CRM Program. Mathematical Biology: Modelling and Differential Equations

        January to June 2009

àngel calsina, Universitat Autònoma de Barcelona

josé antonio carrillo, ICREA and Universitat Autònoma de Barcelona

antoni guillamon, Universitat Politècnica de Catalunya

rafael  bravo, Universidad de Alcalá de Henares

àngel calsina, Universitat Autònoma de Barcelona

josé antonio carrillo,  ICREA and Universitat Autònoma de Barcelona (co-ordinator)

antoni guillamon, Universitat Politècnica de Catalunya

benoit perthame, ENS-Université Paris VI

angela stevens, Heidelberg Universität

Mathematical Biology is a humongous field of application of the applied mathematics motto: “Modelling, Analysis and Simulation”. The variety of different applications: protein folding, signalling in neurons, ionic channels, sensory-motor and cognitive processes, immunology, cell migration, angiogenesis, tumour growth, species evolution, epidemiology, ecology, fisheries resources… and its importance for society and general public has produced an increasing attention from the international applied mathematical community. 

Modelling in biology faces new mathematical challenges due to the numerous spatial and temporal scales involved and the enormous different interactions between “individuals” with chemical, molecular, mechanical and electrical phenomena. An example of this complexity is the understanding of the behaviour of individual cells and populations: chemotaxis, cell movement, cell aggregation, cell behaviour in particular situations as angiogenesis, cell signalling in neurons, functional synaptic architectures… One of the main issues is to understand how to incorporate individual behaviour (single cell models, microscopic modelling) giving rise to a description of their collective behaviour (ensemble behaviour, macroscopic modelling). 

Differential Equations appear naturally in many mathematical biology models since they are the basic language of expression of the different mechanisms taken into account. Ordinary differential systems, either deterministic or stochastic, form usually the bricks of the modelling from which more complicated phenomena can be analysed by means of Partial differential equations. The qualitative properties obtained from the analytical study of these models can be a great deal of information for the modeller in order to assess model’s validity and a way of improving the modelling. Numerical computation and simulation of the models can offer insights on the expected results and complement the information obtained from analytical tools. 

This thematic program wants to explore particular and specific mathematical models in Biology and to open to new biologically important issues in which mathematics have to be developed. Differential Equations are the basic, but not only, brick forming these models and thus, an interaction between different backgrounds has been chosen in the invited people.

             The main themes of this program are:

Computational Neuroscience: One of the main issues in modern computational neuroscience tries to explain the overall behaviour of neurons which interact through synaptic signalling. Usually, the modelling starts with systems of ordinary differential equations, both deterministic and stochastic, to describe the single cell dynamics in each neuron. For the low-dimensional questions, dynamical  systems theory is of great help. On the other hand, collective behaviour of neuron ensembles are obtained by deriving Partial Differential Equations as usually done in kinetic theory and statistical physics. The understanding of these collective behaviour models is only at the beginning and the asymptotic analysis and numerical comparison to simplified models based on averaged quantities is to be done.

Tumour growth and Cancer modelling: Different models have been proposed to study the movement and evolution of the amount of cells leading to precursors of tumours. These models include age-structured coupled to phenotypic trait considerations for cells or chemotactic type terms coming from fluid mechanics models. Microscopic modelling has also been considered by means of kinetic-type equations. In this line of research there is a strong interplay between modelling, simplifying partial differential equations models and numerical simulations and discussions with scientists in contact with real data.

Population Dynamics: These models are one of the most classical applications in mathematical biology. They take into account different mechanisms of interaction between individuals as competition, symbiosis or prey-predator relationships. Spatial dependence is incorporated modelling the emigration depending on the biological scenario. Finally, physiological differences among individuals as age or size are also considered. This latter leads to the so called physiologically structured population dynamics, where basic problems of existence and uniqueness in several formulations as well as of modelling and global behaviour are being addressed. Taking into consideration spatial structure gives rise to systems of reaction-diffusion partial differential equations whose study is yet a paradigm of many interesting mathematical questions such as pattern formation, complex asymptotic behaviour, …

Adaptive Dynamics: Evolution of species has been one of the most important research lines of the life sciences since Darwin’s time. Although population genetics is a very active and successful field of application of mathematics to the understanding of biological evolution, a more ecologically founded mathematical tool called adaptive dynamics has been developed in the last years with similar purposes. A part of adaptive dynamics tries to model the natural selection of phenotypes by means of analysing the long time asymptotics of selection-mutation models for populations structured both at the physiological and phenotypical level. The existence of singular solutions concentrated at particular values of the phenotypic trait leads to the question of the existence of evolutionary stable strategies.

Tentative List of Visiting Researchers

Expected Conferences and Courses:

Advanced Course in Mathematical Biology: Modelling and Differential Equations. From February 2 to 6, 2009.

Conference on Mathematical Biology: Modelling and  Differential Equations. From February 9 to 13, 2009.

Workshop and Advanced Course on Deterministic and Stochastic Modelling in Computational Neuroscience and other Biological Topics. From May 11 to 15, 2009.

Last updated: 05/02/2008