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

A Mathematical Appendices
This appendix is dedicated to rigoristic statements of the important theorems that
have been casually stated in different sections of the thesis. Other important concepts
that pertain to mathematics will also be dealt with in the following sections; e.g. the
Newton-Raphson method.
A.1 Existence and Uniqueness Theorem
A very important thing to determine when working with (systems of) differential equations,
is wether the solution is unique, and in what interval the solution exist. I.e.
given a set of initial conditions on the DuCa model (equations 5.6-5.10), we would like
to know if the solution is unique, and if it exists for all time, t. To determine this we
first need to introduce the property of Lipschitz continuity
Definition 1
Given a function g(x), g : R n → R n , and a domain U ⊂ R n , suppose that
g(x1) − g(x2) ≤ Lx1 − x2 (A.1)
where x1, x2 ∈ U, and L is the Lipschitz constant. Then we say that g is Lipschitz
When g fulfills this condition we say that g is Lipschitz continuous, or that it is continuously
Lipschitz.
Now for the existence and uniqueness theorem itself
Theorem A.1
Suppose that g is Lipschitz continuous. Given an initial point x0 ∈ U, the autonomous
differential equation
˙x = g(x) (A.2)
has a unique solution, φt(x0), that is defined on the maximal interval of existence.
Basically as long as g ∈ C 1 (U), where U is a compact set, we need not be concerned
about pathological behavior of the solution (Lynch, 2004, p.176). Loosely put the C 1
condition guarantees that pathological behavior will not occur.
One thing that may concern the reader is the fact that we need the differential equation
to be autonomous, i.e. not explicitly dependent on time, which the DuCa model is not;
cf. equation 5.8. The apoptotic wave depends explicitly on time, but we can remedy
this by introducing another dimension, and an additional equation, into the system of
equations, by setting t equal to, say, X. By this little trick we have made the DuCa
model autonomous. In our case we have not been able to provide a analytic expression of
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