Mathematical Models

'Before courses in math modeling became de rigueur, Richard Haberman had already demonstrated that mathematical techniques could be unusually effective in understanding elementary mechanical vibrations, population dynamics, and traffic flow, as well as how such intriguing applications could motivate the further study of nonlinear ordinary and partial differential equations. My students and I can attest that this carefully crafted book is perfect for both self-study and classroom use.' Robert E. O'Malley, Jr, University of Washington

• ForewordPreface to the Classics EditionPrefacePart 1: Mechanical Vibrations. Introduction to Mathematical Models in the Physical SciencesNewton's LawNewton's Law as Applied to a Spring-Mass SystemGravityOscillation of a Spring-Mass SystemDimensions and UnitsQualitative and Quantitative Behavior of a Spring-Mass SystemInitial Value ProblemA Two-Mass OscillatorFrictionOscillations of a Damped SystemUnderdamped OscillationsOverdamped and Critically Damped OscillationsA PendulumHow Small is Small?A Dimensionless Time VariableNonlinear Frictionless SystemsLinearized Stability Analysis of an Equilibrium SolutionConservation of EnergyEnergy CurvesPhase Plane of a Linear OscillatorPhase Plane of a Nonlinear PendulumCan a Pendulum Stop?What Happens if a Pendulum is Pushed Too Hard?Period of a Nonlinear PendulumNonlinear Oscillations with DampingEquilibrium Positions and Linearized StabilityNonlinear Pendulum with DampingFurther Readings in Mechanical VibrationsPart 2: Population Dynamics--Mathematical Ecology. Introduction to Mathematical Models in BiologyPopulation ModelsA Discrete One-Species ModelConstant Coefficient First-Order Difference EquationsExponential GrowthDiscrete Once-Species Models with an Age DistributionStochastic Birth ProcessesDensity-Dependent GrowthPhase Plane Solution of the Logistic EquationExplicit Solution of the Logistic EquationGrowth Models with Time DelaysLinear Constant Coefficient Difference EquationsDestabilizing Influence of DelaysIntroduction to Two-Species ModelsPhase Plane, Equilibrium, and LinearizationSystem of Two Constant Coefficient First-Order Differential Equations, Stability of Two-Species Equilibrium PopulationsPhase Plane of Linear SystemsPredator-Prey ModelsDerivation of the Lotka-Volterra EquationsQualitative Solution of the Lotka-Volterra EquationsAverage Populations of Predators and PreysMan's Influence on Predator-Prey EcosystemsLimitations of the Lotka-Volterra EquationTwo Competing SpeciesFurther Reading in Mathematical EcologyPart 3: Traffic Flow. Introduction to Traffic FlowAutomobile Velocities and a Velocity FieldTraffic Flow and Traffic DensityFlow Equals Density Times VelocityConservation of the Number of CarsA Velocity-Density RelationshipExperimental ObservationsTraffic FlowSteady-State Car-Following ModelsPartial Differential EquationsLinearizationA Linear Partial Differential EquationTraffic Density WavesAn Interpretation of Traffic WavesA Nearly Uniform Traffic Flow ExampleNonuniform Traffic - The Method of CharacteristicsAfter a Traffic Light Turns GreenA Linear Velocity-Density RelationshipAn ExampleWave Propagation of Automobile Brake LightsCongestion AheadDiscontinuous TrafficUniform Traffic Stopped by a Red LightA Stationary Shock WaveThe Earliest ShockValidity of LinearizationEffect of a Red Light or an AccidentExits and EntrancesConstantly Entering CarsA Highway EntranceFurther Reading in Traffic FlowIndex.