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

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130 Bibliography Yu, J. and G. S. Eisenbarth (2006). Mechanisms underlying type 1 diabetes and islet transplantation. Drug Discovery Today: Disease Mechanisms 3 (2), 155–162. Yu, P. (1997). Computation of normal forms via a pertubation technique. Journal of Sound and Vibration 211, 19–38. Zeeman, M. (1990). Hopf bifurcation in competetive three-dimensional lotka-volterra systems.
A Mathematical Appendices This appendix is dedicated to rigoristic statements of the important theorems that have been casually stated in different sections of the thesis. Other important concepts that pertain to mathematics will also be dealt with in the following sections; e.g. the Newton-Raphson method. A.1 Existence and Uniqueness Theorem A very important thing to determine when working with (systems of) differential equations, is wether the solution is unique, and in what interval the solution exist. I.e. given a set of initial conditions on the DuCa model (equations 5.6-5.10), we would like to know if the solution is unique, and if it exists for all time, t. To determine this we first need to introduce the property of Lipschitz continuity Definition 1 Given a function g(x), g : R n → R n , and a domain U ⊂ R n , suppose that g(x1) − g(x2) ≤ Lx1 − x2 (A.1) where x1, x2 ∈ U, and L is the Lipschitz constant. Then we say that g is Lipschitz When g fulfills this condition we say that g is Lipschitz continuous, or that it is continuously Lipschitz. Now for the existence and uniqueness theorem itself Theorem A.1 Suppose that g is Lipschitz continuous. Given an initial point x0 ∈ U, the autonomous differential equation ˙x = g(x) (A.2) has a unique solution, φt(x0), that is defined on the maximal interval of existence. Basically as long as g ∈ C 1 (U), where U is a compact set, we need not be concerned about pathological behavior of the solution (Lynch, 2004, p.176). Loosely put the C 1 condition guarantees that pathological behavior will not occur. One thing that may concern the reader is the fact that we need the differential equation to be autonomous, i.e. not explicitly dependent on time, which the DuCa model is not; cf. equation 5.8. The apoptotic wave depends explicitly on time, but we can remedy this by introducing another dimension, and an additional equation, into the system of equations, by setting t equal to, say, X. By this little trick we have made the DuCa model autonomous. In our case we have not been able to provide a analytic expression of 131
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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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5.8 Discussion of Parameters 43 Fig
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46 The Intermediated Model The diff
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48 The Intermediated Model expected
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50 The Intermediated Model Stabilit
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52 The Intermediated Model restrict
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54 The Intermediated Model We see t
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56 The Intermediated Model paramete
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58 The Intermediated Model ✻ Ma(t
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60 The Intermediated Model 6.2 The
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62 The Intermediated Model ✻ Ma(t
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7 A Brief Introduction to Bifurcati
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7.2 The Hopf bifurcation 67 the gen
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7.3 Biological Relevance of Bifurca
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7.5 Locating the Fixed Points 71 Ev
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74 Bifurcation Diagrams and Analysi
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Page 140 and 141: 126 Bibliography Christen, U. and M
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Page 172 and 173: 158 matlab Code w(r+1)=w(r)+dw; g=w
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Page 178 and 179: 164 matlab Code M=M’; % end s_7(r
Page 180 and 181: 166 matlab Code w_7,V_34,’.b’,w
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Page 190 and 191: 176 Index of diabetes, 2 Michaelis
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nr. 477 - 2011 - Institut for Natur, Systemer og Modeller (NSM)

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