Symmetries of Mathematical Models in Biology

Abstract

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.