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2023 | OriginalPaper | Buchkapitel

6. Nonlinear Dynamical Systems

verfasst von : Glenn Ledder

Erschienen in: Mathematical Modeling for Epidemiology and Ecology

Verlag: Springer International Publishing

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Abstract

In Chap. 4, we studied the dynamics of a single variable. Now we look at the dynamics of nonlinear systems. The methods available to us are analogous to the methods we used for single-variable dynamics, but with some important differences. The phase line for one variable scales up to the phase plane for two variables; however, there is no graphical method for discrete systems. The analytical method for determining stability with the derivative scales up to higher dimensions, but with technical complications that rapidly increase as the number of variables increases. When possible, we can greatly simplify the analysis of a model by using an appropriate approximation to reduce the number of dynamic equations.

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Vorheriges Kapitel Discrete Linear Systems

Section 4.3.

Section 3.8.

See also Sect. 4.3.

There are analytical methods that can sometimes be used to prove global stability, but they are far more sophisticated than anything in this book.

See Project 6C.

See Sects. 6.2 and 6.3.

Section 3.7.

Macrophages are a type of white blood cell. See Problem 3.3.10 for more background.

You should see very different outcomes for the two scenarios in spite of their being nearly identical. There is a curve in the phase plane called a separatrix that separates the plane into domains of attraction for the different stable equilibria. The initial conditions were chosen to give two solutions very close to the separatrix but on opposite sides.

Section 3.7.

Problem 6.2.14.

Repeated roots have to contribute multiple solutions. If the eigenvalue equation has a factor \((\lambda -\lambda _1)^2\), then obviously one solution is \(e^{\lambda _1 t}\). It can be shown that the other solution is \(te^{\lambda _1 t}\). This is important if we need solution formulas, but irrelevant to the question of stability.

Complex pairs of roots have to contribute two solutions. The eigenvalue pair \(\alpha \pm i \beta \) corresponds to solutions \(e^{\alpha t} \cos \beta t\) and \(e^{\alpha t} \sin \beta t\). These solutions oscillate, with the rate of oscillation controlled by \(\beta \); however, only the exponential factor affects stability.

Section 3.8.

See Appendix B for the definition of the partial derivative and methods for calculation.

See Appendix F.

This model generalizes the SIS model, which has \(\phi \rightarrow \infty \), as well as the SIR model, although deriving the SIS model from it is nontrivial because the scaling we used is based on the assumption that the duration of immunity is long compared to the duration of the disease.

Problem 6.3.15.

“Routh” rhymes with “mouth.”

Problem 6.3.16.

Problem 6.3.11.

See Appendix G for detailed guidelines and Problem 6.2.1 for a simple example.

Calculation of \(c_3\) is still messy, but the two hardest parts of the stability demonstration are considerably simplified by the assumption that \(\epsilon \) is small.

on-ko-sir-KI-a-sis.

While we could formally demonstrate this in conjunction with real data using AIC (Sect. 2.4), it is clearly acceptable modeling practice to make this decision based on general principles, in the absence of hard data.

Section 6.1.

See Appendix F.

Problem 6.4.2.

Appendix B.

Theorem 4.4.1.

Problem 6.5.7.

Section 6.3.

Using \(L=f \ell /2\), \(Y=by/2\), \(A=ba/2\), \(r=fs_1/b\), and \(s=s_2\).

The patterns are even more complicated than they first appear. If you look for periodicity in the \(r=15\) case, you find what appears to be a cycle of period 171, but in fact there is a slight drifting away from a true periodic solution.

Stumbling across a good set of parameter values by chance can be like throwing a dart at a fog-obscured board and managing to hit the bullseye. Finding good parameter values after the analysis is like walking up to the board and stabbing the bullseye with the dart.

Problem 6.5.1.

PAIR-uh-si-toid.

Section 3.1.

Neutrally stable periodic solutions have an amplitude determined by the initial conditions, whereas the amplitude of a limit cycle is determined entirely by the system parameters.

Note that \(f'<0\) and \(R_0>1\), so both quantities in the inequality are positive.

It is common in practice to use discrete models for cases such as this, where the stage durations are approximately comparable. As noted in Sect. 6.5.3, this is not a good modeling decision. It is not clear whether the complex behavior of the model represents complex biological behavior or is merely an artifact of the use of a discrete model. Nevertheless, the model is very interesting from a mathematical perspective.

The reported values have as many as four significant figures, suggesting a high degree of precision. However, such apparent precision is unwarranted in light of the limited accuracy of the values; for example, the reported values for s differ by a factor of almost 2. One should not put too much faith in reported parameter values in ecology or epidemiology; in particular, it is misleading to use values that appear to indicate a high degree of precision. One should be particularly careful of theoretical results that rely on a narrow range of parameter values.

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Titel: Nonlinear Dynamical Systems
verfasst von: Glenn Ledder
Verlag: Springer International Publishing
Buch: Mathematical Modeling for Epidemiology and Ecology
Print ISBN: 978-3-031-09453-8

Electronic ISBN: 978-3-031-09454-5

Copyright-Jahr: 2023
DOI: https://doi.org/10.1007/978-3-031-09454-5_6

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