Two principles, proposed by Darwin, form the basis for evolution in these models:
- Mutation: The traits of individuals can change slightly when they are passed on to their descendants,
- Selection: Individuals whose traits are better adapted to the environment in which they live reproduce with a higher probability.
Now we can include mutations in the model e.g. by an integral kernel (giving the probability of jumping from one traits to the other) or by the Laplacian (modelling mutations as a diffusion process). Selection is introduced in the model by a growth-death term depending on the trait and on the total number of the population. So competition for a common resource such as food can be included by requiring that the growth-death term becomes negative for a large population. Mathematically speaking, we consider nonlocal Lotka-Volterra models. When regarding mutations as rare events i.e. small diffusion, the solutions of these Lotka-Volterra models concentrate as Dirac masses. In A. Lorz, S. Mirrahimi, B. Perthame we described the dynamics of the limiting concentration points. Also we establish a form of a canonical equation and investigate the long time asymptotics of these Dirac masses. Moreover, numerical simulations show that the trajectories can exhibit unexpected dynamics well explained by this equation.These models from population dynamics can help to understand why resistances against drugs develop in therapeutic processes. E.g. resistance to chemotherapy occurs when cancers that have been responding to a therapy suddenly begin to grow. In general, resistance is a quite common feature in diseases like malaria as well as infections caused by bacteria or viruses. Motivated by the theory of mutation-selection in adaptive evolution, we propose a model based on a continuous variable that represents the expression level of a resistance phenotype in Lorz et al. This phenotype influences birth/death rates, effects of chemotherapies (both cytotoxic and cytostatic) and mutations in healthy and tumor cells. We extend previous work by demonstrating how qualitatively different actions of cytostatic (slowing down cell division) and cytostatic (actively kills cells) treatments may induce different levels of resistance. We also derive the long-term temporal dynamics of the fittest traits in the regime of small mutations.
The picture below shows one of the results from this paper: in our models the combination of cytostatic and cytostatic treatments are much more effective against cancer cells while keeping alive more healthy cells