Asymptotic behaviour of integro-differential equations describing clonal evolution of leukemia

Abstract

This thesis is devoted to the analysis of a system of integro-differential equations describing leukemia, a type of blood cancer. Existence and uniqueness for arbitrary times are shown and the long-term behaviour of the solution is characterised. In order to achieve the asymptotic behaviour of the solution it is proved that a normalized (with respect to total mass) solution forms a Dirac sequence, thus the solution converges for time tending to infinity to a Dirac measure. Moreover the total mass converges, too, which is shown by combining an asymptotic stability result via a Lyapunov function with a perturbation argument. Additionally, the convergence result is generalised to a suitable measure space. Furthermore, the model is extended by an additional integral term with a small multiplicative coefficient in order to capture the idea of mutation. For this newly obtained system it is shown existence and uniqueness of both a solution for arbitrary times and a positive steady state. The latter is achieved by interpreting the steady state equations as an eigenvalue problem and by using the Krein-Rutman theorem. The local asymptotic stability of the steady state is proven by using linearised stability. The spectrum, which is crucial for linearised stability, is investigated with the method of the Weinstein-Aronszajn formula. Moreover, it is proven that the stable steady state of the extended model converges weakly^* to the stable steady state of the original model if the coefficient of the newly introduced integral term tends to zero. Lastly, a numerical scheme, which has been used to simulate the original model, is illustrated and its convergence to the analytical solution is proven.