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

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58 The Intermediated Model ✻ Ma(t) ˙ Ma < 0 ˙ Ba > 0 ❅❘ ❄ ✠ Ma ˙ < 0 Ba ˙ < 0 ˙ Ba = 0 ✛ ❅■ ˙ Ma = 0 Asymp2 Asymp1 ˙ Ma > 0 ˙ Ba < 0 ✲ Ba(t) Figure 6.4 Sketch of the nullclines and some representative arrows that indicate the flow for for the IM with dM/dt = 0. For the healthy rest state, (Ba, Ma) = (0, 0), to be stable the initial slope of the Ma-nullcline must be less than the initial slope of the Ba-nullcline. Then by assuming that the nullcline of Ba reaches its asymptotic point before the nullcline of Ma leaves the stable healthy rest state to be the only existing physiological relevant fixed point. This means that any stimulated inflammation will be nonpermanent.
6.1 Fixed Points of the IM Model 59 unknown, and it also reduces the number of parameters to keep track of. Per automation we also nondimensionalized the IM (including crowding terms) – cf. appendix A.6 – but since we are in the rare situation of having estimates of all the parameters, we have chosen not to work with the nondimensional version. A brief note on linearizing Some newcomers to the field of mathematical modelling may think that we oversimplified things by linearizing the system when we analyzed the stability of the fixed points because we casually threw away the higher order nonlinear terms without making any remarks about these. So what gave us the right to do so? Hartman-Grobman’s theorem 8 informally states that the flow of a nonlinear system is topologically equivalent to that of a linear system in the neighborhood of a steady state solution, provided that the eigenvalues of the Jacobian of the linear system are not purely imaginary or take on zero value. 9 The equivalence here being a homeomorphism, i.e. a continuous deformation with a continuous inverse, that preserves the sense of time (Guckenheimer and Holmes (2002)). More so the eigenspaces attributable to the eigenvectors, of the linearized system, where the eigenvalues are subject to the behavior given above, are tangent to the invariant manifolds of the fixed points of the nonlinear system (Guckenheimer and Holmes (2002)) – provided that the fixed points are hyperbolic. This last statement is the essence of the theorem known as The Stable Manifold Theorem for a Fixed Point; see e.g. Guckenheimer and Holmes (2002) page 13. When we are in three or less dimensions this fact should lend a helping hand to the intuitive understanding of the situation. Let us recapitulate in (pseudo) laymen terms: when the eigenvalues of the Jacobian does not take on zero or purely imaginary values, the behavior obtained from the linear system is in qualitative agreement with that of the nonlinear system. One could hope that as long as we knew all the eigenvalues we could precisely determine what kind of behavior is in play, and in fact it is so when we are in R 2 . But when we enter R 3 and higher dimensions there are no theorems such as e.g. the Poincaré- Bendixson theorem, that yield guarantees about the dynamical behavior, when we are dealing with systems that evolve in more than two dimensions (Strogatz (2000)) – the Poincaré-Bendixson theorem states that given a trajectory confined to a compact subset of the plane, R 2 , that does not contain any fixed points, then said trajectory will either approach a closed orbit, or it is itself a closed orbit (Strogatz, 2000, p.203). When dealing with systems that evolve in R 3 and above a plethora of dynamics, including chaos, can arise. 10 Now that we have elaborated a little on the relationship between a nonlinear system and the linearized version thereof we will return to the analysis of the intermediate model for that story is not completely over. 8 For a formal statement of the Theorem cf. appendix A.3. 9 Note that we are not given any promises regarding the size of said neighborhood. 10 Actually we would find it boring if there was no more to the study of systems of differential equations than determining the eigenvalues.
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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
Page 22 and 23: 8 Type 1 Diabetes and Its Etiology
Page 24 and 25: 10 Type 1 Diabetes and Its Etiology
Page 27 and 28: 3 Prospects for Therapy - A Mini Re
Page 29 and 30: 3.3 Gastrin 15 eration in situ Urus
Page 31 and 32: 4 Mathematical Modelling The langua
Page 33 and 34: 5 The DuCa Model In the following w
Page 35 and 36: 5.3 The DuCa Model - An Appetiser 2
Page 37 and 38: 5.4 The DuCa Model - Compartment Mo
Page 43 and 44: 5.6 Our Compartment Model 29 migrat
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 55 and 56: 5.8 Discussion of Parameters 41 tha
Page 57: 5.8 Discussion of Parameters 43 Fig
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 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
Page 85: 7.5 Locating the Fixed Points 71 Ev
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
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108 Expanding and Modifying the DuC
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118 Discussion of the Expanded Mode
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120 Discussion of the Expanded Mode
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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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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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