

Dirac concentrations in parabolic systems


Piotr Gwiazda (MIM University of Warsaw) and Grzegorz Karch (University of Wroclaw)
in collaboration with Benoit Perthame (Pierre and Marie Curie University, Paris) 

The project is motivated by an interesting issue of the Darwinian evolution of a structured population density. We want to study the population dynamics with a special attention to selection and mutations processes. One can observe that different traits have different capacities to adapt to the environment, which results in a competition between the traits and consequently to the selection process. On the other hand, the new born individuals can have a trait slightly different from their partens, which means that a mutation took place. The population models that are proposed, are structured by a parameter representing a phenotipical trait. The mutation process is a stochastic process (like diffusion), which intuitively should lead after long time to the uniform distribution. However, this is not what we observe in nature. The population consists of different species, we observe the concentration in single points. Understanding this phenomenon is the main objective of the project. A research in this direction is of high importance.
Many therapeutic processes, in cancer, antiviral, antimalarian therapies or antibiotherapy escape control. Using a drug induces the viral, tumor, population can become resistant. Because resistance is very usual in these diseases, it has attracted the attention of medical doctors and human biologists which have gathered observations on the mechanisms explaining it.
An hypothesis is that a resistant subpopulation is selected when a drug targets a specific gene. A possible mechanism that might explain this, is that efflux pumps (ABC transporters) within the cells are multiplied under the effect of enzymes activated by the drug. A gene might enhance this activation and thus be selected when using the drug, [1][6].
The process is described by parabolic equations of diffusive type. The appearance of solutions in form of Dirac concentrations is an unusual situation. The main goal of the project is to explain this phenomenon. Basic approach is to consider rescaled equations. There are various ways of rescaling the equation in accordance to different rates of processes. For further conclusions, singular limits of scaled equations should be studied.
Parabolic systems are known to often exhibit Turing instabilities when the ratio of diffusion is large enough. Among the Turing patterns generated, one often encounters concentration as stiff peaks. This phenomena is challenging in terms of functional analysis, scientific computing and PDEs analysis because it relies on strongly nonlinear effects.
A general tool has been introduced in a series of papers [2][3] to analyze the concentration effect via the HopfCole transform and a new constrained HamiltonJacobi equation in the context of nonlocal LotkaVolterra equations.
The project will be realized at the University of Warsaw, supervised by Piotr Gwiazda and in a close collaboration with Grzegorz Karch from the University of Wroclaw. This allows to benefit from the experience of both researchers in measurevalued solutions, cf. [4] and in the analysis of dynamics in singular settings, cf. [5] respectively. The Ph.D. student will spend 12 months at Pierre and Marie Curie University in Paris, hosted by Benoit Perthame. The Paris team provides an expertise in analysis of mutationselection process.
[1] D'Agata E. M. C., DupontRouzeyrol M., Pierre M. Olivier D. and Ruan S.
The impact of di
erent antibioticregimens on the emergence of antimicrobial
resistant bacteria. PLoS One 3(12): es4036 (2008).
[2] Barles, G. and Perthame, B. Dirac concentrations in LotkaVolterra parabolic
PDEs. Indiana Univ. Math. J. 57(7) 2008, 32753301
[3] Diekmann, O., Jabin, P.E., Mischler S. and Perthame B., The dynamics of
adaptation : an illuminating example and a HamiltonJacobi approach. Th.
Pop. Biol., 67(4) (2005) 257271.
[4] Gwiazda P., Lorenz T. and MarciniakCzochra A. (2010) Flat metric and structural stability of a nonlinear population model. J. Di
. Eq. 248 27032735
[5] Karch, G. and Suzuki, K. Spikes and di
usion waves in onedimensional model
of chemotaxis, Nonlinearity 23 (2010) 3119{3137
[6] Tomasetti Cristian and Levy Doron. An elementary approach to modeling drug
resistance in cancer. Preprint 2009




