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

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94 Bifurcation Diagrams and Analysis Figure 8.17 On the left we have the original five dimensional DuCa model, on the right the reduced three dimensional DuCa model. We see that despite some minor initial differences differences the curves are interchangeable. dM dt = a + (k + b)Ma − cM − f1MBa − e1M(M + Ma) (8.5) dMa dt = f1MBa − kMa − e2Ma(M + Ma) (8.6) dBa dt = W (t) + AmaxdBaMa kb(f1M + f2Ma) − f1MBa − f2MaBa − dBa (8.7) Figure 8.18 shows a bifurcation diagram of this reduced model with f1 as the bifurcation parameter. The similarities with figure 8.1 are striking, but this may not come as a surprise since the positions of the fixed points are expected to be the same because the QSS assumption of course holds in the fixed points. The real test is to see if the solution curves of M, Ma and Ba change under this assumption. As we can see in figure 8.17 there is not much difference, if any. But how else could we justify making a QSS assumption. The answer lies in the eigenvalues of the full system versus those of the reduced system. Let us digress to the example where we looked at the general solution to 7.2; given by equation 7.3. For the sake of argument let us assume that we have a two-dimensional system, and that both the eigenvalues are negative and not too different in magnitude. We know that in reality another component play a part in the behavior of system. Thus in reality we need an extra equation, which describes this extra component. At the same time we are aware that this makes the stability analysis more cumbersome. Thus we would like to know if it would be reasonable to make a QSS assumption on the extra equation, so we would be back in 2D land. To figure this out we add the extra equation, and take a look at the corresponding characteristic polynomial for the now three-dimensional system. If the new eigenvalue is significantly more negative than the others, then the solution curve associated with this eigenvalue will stabilize much faster than the two others. In fact the solution curve associated with this very negative eigenvalue will tend to the stable manifold spanned by the eigenvectors of the two other eigenvalues as an
8.4 Reducing the DuCa-model 95 Figure 8.18 Bifurcation diagram for the three dimensional version of the full model using NOD parameters (see table 5.1). The diagram shows the fixed points of Ma when using f1 as the bifurcation parameter. The solid and dashed lines illustrates stable and unstable fixed points respectively. We see that the system is bistable in the approximate range (0 − 2.57 × 10 −5 ) with a “line” of unstable fixed points to separate the two regions. For f1 > 2.57 × 10 −5 only the healthy rest state remains stable. exponential function raised to the power of the eigenvalue. This means that adding the extra equation does not contribute any new information about the system after a given amount of time (which depends on the magnitude of the eigenvalue) has transpired. By then the dynamics of the system will be governed by the two other eigenvalues. Thus when this is the case we can justify applying a QSS to the equation that describes the rate of change of the variable we introduced lastly. In our case this is exactly the case, though the story is reversed. We started out with 5 equations in the DuCa model, and wanted to know if three would suffice; using QSS assumptions on the last two. Furthermore the value of the remaining fixed points do not change significantly which again indicates that the QSS assumptions are sound approximations. The stability analysis of the fixed points reveals that the behavior of the eigenvalues are less complicated than in the five dimensional case; cf. figures B.18-B.22 in appendix B.5. The unstable fixed points in the LUB are now rid of complex eigenvalues, while the set of large, negative and complex eigenvalues that were associated with USB have also gone. The LUB still serves as a separatrix that divides the region of nonpermanent inflammation from the region of chronic inflammation. The Hopf bifurcation still happens as the stable nontrivial fixed points become unstable and the saddle-node bifurcation is also present. Now that we have learned that the behavior of the reduced DuCa model is very much
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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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5.4 The DuCa Model - Compartment Mo
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5.6 Our Compartment Model 29 migrat
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5.6 Our Compartment Model 31 a ✓
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5.7 Discussion of Marée et al. (20
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5.7 Discussion of Marée et al. (20
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5.8 Discussion of Parameters 37 Fig
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5.8 Discussion of Parameters 39 Par
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5.8 Discussion of Parameters 41 tha
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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
Page 66 and 67: 52 The Intermediated Model restrict
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 88 and 89: 74 Bifurcation Diagrams and Analysi
Page 112 and 113: 98 Discussion of the Bifurcation An
Page 114 and 115: 100 Discussion of the Bifurcation A
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Page 120 and 121: 106 Expanding and Modifying the DuC
Page 132 and 133: 118 Discussion of the Expanded Mode
Page 134 and 135: 120 Discussion of the Expanded Mode
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Page 140 and 141: 126 Bibliography Christen, U. and M
Page 142 and 143: 128 Bibliography Notkins, A. L. and
Page 144 and 145: 130 Bibliography Yu, J. and G. S. E
Page 146 and 147: 132 Mathematical Appendices the com
Page 148 and 149: 134 Mathematical Appendices in that
Page 150 and 151: 136 Mathematical Appendices A.6 Dim
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Page 154 and 155: 140 Additional Figures Figure B.2 P
Page 156 and 157: 142 Additional Figures B.2 Eigenval
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144 Additional Figures Figure B.8 P
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146 Additional Figures B.3 Eigenval
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148 Additional Figures Figure B.14
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152 Additional Figures B.5 Eigenval
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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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