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

6.2 The IM Including Crowding-Terms 61
from the IM without crowding terms (equations 6.50 and the expression for M derived
from the IM including crowding terms (equation 6.65)? Let us approach this matter in
an intuitive fashion by looking at the nullsurface of equation 6.61 (setting dM/dt = 0).
0 = a + (k + b)Ma − cM − f1MBa − e1M(M + Ma) (6.66)
Then from a given point (M, Ma, Ba) on the nullsurface we increase Ma and Ba by a
small amount |δMa| and |δBa|. From this we see that M has to increase less, relative to
the equation without the crowding terms, to maintain the equilibrium. If this argument
seems strange, picture the system as a leaky bucket, where the volume of water is kept
constant by pouring water into it. That is, the inflow of water from the top of the
bucket is the same as the outflow through the holes in the bottom of the bucket. Then
suddenly some additional holes appear in the bottom but the equilibrium remains. This
means that the outflow have more outlets to be distributed among and therefore less
flow through every single outlet. However, in the case above we defined the change of
Ma and Ba so only M was allowed to vary. This leads to the crucial observation that
for a given concentration of Ma and Ba, the concentration of M can be considered to
be “less” relative to if the crowding terms were excluded.
Continuing in the same direction we now take a look at the nullclines 11 of Ma
0 = f1MBa − kMa − e2Ma(M + Ma) (6.67)
This time let Ma increase with at small amount |δMa|. Then because of the inclusion of
the quadratic crowding terms causing a greater efflux and the relatively smaller M, there
must be a greater increase of Ba relatively to the equation without the crowding terms.
In other words, dMa
must be less in the case where the crowding terms are included
dBa
compared to when they are not. This is interesting since it reveals that the slope of
the Ma-nullcline must decrease when the crowding terms are no longer negligible. To
complete the analysis let us turn to the nullcline of Ba
0 = lMa − f1MBa − f2MaBa − dBa
(6.68)
As mentioned the only difference here in respect to the model without the crowding
terms is the decrease of M. This implies that if we increase Ba by a small amount |δBa|
then Ma must increase less, relative to the case where the crowding terms are excluded.
Thus the slope of the nullcline dMa
is again gradually reduced when the crowding terms
dBa
are no longer negligible, so that the asymptotic behavior fails to happen. However the
reduction is less for the nullcline of Ba compared to the nullcline of Ma such that
an inevitable third intersection of the nullclines takes place; cf. figure 6.5. From the
slopefield we see that the fixed point is stable, which makes it an upper bound for
the inflammation as claimed by Marée et al. (2006). Note that when the system is
stimulated above the separatix the concentrations of the macrophages and apoptotic
β-cells will tend towards the concentrations that represent the coordinate of the upper
stable fixed point.
11 We now consider them as nullclines instead of nullsurfaces since M is considered to be a constant
of a size that is less than if the crowding terms were excluded.