How to deal with the multi-scale dynamical
systems: A case study – Model for Photosynthesis, Photoinhibition and Photoacclimation

Š. Papáček^{1},
S. Čelikovský^{4}, D. Štys^{1,2,3} ^{}

^{1}Institute
of Physical Biology, University of South Bohemia,

^{2}
Institute of Systems Biology and Ecology CAS, Zámek 136, 373 33 Nové Hrady,
Czech Republic

^{3}Institute
of Microbiology CAS, 379 81 Třeboň, Czech Republic.

^{4}UTIA
AV ČR (Institute of Information Theory and Automation, Academy of Sciences of
the

Czech Republic), Prague, Czech Republic.

papacek@greentech.cz

The multi-scale modelling and computation became
essential in solution process of multi-scale systems frequently arising in
chemistry and biology. The systems with multiple time-scales are characterised by
the wide separation between the *O*(*e*) time-scale, and the *O*(1), or even *O*(*e*^{-1}).
In this paper we describe three different ways to solve the problem of
microalgal photosynthesis under periodic intermittent light. The lumped
parameter model for photosynthesis, photoinhibition and photoacclimation in
microalgae represents the so-called stiff system, i.e. the dynamical system
with multiple time-scales. The separation of time-scales in the system is
modeled by a small parameter *e*. From a
mathematical point of view, the problem in hand is a system of two ordinary non-linear
differential equations (ODE), and we have assumed that the phase-space can be
decomposed into the slow phase, *x*_{3}, and the fast one, *x*_{2}.
Ordinarily, the ODE system is not amenable to analytical treatments and
requires an efficient computational tool for its solution. However, in our
case, i.e. for the piecewise constant input (light intensity *u*), is the
analytical solution reachable because we deal with the linear ODEs in both
subintervals of light-dark cycles period: *h*_{a} - light-off
interval, and *h*_{b} - light-on interval. Consequently, we are
able to evaluate the average value of the fast phase *x*_{2} (which
is reflecting the rate of photosynthesis in microalgal culture) depending on
the period of light-dark cycles *h = h*_{a} + *h*_{b}.
Next, we can analytically compute the solution of the same ODE system applying
two common approaches for solving the stiff systems: (i) neglecting the fast
system dynamics, and (ii) neglecting the slow dynamics. Once having the exact analytical
solution, we can observe the limits of validity of both approximated solutions.
The presented simulation results based on published data demonstrate the perfect
concordance between the theoretical assumptions and the numerical results and
could serve as the motivation example for applying the multi-scale method, and
at the same time, to eliminate some frequent errors in treatment of stiff
systems.