Deepen students’ understanding of biological phenomena Suitable for courses on differential equations with applications to mathematical biology or as an introduction to mathematical biology, Differential Equations and Mathematical Biology, Second Edition introduces students in the physical, mathematical, and biological sciences to fundamental modeling and analytical techniques used to understand biological phenomena. In this edition, many of the chapters have been expanded to include new and topical material. New to the Second Edition A section on spiral waves Recent developments in tumor biology More on the numerical solution of differential equations and numerical bifurcation analysis MATLAB® files available for download online Many additional examples and exercises This textbook shows how first-order ordinary differential equations (ODEs) are used to model the growth of a population, the administration of drugs, and the mechanism by which living cells divide. The authors present linear ODEs with constant coefficients, extend the theory to systems of equations, model biological phenomena, and offer solutions to first-order autonomous systems of nonlinear differential equations using the Poincar phase plane. They also analyze the heartbeat, nerve impulse transmission, chemical reactions, and predator prey problems. After covering partial differential equations and evolutionary equations, the book discusses diffusion processes, the theory of bifurcation, and chaotic behavior. It concludes with problems of tumor gr2owth and the spread of infectious diseases.

Introduction Population growth Administration of drugs Cell division Differential equations with separable variables Equations of homogeneous type Linear differential equations of the first order Numerical solution of first-order equations Symbolic computation in MATLAB Linear Ordinary Differential Equations with Constant Coefficients Introduction First-order linear differential equations Linear equations of the second order Finding the complementary function Determining a particular integral Forced oscillations Differential equations of order n Uniqueness Systems of Linear Ordinary Differential Equations First-order systems of equations with constant coefficients Replacement of one differential equation by a system The general system The fundamental system Matrix notation Initial and boundary value problems Solving the inhomogeneous differential equation Numerical solution of linear boundary value problems Modelling Biological Phenomena Introduction Heartbeat Nerve impulse transmission Chemical reactions Predator–prey models First-Order Systems of Ordinary Differential Equations Existence and uniqueness Epidemics The phase plane and the Jacobian matrix Local stability Stability Limit cycles Forced oscillations Numerical solution of systems of equations Symbolic computation on first-order systems of equations and higher-order equations Numerical solution of nonlinear boundary value problems Appendix: existence theory Mathematics of Heart Physiology The local model The threshold effect The phase plane analysis and the heartbeat model Physiological considerations of the heartbeat cycle A model of the cardiac pacemaker Mathematics of Nerve Impulse Transmission Excitability and repetitive firing Travelling waves Qualitative behavior of travelling waves Piecewise linear model Chemical Reactions Wavefronts for the Belousov–Zhabotinskii reaction Phase plane analysis of Fisher’s equation Qualitative behavior in the general case Spiral waves and λ − ω systems Predator and Prey Catching fish The effect of fishing The Volterra–Lotka model Partial Differential Equations Characteristics for equations of the first order Another view of characteristics Linear partial differential equations of the second order Elliptic partial differential equations Parabolic partial differential equations Hyperbolic partial differential equations The wave equation Typical problems for the hyperbolic equation The Euler–Darboux equation Visualization of solutions Evolutionary Equations The heat equation Separation of variables Simple evolutionary equations Comparison theorems Problems of Diffusion Diffusion through membranes Energy and energy estimates Global behavior of nerve impulse transmissions Global behavior in chemical reactions Turing diffusion driven instability and pattern formation Finite pattern forming domains Bifurcation and Chaos Bifurcation Bifurcation of a limit cycle Discrete bifurcation and period-doubling Chaos Stability of limit cycles The Poincaré plane Averaging Numerical Bifurcation Analysis Fixed points and stability Path-following and bifurcation analysis Following stable limit cycles Bifurcation in discrete systems Strange attractors and chaos Stability analysis of partial differential equations Growth of Tumors Introduction Mathematical model I of tumor growth Spherical tumor growth based on model I Stability of tumor growth based on model I Mathematical model II of tumor growth Spherical tumor growth based on model II Stability of tumor growth based on model II Epidemics The Kermack–McKendrick model Vaccination An incubation mode Spreading in space Answers to Selected Exercises Index

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