nr. 477 - 2011 - Institut for Natur, Systemer og Modeller (NSM)

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74 Bifurcation Diagrams and Analysis dimensional version of it, suggested by Marée et al. (2006), to see if making the QSS assumptions dBn dt dC dt = 0 (8.1) = 0 (8.2) has any influence on the behavior of the detected fixed points. This is interesting to determine, because it is, ceteris paribus, easier to analyze a three dimensional rather than a 5 dimensional system. 8.2 Using f1 as the bifurcation parameter Figure 8.1 is the bifurcation diagram of the DuCa model with NOD parameters, using f1 as the bifurcation parameter (seen on the x-axis), and Ma as the variable (seen on the y-axis). Let us start by dealing with the obvious features, and then get into more subtle things after that. Basic features of the NOD bifurcation diagram Figure 8.1 Bifurcation diagram for the DuCa model using f1 as the bifurcation parameter and NOD parameters; cf. appendix B.1 for auxiliary figures. The solid and dashed lines illustrate fixed points that have only negative and mixed, i.e. positive and negative, eigenvalues respectively. The system is bistable up to f ∗ 1 ≈ 2.57 × 10 −5 . The NODf1-LUB separates the two stable domains. For f1 > 2.57 × 10 −5 only the HRS remains stable.
8.2 Using f1 as the bifurcation parameter 75 First of all, we notice two kinds of curves: there are dashed, and then there are fully drawn curves. Dashed curves are associated with fixed points whose eigenvalues are not purely negative, while a fully drawn curve signifies fixed points that have strictly negative eigenvalues (or complex eigenvalues with negative real parts). At some point (f1 ≈ 2.57 × 10−5 ) the fully drawn curve that is increasing from the origin becomes dashed, and then proceeds to meet with the dashed curve from beneath it. The curve that starts out being fully drawn will be referred to as the NODf1 upper stable branch, NODf1-USB3 for “short”. The lower dashed line will be referred to as the NODf1 lower unstable branch, or NODf1-LUB (both of these names turn out to be somewhat of a couple of misnomers, but it eases the matter of explaining what is going on). Near the x-axis, or rather through Ma = 0 is another curve, which is colored green, this one is also fully drawn and is referred to as the healthy rest state (HRS). The HRS is the situation where no inflammation is present, hence healthy rest state – the HRS is the same in all the bifurcation diagrams, so we will not give it a prefix. Just as a reminder we state that each branch is actually a line of fixed points. Figure 8.1 tells us that for f1 ∈ (0 : 2.57) × 10 −5 the system is bistable 4 with at least three stability branches in all. 5 The most interesting branches are the NODf1-USB and the NODf1-LUB – the HRS remains stable for all values of f1, i.e. the eigenvalues associated with it are negative for all f1 ∈ R + . Along the NODf1-USB the concentration of active macrophages increases as f1 is increased. This is naturally because f1 is also the activation rate of resting macrophages. The concentration ceases to increase at f1 ≈ 2.067 × 10−5 , the curve flattens, and then slowly decreases (which may be because the amount of apoptotic β-cells stabilizes at a lower concentration, due to the initial faster phagocytosis). As f1 is increased beyond ≈ 2.57×10−5 the fixed points on the NODf1-USB become unstable, and the HRS is the only stable state that remains. The point where the transition occurs is the point where the eigenvalues shown in the bottom left subplot of figure 8.2 have zero real value, i.e. it is a Hopf bifurcation point. After the NODf1-USB has become unstable it descends until it coalesces with the NODf1-LUB. This happens at f1 ≈ 3.45 × 10−5 where they annihilate each other in what, to some extent, resembles a saddle-node bifurcation. From the look of figure 8.1 it does not seem as though there is anything interesting going on with the NODf1-LUB, but in a moment when we look closer at the eigenvalues associated with it, we will learn that this is not true. Eigenvalues, manifolds and behavior of the fixed points on the NODf1 -USB The NODf1 -USB has five distinct, real, and negative, eigenvalues until f1 ≈ 3.4 × 10 −7 . Here two large and negative eigenvalues become complex conjugates. This implies that we should observe spirals, but because the real part of the eigenvalues is large (com- 3 It may be a cumbersome acronym, but it is necessary to avoid confusion when we get to the other bifurcation diagrams 4 Notice that the phagocytosis rate for resting macrophages in NOD-mice, f1 = 1 × 10−5 , lies well within the bistable region on both NOD-based bifurcation diagrams (figure 8.1 and 8.12). This accounts for the development of a chronic inflammation for NOD-mice, because the apoptotic wave stimulates the system to exceed the unstable fixed points along the NODf1 -LUB. 5 Please be aware that other fixed points can exist, as we pointed out in section 7.5.
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- I, OM OG MED MATEMATIK OG FYSIK E
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Exploring Treatment Strategies for
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Preface iii questions that were out
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vi Contents Significance of the Apo
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List of Tables 5.1 summarizes the m
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x List of Figures 8.3 Phase space p
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xii
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2 Introduction In the following dec
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4 Introduction There are several sc
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6 Introduction because I had to mak
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8 Type 1 Diabetes and Its Etiology
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10 Type 1 Diabetes and Its Etiology
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3 Prospects for Therapy - A Mini Re
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3.3 Gastrin 15 eration in situ Urus
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4 Mathematical Modelling The langua
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5 The DuCa Model In the following w
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5.3 The DuCa Model - An Appetiser 2
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Page 45 and 46: 5.6 Our Compartment Model 31 a ✓
Page 47 and 48: 5.7 Discussion of Marée et al. (20
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Page 51 and 52: 5.8 Discussion of Parameters 37 Fig
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Page 60 and 61: 46 The Intermediated Model The diff
Page 62 and 63: 48 The Intermediated Model expected
Page 64 and 65: 50 The Intermediated Model Stabilit
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Page 68 and 69: 54 The Intermediated Model We see t
Page 70 and 71: 56 The Intermediated Model paramete
Page 72 and 73: 58 The Intermediated Model ✻ Ma(t
Page 74 and 75: 60 The Intermediated Model 6.2 The
Page 76 and 77: 62 The Intermediated Model ✻ Ma(t
Page 79 and 80: 7 A Brief Introduction to Bifurcati
Page 81 and 82: 7.2 The Hopf bifurcation 67 the gen
Page 83 and 84: 7.3 Biological Relevance of Bifurca
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Page 90 and 91: 76 Bifurcation Diagrams and Analysi
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Page 120 and 121: 106 Expanding and Modifying the DuC
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126 Bibliography Christen, U. and M
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128 Bibliography Notkins, A. L. and
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130 Bibliography Yu, J. and G. S. E
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132 Mathematical Appendices the com
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134 Mathematical Appendices in that
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136 Mathematical Appendices A.6 Dim
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138
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140 Additional Figures Figure B.2 P
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142 Additional Figures B.2 Eigenval
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144 Additional Figures Figure B.8 P
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148 Additional Figures Figure B.14
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158 matlab Code w(r+1)=w(r)+dw; g=w
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160 matlab Code end V_11(r)=E(1); V
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162 matlab Code n=675; w_5=0; dw_5=
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164 matlab Code M=M’; % end s_7(r
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166 matlab Code w_7,V_34,’.b’,w
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168 matlab Code M=M’; % M must be
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170 matlab Code %options = odeset(
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172
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174 Index value, 73 Bifurcation Ana
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176 Index of diabetes, 2 Michaelis
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nr. 477 - 2011 - Institut for Natur, Systemer og Modeller (NSM)

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