a b s t r a c tComputer 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.

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The aim of this paper is to apply and justify the so-called aggregation of variables method for reduction of a complex system of linear delayed differential equations with two time scales: slow and fast. The difference between these time scales makes a parameter ε > 0 to appear in the formulation, being a mathematical problem of singular perturbations. The main result of this work consists of demonstrating that, under some hypotheses, the solution to the perturbed problem converges when ε → 0 to the solution of an aggregated system whose construction is proposed.

The aim of this paper is contributing to the study of the exploitation of the European eel (Anguilla anguilla) in the basin of the French river 'Adour'. This exploitation constitutes the basis of the economy of local professional fishermen, who have recently noticed a diminution of the hunting of elvers. In order to design a sustainable exploitation strategy, we must know as much as possible about eel life cycle. For that purpose, we build a model to describe a phase of eels evolution, their migration upstream the river. Our model is based on the biological details of eels behaviour and constitutes a virtual laboratory, useful to test different hypotheses about eels migration.

The aim of this work is to develop and study a fully continuous individual-based model (IBM) for cancer tumor invasion into a spatial environment of surrounding tissue. The IBM improves previous spatially discrete models, because it is continuous in all variables (including spatial variables), and thus not constrained to lattice frameworks. The IBM includes four types of individual elements: tumor cells, extracellular macromolecules (MM), a matrix degradative enzyme (MDE), and oxygen. The algorithm underlying the IBM is based on the dynamic interaction of these four elements in the spatial environment, with special consideration of mutation phenotypes. A set of stochastic differential equations is formulated to describe the evolution of the IBM in an equivalent way. The IBM is scaled up to a system of partial differential equations (PDE) representing the limiting behavior of the IBM as the number of cells and molecules approaches infinity. Both models (IBM and PDE) are numerically simulated with two kinds of initial conditions: homogeneous MM distribution and heterogeneous MM distribution. With both kinds of initial MM distributions spatial fingering patterns appear in the tumor growth. The output of both simulations is quite similar.