The differential equation counterpart of an individual-based model for yeast population growth

Computer simulations are increasingly used in biological fields. The amazing power, storage ability, and processing speeds available nowadays have enabled the implementation of computer individual-based models (IbMs) to simulate really complex biological populations. Computers can easily keep track of thousands of individuals (often called 'agents'), whose complex behaviours and large amounts of associated data were daunting only 20 years ago. As such, computer modelling has just entered a field where traditional PDE models used to reign alone. A study of the exchange and non-trivial relationship between these two fields, computer IbMs versus classic partial differential equations (PDEs), is appropriate. The aim of this paper is to compare both approaches through a relevant example, namely the evolution of a yeast population in a batch culture. Thus, this paper deals with the utilization of both classical mathematics and computer science in the solution of problems arising in microbiology. First, an IbM approach to study the evolution of a yeast batch culture is presented. Second, an equivalent PDE model is derived by using some techniques from the interacting particle systems field. Third, a comparison and discussion on the advantages and drawbacks of both modelling tools is given.

[1]  Pablo Gómez-Mourelo,et al.  From individual-based models to partial differential equations ☆: An application to the upstream movement of elvers , 2005 .

[2]  Clara Prats,et al.  Individual-based Modelling: An Essential Tool for Microbiology , 2008, Journal of biological physics.

[3]  Karl Oelschl iger.  On the Derivation of Reaction-Diffusion Equations as Limit Dynamics of Systems of Moderately Interacting Stochastic Processes , 2004 .

[4]  Steven F. Railsback,et al.  Individual-based modeling and ecology , 2005 .

[5]  Angela Stevens,et al.  The Derivation of Chemotaxis Equations as Limit Dynamics of Moderately Interacting Stochastic Many-Particle Systems , 2000, SIAM J. Appl. Math..

[6]  M. J. Hatcher,et al.  Modeling Biological Systems: Principles and Applications , 1997 .

[7]  Steven F. Railsback.  Railsback: Individual-based Modeling and Ecology Chapter One 1.1 Why Individual-based Modeling and Ecology? 1.2 Linking Individual Traits and System Complexity: Three Examples 1.2.1 the Green Woodhoopoe Model , 2005 .

[8]  Bernt Øksendal,et al.  Stochastic differential equations (3rd ed.): an introduction with applications , 1992 .

[9]  B. Øksendal.  Stochastic Differential Equations , 1985 .

[10]  Marta Ginovart,et al.  INDISIM, an individual-based discrete simulation model to study bacterial cultures. , 2002, Journal of theoretical biology.

[11]  M. Silbert,et al.  INDISIM-YEAST, an individual-based model to study yeast , 2007 .

[12]  Iosif Ilitch Gikhman,et al.  Introduction to the theory of random processes , 1969 .

[13]  Graeme M. Walker,et al.  Yeast Physiology and Biotechnology , 1998 .

[14]  Tamás Czárán,et al.  Spatiotemporal models of population and community dynamics , 1998 .

[15]  Karl Oelschläger,et al.  On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes , 1989 .

[16]  D. Williams.  STOCHASTIC DIFFERENTIAL EQUATIONS: THEORY AND APPLICATIONS , 1976 .

[17]  S. Shreve,et al.  Stochastic differential equations , 1955, Mathematical Proceedings of the Cambridge Philosophical Society.

[18]  J. Knight.  Modelling biological systems: principles and applications . By J.W. Haefner (London, Chapman & Hall, 1996). 473 pp. Hard cover £49.00. ISBN 0 412 04201 0. , 1997 .

[19]  C. Cosner,et al.  Spatial Ecology via Reaction-Diffusion Equations , 2003 .

[20]  M Ginovart,et al.  Simulation modelling of bacterial growth in yoghurt. , 2002, International journal of food microbiology.

[21]  明 大久保,et al.  Diffusion and ecological problems : mathematical models , 1980 .

[22]  L. Alberghina,et al.  Structural heterogeneity in populations of the budding yeast Saccharomyces cerevisiae , 1983, Journal of bacteriology.

[23]  Marta Ginovart,et al.  Individual based simulations of bacterial growth on agar plates , 2002 .

[24]  François Dubois,et al.  Equivalent partial differential equations of a lattice Boltzmann scheme , 2008, Comput. Math. Appl..