Symmetries of Mathematical Models in Biology
Publicerad
Författare
Typ
Examensarbete för masterexamen
Master's Thesis
Master's Thesis
Modellbyggare
Tidskriftstitel
ISSN
Volymtitel
Utgivare
Sammanfattning
In biology, a common type of mathematical model is systems of first order ordinary
differential equations (ODE:s). In general, large non-linear systems of ODE:s have no
analytic solutions. Mathematical symmetries can however still be used to analytically
study differential equations, without the need to find explicit solutions. Symmetries are
transformations that map solutions of a differential equation to other solutions, and thus
contain a lot of information about the system. However, due to the historical development
of the theory of symmetries alongside physics, the literature on finding symmetries of the
type of large systems of first order ODE:s usually found in biology is sparse.
In this thesis, four biological models using first order ODE:s are studied using Lie point
symmetries: the Hill equation, the Gompertz model, the Lotka–Volterra predator–prey
model and the Yildirim–Mackey lactose operon model. Symmetries of all models are found
using ansätze. Additionally, symmetries are found using a repurposed method based on
parameter independence. It is also shown that, using both methods for finding symmetries,
sophisticated computer algorithms are needed for the calculation of symmetries of bigger
systems to be viable.
Additionally, the general structure of symmetries is investigated for different formulations
of the Gompertz model. It is shown that the two scalar Gompertz model
formulations found in literature are symmetrically special cases of the original system
formulation, and the consequence for model building in biological systems is discussed.
It is concluded that due to the generality of the mathematical theory, symmetries
show promise of being a useful tool when studying mathematical models in biology.
However, several mathematical problems have to be solved before symmetries can be
used in day–to–day biological modeling.
Beskrivning
Ämne/nyckelord
Lie symmetries, Lie point symmetries, First order ODE:s, Gompertz model, Lotka–Volterra predator–prey model, Yildirim–Mackey lactose operon model, Lie algebra