A simple observation based on (3.1) is

which means that the model is mass-conservative, i. e.\

Let be a numerically computed approximation to
(). In view of property (3.4), it makes
sense to impose either mass conservation (in time) on the computed
approximation or nonnegativity, i. e. the restriction for
all *k*, or both. Mass conservation is a natural concern for environmental
simulations [30, 41]. If nonnegativity does not hold,
physically meaningless mass values are calculated for a time *t*, and
this error might propagate in later time steps. Moreover, the cytodynamic
module in use might not necessarily be able to handle those erroneous values.
In such a case, a filtering technique has to be used to "redistribute"
negative masses such that nonnegativity or mass-consistency
holds [40]. Consequently, implementations based on such
techniques are not very transparent. Numerical issues with respect to
nonnegativity and mass conservation will be discussed in detail in the next
subsection. First, let us focus on the nonnegativity of the exact solution
of (3.3).

Denote by

the system matrix of (3.3), which is also the Jacobian of the
right hand side with respect to *m*. The nonnegativity of the solution
of (3.3) under quite weak conditions
can now be seen as follows. Let *A* be piecewise uniform continuous
and *c* as well as all () be piecewise
continuous. Moreover, let those functions be right-sided
continuous everywhere. Interpolate *A* by a piecewise constant
function on a grid containing the
points of discontinuity of both *A*, *c*, and
() and such that
for all
().
If *A* is replaced by in (3.3), the corresponding
solution on the interval can be written as

(), where, as usually, the matrix exponential function is
given by

for arbitrary square matrices *M*. Since the nondiagonal elements of *A*
are nonnegative, all the elements of are
nonnegative for all . If
(,
) holds, the nonnegativity of the
functions () for follows
if, in addition, *c* is nonnegative. But converges uniformly to the
solution *m* of (3.3) since and converge
uniformly to *A* and *c* when the grid is refined.

Wed Dec 22 12:25:31 CET 1999