List of Figures

Figure 1.1. Decline in Coho salmon stocks on the Thomson River, BC

Figure 1.2. Rate of decline on individual streams related to habitat variables

Figure 1.3. Frequency distribution of recruits per spawner for the Bear, Marsh, and Sulphur Creek stocks of Chinook salmon

Figure 1.4. Compartment diagram of the bathtub model

Figure 1.5. Data on the rate of hydrolysis of sucrose by invertase fitted to the Michaelis-Menten equation

Figure 1.6. Compartment diagram for the enzyme-mediated biochemical reaction

Figure 1.7. Graphical analysis of dv/dt = k₁s − (k₋₁ + k₂ + k₁s)v

Figure 1.8. Compartment model for arsenic in rat hepatocytes

Figure 1.9. Compartment diagram for the IEUBK model for lead in children

Figure 1.10. A cartoon of the pogo stick or monopode

Figure 1.11. The monopode in stance phase

Figure 1.12. Diagram of the experimental fishpond system showing water flows between compartments

Figure 1.13. Compartment diagrams for Exercise 1.3

Figure 2.1. The standard size-class model

Figure 2.2. Stage-structured model for killer whales

Figure 2.3. Eigenvalue elasticities for stage-structured model for desert tortoise

Figure 2.4. Eigenvalue elasticities for stage-structured model for loggerhead sea turtles

Figure 2.5. The LPA model

Figure 2.6. Eigenvalue elasticities for matrix models of northern monkshood using two different size populations

Figure 2.7. Elasticity surface for the integral projection model for northern monkshood

Figure 3.1. Membrane-spanning channel with equivalent electrical circuit

Figure 3.2. Single-channel recording of the nicotinic acetylcholine receptor channel in the neuromuscular junction

Figure 3.3. Hypothetical energy landscape

Figure 3.4. Histogram from a computer experiment of tossing an unfair coin

Figure 3.5. Sums of exponentials fit to dwell time distributions of open states in an nAChR channel

Figure 3.6. Diagram of the states and transitions in the Markov chain model used to model the nAChR channel

Figure 3.7. Data from a series of voltage clamp measurements on a squid giant axon

Figure 3.8. Separation of voltage clamp currents in the squid axon into sodium and potassium currents

Figure 3.9. Plots of the functions 1/[1 + exp(−v)] and 1/cosh(−v/2)

Figure 3.10. Voltage clamp currents in a barnacle muscle fiber

Figure 4.1. Schematic diagram of the repressilator

Figure 4.2. Solutions of the repressilator equations with initial conditions (m_lacI, m_tetR, m_cl, p_IacI, p_tetR, p_cl) = (0.2, 0.1, 0.3, 0.1, 0.4, 0.5) and parameters (a₀, α, β, n) = (0, 50, 0.2, 2)

Figure 4.3. Solutions of the repressilator equations with initial conditions (m_lacI, m_tetR, m_cl, P_lacI, p_tetR, p_cl) = (0.2, 0.1, 0.3, 0.1, 0.4, 0.5) and parameters (α₀, α, β, n) = (0, 1, 0.2, 2)

Figure 4.4. Solutions of the repressilator equations with initial conditions (m_lacI, m_tetR, m_cl, p_lacI, p_tetR, p_cl) = (0.2, 0.1, 0.3, 0.1, 0.4, 0.5) and parameters (α₀, α, β, n) = (1, 50, 0.2, 2)

Figure 4.5. Graph of the function E(p) = −p + 18/(1 + p²) + 1

Figure 4.6. Diagram of the stability region for the repressilator system

Figure 4.7. Schematic diagram of bistable gene regulatory networks in bacteriophage λ

Figure 4.8. Plots of the nullclines for the model [4.5] in the (u, v) plane for values of a below, above, and exactly at the bifurcation between one and three equilibria for the model

Figure 4.9. Bifurcation diagram of the (b, a) plane for the model [4.5]

Figure 4.10. Phase portrait of the (u, v) plane for the model [4.5] with (b, a) = (3, 1)

Figure 4.11. Phase portrait of the (u, v) plane for the model [4.5] with (b, a) = (3, 2)

Figure 5.1. Phase line of a one-dimensional differential equation with three equilibrium points

Figure 5.2. Plots of solutions x(t) to the differential equation

Figure 5.3. “Quiver plot” of the vector field defined by the differential equations [5.5]

Figure 5.4. Log-log plot of the residual (error) |r(h)| obtained in calculating the derivative of the function f(x) = 1 + x + 3x² with a finite-difference formula

Figure 5.5. Phase portraits of two-dimensional linear vector fields with a stable node and a saddle

Figure 5.6. Phase portraits of two-dimensional linear vector fields with a stable node and a saddle with eigenvectors that are not orthogonal

Figure 5.7. Phase portraits of two-dimensional linear vector fields with a stable focus and a center

Figure 5.8. Limit set in a two-dimensional vector field

Figure 5.9. Nullclines for the Morris-Lecar model

Figure 5.10. Phase portrait of the Morris-Lecar model for parameter values given by Set 1 in Table 5.1

Figure 5.11. Time series of (v, w) for the large periodic orbit of the Morris-Lecar model shown in Figure 5.10

Figure 5.12. Phase portrait of the Morris-Lecar model for parameter values given by Set 2 in Table 5.1

Figure 5.13. Dynamics of system [5.22] along the x-axis for varying μ

Figure 5.14. The determinant of the Jacobian plotted as a function of v

Figure 5.15. Nullclines for the Morris-Lecar system for saddle-node parameter values

Figure 5.16. Phase portraits of the supercritical Hopf bifurcation for μ = −0.1 and 0.2

Figure 5.17. Phase portraits of the subcritical Hopf bifurcation for μ = −0.2 and 0.1

Figure 5.18. Phase portrait of the Morris-Lecar model with a single periodic orbit

Figure 5.19. Phase portraits of the Morris-Lecar system close to a saddle-node of periodic orbits

Figure 5.20. Phase portraits of the Morris-Lecar system close to a homoclinic orbit

Figure 5.20. [

Figure 5.21. Bifurcation curves of saddle-node and Hopf bifurcation in the plane of the parameters (g_Ca, )

Figure 5.22. Phase portraits of the Morris-Lecar system close to a Takens-Bogdanov codimension-2 bifurcation

Figure 5.23. (a) Nullclines of the Morris-Lecar system at a cusp point; (b) saddle-node curve in the (g_Ca, i) parameter plane

Figure 6.1. Examples of epidemic curves

Figure 6.2. Deaths per week from plague in the island of Bombay 1905–1906

Figure 6.3. Graphical illustration that equation [6.5] has a unique solution between x = 0 and x = 1

Figure 6.4. Monthly case report time series for measles and chickenpox in New York City prior to vaccination, and their power spectra

Figure 6.5. Monthly case report totals for measles and chickenpox in New York City prior to vaccination and power spectra of the case report

Figure 6.6. Output from a finite-population SEIR model with parameters appropriate for prevaccination dynamics of measles

Figure 6.7. Compartment diagram for discrete-event SIR model

Figure 6.8. Reported gonorrhea rate in the United States

Figure 6.9. Compartment diagram for HIV transmission dynamics in the presence of antiretrovivral therapy

Figure 6.10. Fraction of new HIV infections that are resistant to combination ARV treatment

Figure 6.11. Schematic depiction of the typical course of HIV infection in an adult

Figure 6.12. Decay of plasma viral load in two patients following treatment with a protease inhibitor

Figure 6.13. Viral load data versus model predictions during first phase of viral decay after onset of treatment

Figure 7.1. Depiction of particles undergoing a random walk on the line

Figure 7.2. Animal coat pattern of the eastern chipmunk

Figure 7.3. Insect color patterns: two beetles from the genus Anoplophora

Figure 7.4. Trajectories of the vector field [7.17]

Figure 7.5. Spiral patterns of the BZ reaction

Figure 7.6. Map of spiral wave in the (x, y) plane into the (c₁, c₂) concentration plane

Figure 7.7. Meandering paths of spiral cores in simulations of a reaction-diffusion system

Figure 8.1. Results from simulations of the individual-based model for larval fish growth and survival in the presence of size-dependent predation

Figure 8.2. Genealogies of successive dominant genotypes under low and high mutation rates

Figure 8.3. Relationship between resource supply rate and the number of distinct species descended from the ancestor

Figure 8.4. Relaxation oscillations in the van der Pol model [8.17] with C = 20

Figure 8.5. A simulation of the Helbing et al. (2000) model for exit panic

Figure 9.1. Outline of the modeling process

Figure 9.2. Outline of the steps in developing a model

Figure 9.3. Diagram of the compartments for soil organic matter in the CENTURY model

Figure 9.4. Compartment diagram of the model for TAME in the rat

Figure 9.5. Fitting a parametric rate equation model

Figure 9.6. Some widely used equations to fit nonlinear functional relationships

Figure 9.7. Cross-validation for linear versus quadratic regression

Figure 9.8. Examples of nonparametric rate equations based on data

Figure 9.9. Fitting and testing a parametric model for the distribution of size in surviving thistle plants

Figure 9.10. Results from calibration of the θ-logistic population model to data on population growth of Paramecium aurelia feeding on bacteria

Figure 9.11. The Doomsday model fitted to historical data on human population growth