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

8.3 Using f2 as the bifurcation parameter 89
for NOD-mice is reached, when the apoptotic wave stimulates the system to exceed the
threshold concentration that corresponds to unstable fixed points. We notice that the
Figure 8.12 Bifurcation diagram for the DuCa model using f2 as the bifurcation parameter
and NOD parameters. The solid and dashed lines illustrate stable and unstable fixed points
respectively. The x-axis has been chosen such that the change in stability is apparent –
leaving out the monotonic part for f2 < 4 × 10 −5 . The system is bistable in the (approximate)
range [0; 8.5 × 10 −5 ] with a “line” of unstable fixed points to separate the two regions. For
f2 > 8.5 × 10 −5 only the healthy rest state remains stable.
behavior has some similarities as well as differences from its f1-counterpart in figure
8.1. Let us start with the resemblances.
Again we start out by having three stability branches; two stable and one unstable –
Ma = 0 being one of the stable branches. These change into two unstable and one
stable (at f2 ≈ 8.5 × 10 −5 ), and eventually the two unstable branches coalesce in a
saddle-node bifurcation (at f2 ≈ 9.25 × 10 −5 ) after which only the healthy rest state
remains. Finally both figures exhibit hysteresis. These are the superficial similarities
we can gather from a quick glance at figure 8.12.
The dissimilarities are easily spotted. First of all, the nontrivial line of stable fixed
points decreases as f2 is increased, and changes stability at a much lower Ma-value than
was observed when we used f1 as the bifurcation parameter. If we return to the biological
setting for a moment then this makes excellent sense. The active macrophages still
come from the pool of resting macrophages. Now, since f2 is the phagocytosis rate of the
active macrophages, turning up f2 means that a fewer amount of active macrophages
are needed to handle the same amount of apoptotic β-cells, but during the initial apoptotic
wave a significant recruitment of active macrophages from the resting pool will
still take place. This together with the enhanced phagocytic ability implies that the
apoptotic β-cells will be decimated quickly, thus leading to a drop in the number of