BASIC MATHEMATICS FOR THE BIOLOGICAL AND SOCIAL SCIENCES
CONTENTS PART I CALCULUS AND MODELING A BRIEF SUMMARY OF CALCULUS WORKING WITH PARAMETERS SCALING PARAMETERS NONLINEAR PARAMETERS BIFURCATIONS RATES OF CHANGE AND THE DERIVATIVE RATE OF CHANGE FOR A FUNCTION OF DISCRETE TIME RATE OF CHANGE FOR A FUNCTION OF CONTINUOUS TIME THE DERIVATIVE SLOPE OF A TANGENT TO A GRAPH COMPUTING DERIVATIVES TWO NOTATIONS ELEMENTARY DERIVATIVE FORMULAS GENERAL DERIVATIVE RULES PARTIAL DERIVATIVES LOCAL BEHAVIOR AND LINEAR APPROXIMATION TANGENT LINES LOCAL EXTREMA LINEAR APPROXIMATION OPTIMIZATION THE MARGINAL VALUE THEOREM RELATED RATES DIFFERENTIAL EQUATIONS THE CHAIN RULE ACCUMULATION AND THE DE?NITE INTEGRAL ESTIMATING TOTAL VOLUME FROM FLOW RATE DATA THE DE?NITE INTEGRAL APPLICATIONS OF THE DE?NITE INTEGRAL PROPERTIES OF THE DE?NITE INTEGRAL THE FUNDAMENTAL THEOREM OF CALCULUS COMPUTING DE?NITE INTEGRALS WITH THE FUNDAMENTAL THEOREM COMPUTING ANTIDERIVATIVES AND DE?NITE INTEGRALS SUBSTITUTION CONSTRUCTING ANTIDERIVATIVES WITH THE FUNDAMENTAL THEOREM OBTAINING A GRAPH OF F FROM A GRAPH OF F ? REFERENCES MATHEMATICAL MODELING MATHEMATICS IN BIOLOGY BIOLOGICAL DATA OVERALL PATTERNS IN A RANDOM WORLD DETERMINING RELATIONSHIPS BASIC CONCEPTS OF MODELING MECHANISTIC AND EMPIRICAL MODELING AIMS OF MATHEMATICAL MODELING THE NARROW AND BROAD VIEWS OF MATHEMATICAL MODELS ACCURACY, PRECISION, AND INTERPRETATION OF RESULTS EMPIRICAL MODELING I: FITTING LINEAR MODELS TO DATA THE BASIC LINEAR LEAST SQUARES METHOD (Y = MX) ADAPTING THE METHOD TO THE GENERAL LINEAR MODEL IMPLIED ASSUMPTIONS OF LEAST SQUARES EMPIRICAL MODELING II: FITTING SEMILINEAR MODELS TO DATA FITTING THE EXPONENTIAL MODEL BY LINEAR LEAST SQUARES LINEAR LEAST SQUARES FIT FOR THE POWER FUNCTION MODEL Y = AXP SEMILINEAR LEAST SQUARES MECHANISTIC MODELING I: CREATING MODELS FROM BIOLOGICAL PRINCIPLES CONSTRUCTING MECHANISTIC MODELS DIMENSIONAL ANALYSIS A MECHANISTIC MODEL FOR RESOURCE CONSUMPTION A MORE SOPHISTICATED MODEL FOR FOOD CONSUMPTION A COMPARTMENT MODEL FOR POLLUTION IN A LAKE “LET THE BUYER BEWARE” MECHANISTIC MODELING II: EQUIVALENT FORMS NOTATION ALGEBRAIC EQUIVALENCE DIFFERENT PARAMETERS VISUALIZING MODELS WITH GRAPHS DIMENSIONLESS VARIABLES DIMENSIONLESS FORMS EMPIRICAL MODELING III: CHOOSING AMONG MODELS QUANTITATIVE ACCURACY COMPLEXITY THE AKAIKE INFORMATION CRITERION CHOOSING AMONG MODELS SOME RECOMMENDATIONS PART II PROBABILITY PROBABILITY DISTRIBUTIONS CHARACTERIZING DATA TYPES OF DATA DISPLAYING DATA MEASURES OF CENTRAL TENDENCY MEASURING VARIABILITY CONCEPTS OF PROBABILITY EXPERIMENTS, OUTCOMES, AND RANDOM VARIABLES SEQUENCES AND COMPLEMENTS DISCRETE PROBABILITY DISTRIBUTIONS DISTRIBUTION FUNCTIONS EXPECTED VALUE, MEAN, AND STANDARD DEVIATION THE BINOMIAL DISTRIBUTION BERNOULLI TRIALS AND BINOMIAL DISTRIBUTIONS CONTINUOUS PROBABILITY DISTRIBUTIONS CUMULATIVE DISTRIBUTION FUNCTIONS PROBABILITY DENSITY FUNCTIONS EXPECTED VALUE, MEAN, AND VARIANCE THE NORMAL DISTRIBUTION THE STANDARD NORMAL DISTRIBUTION STANDARD PROBABILITY INTERVALS THE POISSON AND EXPONENTIAL DISTRIBUTIONS THE POISSON DISTRIBUTION THE EXPONENTIAL DISTRIBUTION MEMORY IN PROBABILITY DISTRIBUTIONS WORKING WITH PROBABILITY AN INTRODUCTION TO STATISTICAL INFERENCE TESTS ON PROBABILITY DISTRIBUTIONS A GRAPHICAL TEST FOR DISTRIBUTION TYPE THE CRAMER–VON MISES TEST OUTLIERS PROBABILITY DISTRIBUTIONS OF SAMPLES SUMS AND MEANS OF TWO RANDOM VARIABLES GENERAL CHARACTERISTICS OF SAMPLES MEANS AND STANDARD DEVIATIONS OF SAMPLE SUMS AND MEANS DISTRIBUTIONS OF SAMPLE MEANS INFERENCES ABOUT POPULATIONS INFERENCES ABOUT SAMPLE MEANS INFERENCES ABOUT SAMPLE PROPORTIONS ESTIMATING PARAMETERS FOR DISTRIBUTIONS CON?DENCE INTERVALS ESTIMATING SUCCESS PROBABILITY WITH A LIKELIHOOD FUNCTION ESTIMATING POPULATION SIZE BY MARK AND RECAPTURE A FINAL OBSERVATION CONDITIONAL PROBABILITY THE CONCEPT OF CONDITIONAL PROBABILITY FORMULAS FOR CONDITIONAL PROBABILITY THE PARTITION RULE CONDITIONAL PROBABILITY APPLIED TO DIAGNOSTIC TESTS THE STANDARD TEST INTERPRETATION PROBLEM PART III DYNAMICAL SYSTEMS DYNAMICS OF SINGLE POPULATIONS DISCRETE POPULATION MODELS DISCRETE EXPONENTIAL GROWTH THE DISCRETE LOGISTIC MODEL SIMULATIONS FIXED POINTS COBWEB ANALYSIS COBWEB DIAGRAMS STABILITY ANALYSIS CONTINUOUS POPULATION MODELS EXPONENTIAL GROWTH LOGISTIC GROWTH EQUILIBRIUM SOLUTIONS A RENEWABLE RESOURCE MODEL EQUILIBRIA OF THE RENEWABLE RESOURCE MODEL A GRAPHICAL METHOD PHASE LINE ANALYSIS THE PHASE LINE THE PHASE LINE FOR THE HOLLING TYPE III RENEWABLE RESOURCE MODEL COMPARISON OF GRAPHICAL METHODS LINEARIZED STABILITY ANALYSIS STABILITY ANALYSIS FOR DISCRETE MODELS: A MOTIVATING EXAMPLE STABILITY ANALYSIS FOR DISCRETE MODELS: THE GENERAL CASE STABILITY ANALYSIS FOR CONTINUOUS MODELS SIMILARITIES AND DIFFERENCES DISCRETE DYNAMICAL SYSTEMS DISCRETE LINEAR SYSTEMS SIMPLE STRUCTURED MODELS FINDING THE GROWTH RATE AND STABLE DISTRIBUTION GENERAL PROPERTIES OF DISCRETE LINEAR STRUCTURED MODELS A MATRIX ALGEBRA PRIMER MATRICES AND VECTORS POPULATION MODELS IN MATRIX NOTATION THE CENTRAL PROBLEM OF MATRIX ALGEBRA THE DETERMINANT THE EQUATION AX = 0 LONG-TERM BEHAVIOR OF LINEAR MODELS EIGENVALUES AND EIGENVECTORS SOLUTIONS OF XT+1 = MXT LONG-TERM BEHAVIOR CONTINUOUS DYNAMICAL SYSTEMS PHARMACOKINETICS AND COMPARTMENT MODELS COMPARTMENT MODELS A SIMPLI?ED MODEL THE DIMENSIONLESS MODEL ENZYME KINETICS NONDIMENSIONALIZATION SIMULATION THE BRIGGS–HALDANE APPROXIMATION PHASE PLANE ANALYSIS SOLUTION CURVES IN THE PHASE PLANE NULLCLINES NULLCLINE ANALYSIS USING SMALL PARAMETERS NULLCLINE ANALYSIS IN GENERAL STABILITY IN LINEAR SYSTEMS LINEAR SYSTEMS EIGENVALUES AND STABILITY THE ROUTH–HURWITZ CONDITIONS THE ROUTH–HURWITZ CONDITIONS FOR A THREE-DIMENSIONAL SYSTEM STABILITY IN NONLINEAR SYSTEMS APPROXIMATING A NONLINEAR SYSTEM AT AN EQUILIBRIUM POINT THE JACOBIAN AND STABILITY PRIMARY HIV INFECTION REDUCTION TO TWO VARIABLES EQUILIBRIA AND STABILITY