Core models

Several models are used repeatedly in one or more chapters to provide a common reference point for the introduction and testing of new methods and ideas. To help you decide how to allocate your time, these core models, their point of first introduction, and their subsequent appearances are described below (see p. ) and summarized in table 1.

 

Introduced Used

in section in sections

Rock model: v' = -g 1.2 1.3; 3.2; 4.1, 4.4; 6.6

Simple: P' = kP 2.1 3.1-3.2; 4.1-4.3

Emigration: P' = kP - E 2.2.1 3.2-3.3; 4.2-4.4 10.4-10.5

Logistic: P' = aP - s P² 2.2.2 3.2-3.3; 4.1-4.3

2.3 3.1-3.3; 4.1, various

$$T' = -(Ak/cm)(T - T_{\mathrm{out}})$$ exercises in 4.2-4.5 10.4; 10.5, exercise 5

Predator-prey (2.24-2.25), p. 67 2.4.1 7.1, exercises 2-3 7.2, exercises 12-13 8.1; 8.3, exercises 9, 33

Epidemic (SIR, SIRS) 2.4.2 7.1, exercises 2-3

(2.29-2.30), p. 71 7.2.3; 7.2, exercises 9-11 8.1; 8.3, exercises 9, 26 cover of text

Competition (2.35-2.36), p. 73 2.4.3 7.1, exercises 2-3 7.2, exercises 14-17 8.3, exercise 9

Spring-mass: 5.1 6.1, 6.4-6.7

mx'' + px' + kx = F(t) 10.5, exercises 6, 10

$$L \theta'' + g \sin \theta = 0$$ 5.2 6.1, 6.5.2, 6.8 7.1-7.2; 8.3 10.5, exercise 9

-D(A c')' +(A V c)' = 0 9.1 9.2-9.3; 11.2

9.4 9.5-9.6

$$\partial T/\partial t = \kappa \partial T^2/\partial x^2$$ 11.5; 11.7, exercise 25

 

As suggested in the preceding section, your options for depth of coverage range from ``Take my word for it.'' to deriving each model in painstaking detail. I advocate the middle ground, plausibility arguments with back-up reading and homework assignments, in addition to a few complete derivations and a few models issued by decree from higher authority.

Briefly, the first-order scalar population models (Malthus, emigration, logistic) and heat-flow derived in chapter 2, Models from Conservation Laws, appear repeatedly in chapters 3 and 4, which develop numerical, graphical, and analytical ideas for first-order scalar equations. The simple harmonic oscillator (spring-mass model) and the linear and nonlinear pendulum derived in sections 5.1 and 5.2 are the standard examples in chapter 6, which treats second-order analytic solution methods and applies numerical methods. The pendulum and the SIRS (epidemic) system (subsection 2.4.2) motivate phase plane analysis in chapter 7, Graphical Tools for Two Dimensions, and linearized phase plane analysis in section 8.3, Connections with the Phase Plane (after sections 8.1 and 8.2 have solved constant-coefficient, homogeneous linear systems.)

Second-order boundary-value problems, ordinary and partial respectively, build upon the steady-state diffusion model of section 9.1, Diffusion Models, and upon the heat equation developed in section 9.4, Time-dependent Diffusion.

Chapter 10, The Laplace Transform, uses as examples linear first- and second-order initial-value problems involving population and heat-flow from chapter 2 and spring-mass systems from section 5.1. Some of the additional analytic methods of chapter 11, More Analytic Tools for Two Dimensions, also draw upon the diffusion equations of sections 9.1 and 9.4 for examples of equations with nonconstant coefficients or singular points.

For more detail, the point of first introduction, and subsequent uses of the primary models, see table 1 or the following paragraphs. For an overview of the connections between mathematical ideas and models, see tables 3, 4, and 5.