General Solutions and Transformations—Models with

Multiple Variables

In this chapter, we describe recipes and techniques for finding general solutions of models with multiple variables. As we shall see, there is a recipe for finding the general solution to any linear model, regardless of the number of variables. General solutions can also sometimes be obtained for nonlinear models using transformations, but there is no single recipe. When a general solution cannot be found using transformations, we can sometimes use another technique known as a separation of time scales to approximate the dynamics. Although this technique is only approximate, it often provides a more thorough understanding of a model's behavior.

In section 9.2 we derive the general solution to linear models with multiple variables. Section 9.3 then considers nonlinear models with multiple variables. There we revisit the use of transformations (Chapter 6) in the context of models with multiple variables. Then we describe how a separation of times scales can be used to approximate the general solution.

Linear equations involving multiple variables commonly arise in models of age- or class-structured populations (Chapters 7 and 10), and they are also used to approximate the behavior of nonlinear models in the vicinity of equilibria (Chapter 8). Here, we derive the exact general solution to linear models with multiple variables. We focus on discrete-time models and provide the general solution for continuous-time models in Box 9.3 below. General solutions allow us to generate exact predictions when accuracy matters, and they also provide greater insight into why a model behaves the way that it does.

Consider a set of linear recursion equations describing the state of a population in the next generation, n,(t + 1), as a function of its state in the current generation, n,(f). As with single-variable models, linear multivariable models have one of two forms. The model might involve d equations in d variables without constant terms; i.e., ni(t + 1) = mn"i(t) + tn12n2(t) + ••• + mund{t), n2(t + 1) = mzlnx{t) + m22n2(t) + ■■• + m2dnd(t), (g ^

Chapter Coals:

• To derive the general solution to linear models involving multiple variables

• To examine methods for solving nonlinear models with multiple variables

• To introduce transformations for nonlinear models with multiple variables

• To obtain approximate general solutions using a separation of time scales

Chapter Concepts:

• Long-term dynamics

• Proportional transformation

• Separation of time scales

• Quasi-linkage equilibrium nd(t + 1) = mdi nx(f) + md2n2(t) + ••• + mddnd(t).

In Chapter 7, we explored several models of this form, such as model (7.27) describing the number of occupied sea anemones of different colors.

Alternatively, the model might be affine and involve d equations in d variables with the addition of constant terms. As in the one-variable case, we can apply Steps 1-3 of Recipe 6.1 to every variable in a multiple-variable affine model, transforming it into the form (9.1) (see Problem 9.2). For example, we can write a discrete-time affine model involving two variables as

«i(t + 1) = an^t) + bn2(t) + a, n2(t + 1) = cn,{t) + dn2{t) + p.

If we transform the system into the new variables Sj = nx - ft, and S2 = n2 - h2, describing the distance of the system from this equilibrium, the recursions then become

8x{t + 1) = a8x{t) + b82(t), Sz(t + 1) = cS1(t) + dS2(t)

(Recipe 6.1). These recursions are linear and no longer contain constant terms. Therefore, they can be analyzed in the same way that we analyze nonaffine models (9.1).

The recursion equations (9.1) can be written in matrix notation as h(t + 1) = M«(f), where

and M

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