.1 Introduction......Page 220 7.2 Turing Instability......Page 221 7.3 Turing Bifurcations......Page 226 7.4.1 Conditions for Turing Instability......Page 229 7.4.2 Short-range Activation, Long-range Inhibition......Page 234 7.4.3 Do Activator-inhibitor Systems Explain Biological Pattern Formation?......Page 238 7.5 Bifurcations with Domain Size......Page 239 7.6 Incorporating Biological Movement......Page 244 7.8 Conclusions......Page 248 8.1 Introduction......Page 250 8.2 Phenomenological Models......Page 252 8.3 Nutrients: the Diffusion-limited Stage......Page 255 8.4 Moving Boundary Problems......Page 257 8.5 Growth Promoters and Inhibitors......Page 260 8.6 Vascularisation......Page 262 8.7 Metastasis......Page 263 8.8 Immune System Response......Page 264 8.9 Conclusions......Page 266 Further Reading......Page 268 A.1.1 Graphical Analysis......Page 272 A.1.2 Linearisation......Page 273 A.2.1 Saddle-node Bifurcations......Page 275 A.2.2 Transcritical Bifurcations......Page 276 A.2.3 Pitchfork Bifurcations......Page 277 A.2.4 Period-doubling or Flip Bifurcations......Page 278 A.3 Systems of Linear Equations: Jury Conditions......Page 281 A.4 Systems of Nonlinear Difference Equations......Page 282 A.4.2 Bifurcation for Systems......Page 283 B.1.1 Geometric Analysis......Page 286 B.1.3 Linearisation......Page 287 B.2.1 Geometric Analysis (Phase Plane)......Page 288 B.2.2 Linearisation......Page 289 B.2.3 Poincare-Bendixson Theory......Page 291 B.3 Some Results and Techniques for mth Order Systems......Page 292 B.3.2 Lyapunov Functions......Page 293 B.4.1 Bifurcations with Eigenvalue Zero......Page 294 B.4.2 Hopf Bifurcations......Page 295 C.1 First-order Partial Differential Equations and Characteristics......Page 298 C.2.1 The Fundamental Solution......Page 299 C.2.2 Connection with Probabilities......Page 302 C.2.3 Other Coordinate Systems......Page 303 C.3 Some Spectral Theory for Laplace's Equation......Page 304 C.4 Separation of Variables in Partial Differential Equations......Page 306 C.5 Systems of Diffusion Equations with Linear Kinetics......Page 310 C.6 Separating the Spatial Variables from Each Other......Page 312 D.1 Perron-Frobenius Theory......Page 314 E. Hints for Exercises......Page 316 Index......Page 344 Back Cover......Page 351 Essential Mathematical Biology Is A Self-contained Introduction To The Fast-growing Field Of Mathematical Biology. Written For Students With A Mathematical Background, It Sets The Subject In Its Historical Context And Then Guides The Reader Towards Questions Of Current Research Interest, Providing A Comprehensive Overview Of The Field And A Solid Foundation For Interdisciplinary Research In The Biological Sciences. A Broad Range Of Topics Is Covered Including: Population Dynamics, Infectious Diseases, Population Genetics And Evolution, Dispersal, Molecular And Cellular Biology, Pattern Formation, And Cancer Modelling. This Book Will Appeal To 3rd And 4th Year Undergraduate Students Studying Mathematical Biology. A Background In Calculus And Differential Equations Is Assumed, Although The Main Results Required Are Collected In The Appendices. A Dedicated Website At Www.springer.co.uk/britton/ Accompanies The Book And Provides Further Exercises, More Detailed Solutions To Exercises In The Book, And Links To Other Useful Sites. Single Species Population Dynamics -- Population Dynamics Of Interacting Species -- Infectious Diseases -- Population Genetics And Evolution -- Biological Motion -- Molecular And Cellular Biology -- Pattern Formation -- Tumour Modelling -- Further Reading -- Some Techniques For Difference Equations -- Some Techniques For Ordinary Differential Equations -- Some Techniques For Partial Differential Equations -- Non-negative Matrices -- Hints For Exercises -- Index. N.f. Britton. Includes Index. Includes Bibliographical References (pages 253-256) And Index. This self-contained introduction to the fast-growing field of Mathematical Biology is written for students with a mathematical background. It sets the subject in a historical context and guides the reader towards questions of current research interest. A broad range of topics is covered including: Population dynamics, Infectious diseases, Population genetics and evolution, Dispersal, Molecular and cellular biology, Pattern formation, and Cancer modelling. Particular attention is paid to situations where the simple assumptions of homogenity made in early models break down and the process of mathematical modelling is seen in action.