Mathematics in Population Biology

Cover
Title Page
Copyright Page
Contents
Preface
Chapter 1. Some General Remarks on Mathematical Modeling
	Bibliographic Remarks
PART 1. BASIC POPULATION GROWTH MODELS
	Chapter 2. Birth, Death, and Migration
		2.1 The Fundamental Balance Equation of Population Dynamics
		2.2 Birth Date Dependent Life Expectancies
		2.3 The Probability of Lifetime Emigration
	Chapter 3. Unconstrained Population Growth for Single Species
		3.1 Closed Populations
			3.1.1 The Average Intrinsic Growth Rate for Periodic Environments
			3.1.2 The Average Intrinsic Growth Rate for Nonperiodic Environments
		3.2 Open Populations
			3.2.1 Nonzero Average Intrinsic Growth Rate
			3.2.2 Zero Average Intrinsic Growth Rate
	Chapter 4. Von Bertalanffy Growth of Body Size
	Chapter 5. Classic Models of Density-Dependent Population Growth for 37 Single Species
		5.1 The Bernoulli and the Verhulst Equations
		5.2 The Beverton–Holt and Smith Differential Equation
			5.2.1 Derivation from a Resource–Consumer Model
			5.2.2 Derivation from Cannibalism of Juveniles by Adults
		5.3 The Ricker Differential Equation
		5.4 The Gompertz Equation
		5.5 A First Comparison of the Various Equations
	Chapter 6. Sigmoid Growth
		6.1 General Conditions for Sigmoid Growth
		6.2 Fitting Sigmoid Population Data
	Chapter 7. The Allee Effect
		7.1 First Model Derivation: Search for a Mate
		7.2 Second Model Derivation: Impact of a Satiating Generalist Predator
		7.3 Model Analysis
	Chapter 8. Nonautonomous Population Growth:Asymptotic Equality of 75 Population Sizes
	Chapter 9. Discrete-Time Single-Species Models
		9.1 The Discrete Analog of the Verhulst (Logistic) and the Bernoulli Equation: the Beverton–Holt Difference Equation and Its Generalization
		9.2 The Ricker Difference Equation
		9.3 Some Analytic Results for Scalar Difference Equations
		9.4 Some Remarks Concerning the Quadratic Difference Equation
		Bibliographic Remarks
	Chapter 10. Dynamics of an Aquatic Population Interacting with a Polluted Environment
		10.1 Modeling Toxicant and Population Dynamics
		10.2 Open Loop Toxicant Input
		10.3 Feedback Loop Toxicant Input
		10.4 Extinction and Persistence Equilibria and a Threshold Condition for Population Extinction
		10.5 Stability of Equilibria and Global Behavior of Solutions
		10.6 Multiple Extinction Equilibria, Bistability and Periodic Oscillations
		10.7 Linear Dose Response
		Bibliographic Remarks
	Chapter 11. Population Growth Under Basic Stage Structure
		11.1 A Most Basic Stage-Structured Model
		11.2 Well-Posedness and Dissipativity
		11.3 Equilibria and Reproduction Ratios
		11.4 Basic Reproduction Ratios and Threshold Conditions for Extinction versus Persistence
		11.5 Weakly Density-Dependent Stage-Transition Rates and Global Stability of Nontrivial Equilibria
		11.6 The Number and Nature of Possible Multiple Nontrivial Equilibria
		11.7 Strongly Density-Dependent Stage-Transition Rates and Periodic Oscillations
		11.8 An Example for Multiple Periodic Orbits and Both Supercritical and Subcritical Hopf Bifurcation
		11.9 Multiple Interior Equilibria, Bistability, and Many Bifurcations for Pure Intrastage Competition
		Bibliographic Remarks
PART 2. STAGETRANSITIONS AND DEMOGRAPHICS
	Chapter 12. TheTransitionThrough a Stage
		12.1 The Sojourn Function
		12.2 Mean Sojourn Time, Expected Exit Age, and Expectation of Life
		12.3 The Variance of the Sojourn Time, Moments and Central Moments
		12.4 Remaining Sojourn Time and Its Expectation
		12.5 Fixed Stage Durations
		12.6 Per Capita Exit Rates (Mortality Rates)
		12.7 Exponentially Distributed Stage Durations
		12.8 Log-Normally Distributed Stage Durations
		12.9 A Stochastic Interpretation of Stage Transition
		Bibliographic Remarks
	Chapter 13. Stage Dynamics with Given Input
		13.1 Input and Stage-Age Density
		13.2 The Partial Differential Equation Formulation
		13.3 Stage Content and Average Stage Duration
		13.4 Average Stage Age
		13.5 Stage Exit Rates
			13.5.1 The Fundamental Balance Equation of Stage Dynamics
			13.5.2 Average Age at Stage Exit
		13.6 Stage Outputs
		13.7 Which Recruitment Curves Can Be Explained by Cannibalism of Newborns by Adults?
		Bibliographic Remarks
	Chapter 14. Demographics in an Unlimiting Constant Environment
		14.1 The Renewal Equation
		14.2 Balanced Exponential Growth
		14.3 The Renewal Theorem: Approach to Balanced Exponential Growth
	Chapter 15. Some Demographic Lessons from Balanced Exponential Growth
		15.1 Inequalities and Estimates for the Malthusian Parameter
		15.2 Average Age and Average Age at Death in a Population at Balanced Exponential Growth. Average Per Capita Death Rate
		15.3 Ratio of Population Size and Birth Rate
		15.4 Consequences of an Abrupt Shift in Maternity: Momentum of Population Growth
		Bibliographic Remarks
	Chapter 16. Some Nonlinear Demographics
		16.1 A Demographic Model with a Juvenile and an Adult Stage
		16.2 A Differential Delay Equation
		Bibliographic Remarks
PART 3. HOST–PARASITE POPULATION GROWTH: EPIDEMIOLOGY OF INFECTIOUS DISEASES
	Chapter 17. Background
		17.1 Impact of Infectious Diseases in Past and Present Time
		17.2 Epidemiological Terms and Principles
		Bibliographic Remarks
	Chapter 18. The Simplifed Kermack–McKendrick Epidemic Model
		18.1 A Model with Mass-Action Incidence
		18.2 Phase-Plane Analysis of the Model Equations. The Epidemic Threshold Theorem
		18.3 The Final Size of the Epidemic. Alternative Formulation of the Threshold Theorem
	Chapter 19. Generalization of the Mass-Action Law of Infection
		19.1 Population-Size Dependent Contact Rates
		19.2 Model Modi?cation
		19.3 The Generalized Epidemic Threshold Theorem
	Chapter 20. The Kermack–McKendrick Epidemic Model with Variable Infectivity
		20.1 A Stage-Age Structured Model
		20.2 Reduction to a Scalar Integral Equation
		Bibliographic Remarks
	Chapter 21. SEIR (-> S)Type Endemic Models for“Childhood Diseases”
		21.1 The Model and Its Well-Posedness
		21.2 Equilibrium States and the Basic Replacement Ratio
		21.3 The Disease Dynamics in the Vicinities of the Disease-Free and the Endemic Equilibrium: Local Stability and the Interepidemic Period
		21.4 Some Global Results: Extinction, Persistence of the Disease; Conditions for Attraction to the Endemic Equilibrium
		Bibliographic Remarks
	Chapter 22. Age-Structured Models for Endemic Diseases and 341 OptimalVaccination Strategies
		22.1 A Model with Chronological Age-Structure
		22.2 Disease-Free and Endemic Equilibrium: the Replacement Ratio
		22.3 The Net Replacement Ratio, and Disease Extinction and Persistence
		22.4 Cost of Vaccinations and Optimal Age Schedules
		22.5 Estimating the Net Replacement Ratio: Average Duration of Susceptibility and Average Age at Infection. Optimal Vaccination Schedules Revisited
		Bibliographic Remarks
	Chapter 23. Endemic Models with Multiple Groups or Populations
		23.1 The Model
		23.2 Equilibrium Solutions
		23.3 Local Asymptotic Stability of Strongly Endemic Equilibria
		23.4 Extinction or Persistence of the Disease?
		23.5 The Basic Replacement Matrix, Alias Next-Generation Matrix
		23.6 The Basic Replacement Ratio as Spectral Radius of the Basic Replacement Matrix
		23.7 Some Special Cases of Mixing
		Bibliographic Remarks
PART 4. TOOLBOX
	Appendix A Ordinary Differential Equations
		A.1 Conservation of Positivity and Boundedness
		A.2 Planar Ordinary Differential Equation Systems
		A.3 The Method of Fluctuations
		A.4 Behavior in the Vicinity of an Equilibrium
		A.5 Elements of Persistence Theory
		Bibliographic Remarks
		A.6 Global Stability of a Compact Minimal Set
		A.7 Hopf Bifurcation
		A.8 Perron–Frobenius Theory of Positive Matrices andAssociated Linear Dynamical Systems
		Bibliographic Remarks
	Appendix B Integration, Integral Equations, and Some Convex Analysis
		B.1 The Stieltjes Integral of Regulated Functions
		B.2 Some Elements from Measure Theory
		B.3 Some Elements from Convex Analysis
		B.4 Lebesgue–Stieltjes Integration
		B.5 Jensen’s Inequality and Related Material
		B.6 Volterra Integral Equations
		B.7 Critical and Regular Values of a Function
		Bibliographic Remarks
	Appendix C Some MAPLE Worksheets with Comments for Part 1
		C.1 Fitting the Growth of the World Population (Figure 3.1)
		C.2 Periodic Modulation of Exponential Growth in Closed Populations (Figures 3.2 and 3.3)
		C.3 Fitting Sigmoid Population-Growth Curves (Figures 6.1 and 6.2)
		C.4 Fitting Bernoulli’s Equation to Population Data of Sweden (Figure 6.3)
		C.5 Illustrating the Allee Effect (Figures 7.2–7.4)
		C.6 Dynamics of an Aquatic Population Interacting with a Polluted Environment (Figure 10.3E)
References
Index