Mathematical Modeling

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This book introduces modeling by a collection of ordinary differential or difference equations, calibrating the equations against data, checking quantitative predictions against events and understanding the qualitative patterns suggested by the model. One term of differential calculus is enough to get started and two terms are enough to finish. The topics covered are: Richardson's Model of Arms Races, Phase Portraits: Sketching the Phase Plane, Numerical Methods for Initial Value Problems, Modeling Population: Malthus' exponential model, growth rates, the Logistic Model, Discrete Time Reproduction Models, Overshoot and collapse models, Errors: Regression, Conditioning, Sensitivity and Predictability in Models, The Lotka-Volterra Model: Population Oscillations, Conservative Systems, Harvesting, General Models of Interacting Populations, Epidemics: SIR Models, Temporary Immunity, Latency and Asymptomatic Carriers, Persistent Oscillations: Limit cycles, Examples via polar coordinates, Poincaré-Bendixon Theory, Hopf bifurcations, Oscillations in the Holling-Tanner Model: The Development of Predator-Prey Models, Analysis of the Holling-Tanner model, Testing the model, and Business Cycles: Business cycle theories, basic difficulties, Goodwin's model, Conclusions from Goodwin's model.