A stochastic operator is a positive linear contraction, P : L1 --+ L1,
such that
llPfII2 = llfll1 for f > 0. It is called asymptotically stable if the iterates pn f of
each density converge in the norm to a fixed density. Pf(x) = f K(x,y)f(y)dy,
where K( ยท, y) is a density, defines a stochastic kernel operator. A general probabilistic/
deterministic model for biological systems is considered. This leads to the
LMT operator
P f(x) = Jo - Bx H(Q(>.(x)) - Q(y)) dy,
where -H'(x) = h(x) is a density. Several particular examples of cell cycle models
are examined. An operator overlaps supports iffor all densities f,g, pn f APng of 0
for some n. If the operator is partially kernel, has a positive invariant density and
overlaps supports, it is asymptotically stable. It is found that if h( x) > 0 for
x ~ xo ~ 0 and
["'" x"h(x) dx < liminf(Q(A(x))" - Q(x)") for a E (0, 1] lo x-oo
then P is asymptotically stable, and an opposite condition implies P is sweeping.
Many known results for cell cycle models follow from this. / Mathematical Science / M. Sc. (Mathematics)
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:unisa/oai:umkn-dsp01.int.unisa.ac.za:10500/16068 |
| Date | 06 1900 |
| Creators | Brown, Thomas John |
| Contributors | Bartoszek, W. K. |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Dissertation |
| Format | 1 online resource (102 leaves) |
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