A. General Results for the Laplacian Operator in Bounded Domains
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Table of Contents (continued)
CONTENTS, VOLUME I
J.D. Murray: Mathematical Biology, I: An Introduction
Preface to the Third Edition
vii
Preface to the First Edition
xi
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40
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44
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49
53
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62
67
69
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Models for Interacting Populations
3.1
Predator–Prey Models: Lotka–Volterra Systems . . . . . . . . . . . .
3.2
Complexity and Stability . . . . . . . . . . . . . . . . . . . . . . . .
79
79
83
2.
3.
Continuous Population Models for Single Species
1.1
Continuous Growth Models . . . . . . . . . . . . . . . . . . . .
1.2
Insect Outbreak Model: Spruce Budworm . . . . . . . . . . . .
1.3
Delay Models . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4
Linear Analysis of Delay Population Models: Periodic Solutions
1.5
Delay Models in Physiology: Periodic Dynamic Diseases . . . .
1.6
Harvesting a Single Natural Population . . . . . . . . . . . . .
1.7
Population Model with Age Distribution . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Discrete Population Models for a Single Species
2.1
Introduction: Simple Models . . . . . . . . . . .
2.2
Cobwebbing: A Graphical Procedure of Solution
2.3
Discrete Logistic-Type Model: Chaos . . . . . .
2.4
Stability, Periodic Solutions and Bifurcations . .
2.5
Discrete Delay Models . . . . . . . . . . . . . .
2.6
Fishery Management Model . . . . . . . . . . .
2.7
Ecological Implications and Caveats . . . . . . .
2.8
Tumour Cell Growth . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . .
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xxii
Contents, Volume I
3.3
3.4
Realistic Predator–Prey Models . . . . . . . . . . . . .
Analysis of a Predator–Prey Model with Limit Cycle
Periodic Behaviour: Parameter Domains of Stability . .
3.5
Competition Models: Competitive Exclusion Principle
3.6
Mutualism or Symbiosis . . . . . . . . . . . . . . . .
3.7
General Models and Cautionary Remarks . . . . . . .
3.8
Threshold Phenomena . . . . . . . . . . . . . . . . .
3.9
Discrete Growth Models for Interacting Populations . .
3.10 Predator–Prey Models: Detailed Analysis . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Temperature-Dependent Sex Determination (TSD)
4.1
Biological Introduction and Historical Asides on the Crocodilia .
4.2
Nesting Assumptions and Simple Population Model . . . . . . .
4.3
Age-Structured Population Model for Crocodilia . . . . . . . .
4.4
Density-Dependent Age-Structured Model Equations . . . . . .
4.5
Stability of the Female Population in Wet Marsh Region I . . . .
4.6
Sex Ratio and Survivorship . . . . . . . . . . . . . . . . . . . .
4.7
Temperature-Dependent Sex Determination (TSD) Versus
Genetic Sex Determination (GSD) . . . . . . . . . . . . . . . .
4.8
Related Aspects on Sex Determination . . . . . . . . . . . . . .
Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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119
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135
137
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Modelling the Dynamics of Marital Interaction: Divorce Prediction
and Marriage Repair
5.1
Psychological Background and Data:
Gottman and Levenson Methodology . . . . . . . . . . . . . . .
5.2
Marital Typology and Modelling Motivation . . . . . . . . . . .
5.3
Modelling Strategy and the Model Equations . . . . . . . . . .
5.4
Steady States and Stability . . . . . . . . . . . . . . . . . . . .
5.5
Practical Results from the Model . . . . . . . . . . . . . . . . .
5.6
Benefits, Implications and Marriage Repair Scenarios . . . . . .
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147
150
153
156
164
170
Reaction Kinetics
6.1
Enzyme Kinetics: Basic Enzyme Reaction . . . . . .
6.2
Transient Time Estimates and Nondimensionalisation
6.3
Michaelis–Menten Quasi-Steady State Analysis . . .
6.4
Suicide Substrate Kinetics . . . . . . . . . . . . . .
6.5
Cooperative Phenomena . . . . . . . . . . . . . . .
6.6
Autocatalysis, Activation and Inhibition . . . . . . .
6.7
Multiple Steady States, Mushrooms and Isolas . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
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175
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178
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188
197
201
208
215
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146
Biological Oscillators and Switches
218
7.1
Motivation, Brief History and Background . . . . . . . . . . . . . . . 218
7.2
Feedback Control Mechanisms . . . . . . . . . . . . . . . . . . . . . 221
Contents, Volume I
Oscillators and Switches with Two or More Species:
General Qualitative Results . . . . . . . . . . . . . .
7.4
Simple Two-Species Oscillators: Parameter Domain
Determination for Oscillations . . . . . . . . . . . .
7.5
Hodgkin–Huxley Theory of Nerve Membranes:
FitzHugh–Nagumo Model . . . . . . . . . . . . . .
7.6
Modelling the Control of Testosterone Secretion and
Chemical Castration . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
xxiii
7.3
8.
9.
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BZ Oscillating Reactions
8.1
Belousov Reaction and the FieldKăorăosNoyes (FKN) Model
8.2
Linear Stability Analysis of the FKN Model and Existence
of Limit Cycle Solutions . . . . . . . . . . . . . . . . . . . .
8.3
Nonlocal Stability of the FKN Model . . . . . . . . . . . . .
8.4
Relaxation Oscillators: Approximation for the
Belousov–Zhabotinskii Reaction . . . . . . . . . . . . . . . .
8.5
Analysis of a Relaxation Model for Limit Cycle Oscillations
in the Belousov–Zhabotinskii Reaction . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Perturbed and Coupled Oscillators and Black Holes
9.1
Phase Resetting in Oscillators . . . . . . . . . . . . . . . .
9.2
Phase Resetting Curves . . . . . . . . . . . . . . . . . . . .
9.3
Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4
Black Holes in Real Biological Oscillators . . . . . . . . . .
9.5
Coupled Oscillators: Motivation and Model System . . . . .
9.6
Phase Locking of Oscillations: Synchronisation in Fireflies .
9.7
Singular Perturbation Analysis: Preliminary Transformation
9.8
Singular Perturbation Analysis: Transformed System . . . .
9.9
Singular Perturbation Analysis: Two-Time Expansion . . . .
9.10 Analysis of the Phase Shift Equation and Application
to Coupled Belousov–Zhabotinskii Reactions . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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257
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278
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10. Dynamics of Infectious Diseases
10.1 Historical Aside on Epidemics . . . . . . . . . . . . . . . . .
10.2 Simple Epidemic Models and Practical Applications . . . . .
10.3 Modelling Venereal Diseases . . . . . . . . . . . . . . . . . .
10.4 Multi-Group Model for Gonorrhea and Its Control . . . . . . .
10.5 AIDS: Modelling the Transmission Dynamics of the Human
Immunodeficiency Virus (HIV) . . . . . . . . . . . . . . . . .
10.6 HIV: Modelling Combination Drug Therapy . . . . . . . . . .
10.7 Delay Model for HIV Infection with Drug Therapy . . . . . .
10.8 Modelling the Population Dynamics of Acquired Immunity to
Parasite Infection . . . . . . . . . . . . . . . . . . . . . . . .
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315
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327
331
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xxiv
Contents, Volume I
10.9
10.10
10.11
10.12
Age-Dependent Epidemic Model and Threshold Criterion .
Simple Drug Use Epidemic Model and Threshold Analysis
Bovine Tuberculosis Infection in Badgers and Cattle . . .
Modelling Control Strategies for Bovine Tuberculosis
in Badgers and Cattle . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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11. Reaction Diffusion, Chemotaxis, and Nonlocal Mechanisms
11.1 Simple Random Walk and Derivation of the Diffusion Equation
11.2 Reaction Diffusion Equations . . . . . . . . . . . . . . . . . . .
11.3 Models for Animal Dispersal . . . . . . . . . . . . . . . . . . .
11.4 Chemotaxis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.5 Nonlocal Effects and Long Range Diffusion . . . . . . . . . . .
11.6 Cell Potential and Energy Approach to Diffusion
and Long Range Effects . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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395
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12. Oscillator-Generated Wave Phenomena
12.1 Belousov–Zhabotinskii Reaction Kinematic Waves . . . . . . .
12.2 Central Pattern Generator: Experimental Facts in the Swimming
of Fish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3 Mathematical Model for the Central Pattern Generator . . . . .
12.4 Analysis of the Phase Coupled Model System . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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13. Biological Waves: Single-Species Models
13.1 Background and the Travelling Waveform . . . . . . . . . . .
13.2 Fisher–Kolmogoroff Equation and Propagating Wave Solutions
13.3 Asymptotic Solution and Stability of Wavefront Solutions
of the Fisher–Kolmogoroff Equation . . . . . . . . . . . . . .
13.4 Density-Dependent Diffusion-Reaction Diffusion Models
and Some Exact Solutions . . . . . . . . . . . . . . . . . . .
13.5 Waves in Models with Multi-Steady State Kinetics:
Spread and Control of an Insect Population . . . . . . . . . .
13.6 Calcium Waves on Amphibian Eggs: Activation Waves
on Medaka Eggs . . . . . . . . . . . . . . . . . . . . . . . .
13.7 Invasion Wavespeeds with Dispersive Variability . . . . . . .
13.8 Species Invasion and Range Expansion . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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467
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484
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14. Use and Abuse of Fractals
14.1 Fractals: Basic Concepts and Biological Relevance . . .
14.2 Examples of Fractals and Their Generation . . . . . . .
14.3 Fractal Dimension: Concepts and Methods of Calculation
14.4 Fractals or Space-Filling? . . . . . . . . . . . . . . . . .
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Contents, Volume I
xxv
Appendices
501
A. Phase Plane Analysis
501
B. Routh-Hurwitz Conditions, Jury Conditions, Descartes’
Rule of Signs, and Exact Solutions of a Cubic
B.1 Polynomials and Conditions . . . . . . . . . . . . . . . . . . . . . .
B.2 Descartes’ Rule of Signs . . . . . . . . . . . . . . . . . . . . . . . .
B.3 Roots of a General Cubic Polynomial . . . . . . . . . . . . . . . . .
507
507
509
510
Bibliography
513
Index
537
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1. Multi-Species Waves and
Practical Applications
1.1 Intuitive Expectations
In Volume 1 we saw that if we allowed spatial dispersal in the single reactant or species,
travelling wavefront solutions were possible. Such solutions effected a smooth transition
between two steady states of the space independent system. For example, in the case
of the Fisher–Kolmogoroff equation (13.4), Volume I, wavefront solutions joined the
steady state u = 0 to the one at u = 1 as shown in the evolution to a propagating wave in
Figure 13.1, Volume I. In Section 13.5, Volume I, where we considered a model for the
spatial spread of the spruce budworm, we saw how such travelling wave solutions could
be found to join any two steady states of the spatially independent dynamics. In this and
the next few chapters, we shall consider systems where several species—cells, reactants,
populations, bacteria and so on—are involved, concentrating, but not exclusively, on
reaction diffusion chemotaxis mechanisms, of the type derived in Sections 11.2 and
11.4, Volume I. In the case of reaction diffusion systems (11.18), Volume I, we have
∂u
= f(u) + D∇ 2 u,
∂t
(1.1)
where u is the vector of reactants, f the nonlinear reaction kinetics and D the matrix of
diffusivities, taken here to be constant.
Before analysing such systems let us try to get some intuitive idea of what kind of
solutions we might expect to find. As we shall see, a very rich spectrum of solutions it
turns out to be. Because of the analytical difficulties and algebraic complexities that can
be involved in the study of nonlinear systems of reaction diffusion chemotaxis equations, an intuitive approach can often be the key to getting started and to what might be
expected. In keeping with the philosophy in this book such intuition is a crucial element
in the modelling and analytical processes. We should add the usual cautionary caveat,
that it is mainly stable travelling wave solutions that are of principal interest, but not always. The study of the stability of such solutions is not usually at all simple, particularly
in two or more space dimensions, and in many cases has still not yet been done.
Consider first a single reactant model in one space dimension x, with multiple
steady states, such as we discussed in Section 13.5, Volume I, where there are 3 steady
states u i , i = 1, 2, 3 of which u 1 and u 3 are stable in the spatially homogeneous situation. Suppose that initially u is at one steady state, u = u 1 say, for all x. Now suppose
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1. Multi-Species Waves and Practical Applications
we suddenly change u to u 3 in x < 0. With u 3 dominant the effect of diffusion is to
initiate a travelling wavefront, which propagates into the u = u 1 region and so eventually u = u 3 everywhere. As we saw, the inclusion of diffusion effects in this situation
resulted in a smooth travelling wavefront solution for the reaction diffusion equation.
In the case of a multi-species system, where f has several steady states, we should reasonably expect similar travelling wave solutions that join steady states. Although mathematically a spectrum of solutions may exist we are, of course, only interested here in
nonnegative solutions. Such multi-species wavefront solutions are usually more difficult to determine analytically but the essential concepts involved are more or less the
same, although there are some interesting differences. One of these can arise with interacting predator–prey models with spatial dispersal by diffusion. Here the travelling
front is like a wave of pursuit by the predator and of evasion by the prey: we discuss one
such case in Section 1.2. In Section 1.5 we consider a model for travelling wavefronts in
the Belousov–Zhabotinskii reaction and compare the analytical results with experiment.
We also consider practical examples of competition waves associated with the spatial
spread of genetically engineered organisms and another with the red and grey squirrel.
In the case of a single reactant or population we saw in Chapter 13, Volume I that
limit cycle periodic solutions are not possible, unless there are delay effects, which we
do not consider here. With multi-reactant kinetics or interacting species, however, as
we saw in Chapter 3, Volume I we can have stable periodic limit cycle solutions which
bifurcate from a stable steady state as a parameter, γ say, increases through a critical γc .
Let us now suppose we have such reaction kinetics in our reaction diffusion system (1.1)
and that initially γ > γc for all x; that is, the system is oscillating. If we now locally
perturb the oscillation for a short time in a small spatial domain, say, 0 < | x | ≤ ε
1,
then the oscillation there will be at a different phase from the surrounding medium. We
then have a kind of localised ‘pacemaker’ and the effect of diffusion is to try to smooth
out the differences between this pacemaker and the surrounding medium. As we noted
above, a sudden change in u can initiate a propagating wave. So, in this case as u regularly changes in the small circular domain relative to the outside domain, it is like
regularly initiating a travelling wave from the pacemaker. In our reaction diffusion situation we would thus expect a travelling wave train of concentration differences moving
through the medium. We discuss such wave train solutions in Section 1.7.
It is possible to have chaotic oscillations when three or more equations are involved, as we noted in Chapter 3, Volume I, and indeed with only a single delay equation in Chapter 1, Volume I. There is thus the possibility of quite complicated wave
phenomena if we introduce, say, a small chaotic oscillating region in an otherwise regular oscillation. These more complicated wave solutions can occur with only one space
dimension. In two or three space dimensions the solution behaviour can become quite
baroque. Interestingly, chaotic behaviour can occur without a chaotic pacemaker; see
Figure 1.23 in Section 1.9.
Suppose we now consider two space dimensions. If we have a small circular domain, which is oscillating at a different frequency from the surrounding medium, we
should expect a travelling wave train of concentric circles propagating out from the
pacemaker centre; they are often referred to as target patterns for obvious reasons. Such
waves were originally found experimentally by Zaikin and Zhabotinskii (1970) in the
Belousov–Zhabotinskii reaction: Figure 1.1(a) is an example. Tyson and Fife (1980)
1.1 Intuitive Expectations
3
(a)
(b)
(c)
Figure 1.1. (a) Target patterns (circular waves) generated by pacemaker nuclei in the Belousov–Zhabotinskii
reaction. The photographs are about 1 min apart. (b) Spiral waves, initiated by gently stirring the reagent. The
spirals rotate with a period of about 2 min. (Reproduced with permission of A. T. Winfree) (c) In the slime
mould Dictyostelium, the cells (amoebae) at a certain state in their group development, emit a periodic signal
of the chemical, cyclic AMP, which is a chemoattractant for the cells. Certain pacemaker cells initiate targetlike and spiral waves. The light and dark bands arise from the different optical properties between moving and
stationary amoebae. The cells look bright when moving and dark when stationary. (Courtesy of P. C. Newell
from Newell 1983)
discuss target patterns in the Field–Noyes model for the Belousov–Zhabotinskii reaction, which we considered in detail in Chapter 8. Their analytical methods can also be
applied to other systems.
We can think of an oscillator as a pacemaker which continuously moves round a
circular ring. If we carry this analogy over to reaction diffusion systems, as the ‘pace-
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1. Multi-Species Waves and Practical Applications
maker’ moves round a small core ring it continuously creates a wave, which propagates
out into the surrounding domain, from each point on the circle. This would produce, not
target patterns, but spiral waves with the ‘core’ the limit cycle pacemaker. Once again
these have been found in the Belousov–Zhabotinskii reaction; see Figure 1.1(b) and, for
example, Winfree (1974), Măuller et al. (1985) and Agladze and Krinskii (1982). See
also the dramatic experimental examples in Figures 1.16 to 1.20 in Section 1.8 on spiral waves. Kuramoto and Koga (1981) and Agladze and Krinskii (1982), for example,
demonstrate the onset of chaotic wave patterns; see Figure 1.23 below. If we consider
such waves in three space dimensions the topological structure is remarkable; each part
of the basic ‘two-dimensional’ spiral is itself a spiral; see, for example, Winfree (1974),
Welsh et al., (1983) for photographs of actual three-dimensional waves and Winfree and
Strogatz (1984) and Winfree (2000) for a discussion of the topological aspects. Much
work (analytical and numerical) on spherical waves has also been done by Mimura and
his colleagues; see, for example, Yagisita et al. (1998) and earlier references there.
Such target patterns and spiral waves are common in biology. Spiral waves, in particular, are of considerable practical importance in a variety of medical situations, particularly in cardiology and neurobiology. We touch on some of these aspects below.
A particularly good biological example is provided by the slime mould Dictyostelium
discoideum (Newell 1983) and illustrated in Figure 1.1(c); see also Figure 1.18.
Suppose we now consider the reaction diffusion situation in which the reaction
kinetics has a single stable steady state but which, if perturbed enough, can exhibit
a threshold behaviour, such as we discussed in Section 3.8, Volume I, and also in
Section 7.5; the latter is the FitzHugh–Nagumo (FHN) model for the propagation of
Hodgkin–Huxley nerve action potentials. Suppose initially the spatial domain is everywhere at the stable steady state and we perturb a small region so that the perturbation
locally initiates a threshold behaviour. Although eventually the perturbation will disappear it will undergo a large excursion in phase space before doing so. So, for a time the
situation will appear to be like that described above in which there are two quite different states which, because of the diffusion, try to initiate a travelling wavefront. The
effect of a threshold capability is thus to provide a basis for a travelling pulse wave. We
discuss these threshold waves in Section 1.6.
When waves are transversely coupled it is possible to analyse a basic excitable
model system, as was done by G´asp´ar et al. (1991). They show, among other things,
how interacting circular waves can give rise to spiral waves and how complex planar
wave patterns can evolve. Petrov et al. (1994) also examined a model reaction diffusion
system with cubic autocatalysis and investigated such things as wave reflection and
wave slitting. Pascual (1993) demonstrated numerically that certain standard predator–
prey models that diffuse along a spatial gradient can exhibit temporal chaos at a fixed
point in space and presented evidence for a quasiperiodic route to it as the diffusion increase. Sherratt et al. (1995) studied a caricature of a predator–prey system in one space
dimension and demonstrated that chaos can arise in the wake of an invasion wave. The
appearance of seemingly chaotic behaviour used to be considered an artifact of the
numerical scheme used to study the wave propogation. Merkin et al. (1996) also investigated wave-induced chaos using a two-species model with cubic reaction terms.
Epstein and Showalter (1996) gave an interesting overview of the complexity in oscillations, wave pattern and chaos that are possible with nonlinear chemical dynamics.