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

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68 A Brief Introduction to Bifurcation Analysis and Numerical Methods I The supercritical Hopf bifurcation. Stable spirals exist for r < r ∗ as α(r) < 0, up until r = r ∗ where α(r) = 0. At the bifurcation point, r = r ∗ , the spiral is still stable but very weakly so (Strogatz, 2000, p.250). For r > r ∗ the spirals are unstable, with stable limit cycles surrounding them, i.e. α(r) > 0. II The subcritical Hopf bifurcation. For r < r ∗ we have α(r) < 0 thus stable spirals coexists with unstable limit cycles. As r → r ∗ the radius of the unstable limit cycles decrease. Ultimately, at r = r ∗ , the limit cycle engulfs the fixed point that was the focus of the stable spiral, and for r > r ∗ the fixed point is unstable, thus making the spirals unstable. III The degenerate Hopf bifurcation. This type of Hopf bifurcation is characterized by stable spirals for r < r ∗ , and unstable spirals for r > r ∗ , while at r = r ∗ infinitely many neutrally stable concentric closed orbits encircle the fixed point that is the focus of the spirals. Figure 7.2 provides an example in R 2 of the three bifurcation types. The limit cycles I II III 2 rr* 2 2 2 rr* 2 2 2 rr* Figure 7.1 A supercritical (I), a subcritical (II) and degenerate Hopf bifurcation (III). From Zeeman (1990) with minor modifications. The dashed lines denote unstable fixed points or limit cycles. that arise from the supercritical version (I) are easiest identified by means of computergenerated phase plots. The unstable limit cycle that exists before the bifurcation point in the subcritical case (II) can in theory be found by reversing the time when one does simulations of the system of interest. This is tantamount to multiplying the right-handside of the differential equations in question by minus one. At the instance where the 2 2
7.3 Biological Relevance of Bifurcation Analysis 69 focus of the stable spiral in case II changes stability, from attracting to repelling, the flow will make a sudden jump, possibly to another stable fixed point, another limit cycle or it may even diverge. This extreme change in behavior is known as hysteresis. Hysteresis also implies that we cannot make the system return to the behavior before the bifurcation point by simply turning the bifurcation parameter back to before the bifurcation value (Strogatz, 2000, p.252). In other words the behavior is irreversible in regards to changes in the bifurcation parameter. 7.3 Biological Relevance of Bifurcation Analysis Now returning to the biological relevance of bifurcation analysis. What biological conclusions can we draw from a model that exhibits bifurcations? The answer is, naturally, that this depends on what the model is supposed to model! If we have data that suggests that our biological system goes from having, say, one to two stable states as a given parameter value is changed, then it is natural that our mathematical model exhibits a bifurcation. But if the stable states in the mathematical model appear at a parameter value that is completely irreconcilable with the actual value, then we must reassess the model. On the other hand if some behavior, that is biologically reasonable, is observed in a mathematical model, then one can use this to make predictions/conclusions about the biological system. In our case it is particularly interesting to see if the qualitative behavior of the system changes close to, or maybe within the span of uncertainties associated with the parameter values; cf. sections 5.8 and 5.8. Should the qualitative behavior change within, or close to, the parameter range of a given parameter then the model is parameter sensitive. This does not necessarily indicate that the model is ill-conceived, e.g. not all NOD mice develop diabetes, so these mice may have a slightly different parameter composition. 7.4 Choice of Bifurcation Parameters The DuCa model contains 12 parameters which makes it a daunting task to conduct a thorough investigation. This amount of parameters also opens up the possibility of co-dimension 12 bifurcations 4 , which is way beyond the scope of this work. Thus we must limit our analysis to a few important parameters. We have gone with f1 and f2 for two reasons: 1. Wang et al. (2006) (see section 1) remark that an analogue of our phagocytosis rates is the most important for the stability in their model 2. The other, and perhaps most important, reason is that the model was originally intended to investigate whether the difference in the values of f1 and f2 alone, could lead to different dynamics of the system – permanent or nonpermanent inflammation Now that we have chosen our bifurcation parameters we must decide what intervals we want f1 and f2 to take values in. Of course the whole negative range of R is out 4 The amount of parameters could naturally be brought down by nondimensionalizing the system, but even after this 9 dimensionless groups remain, which are still too many.
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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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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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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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