STRUCTURAL STABILITY OF A STAGE STRUCTURED MODEL OF FISH: THE CASE OF THE ANCHOVY (ENGRAULIS ENCRASICOLUS L.) IN THE BAY OF BISCAY Val` ere Calaud 1 and Yvan Lagadeuc 2 1 Laboratoire de Math´ ematiques Appliqu´ ees, Bˆ atiment I.P.R.A., Universit´ e de Pau et des Pays de l’Adour, Avenue de l’Universit´ e BP 1155, 64013 Pau Cedex, France. E-mail: Valere.Calaud@univ-pau.fr 2 Universit´ e de Rennes 1, UMR CNRS ECOBIO, Bˆ at. 14B, Campus de Beaulieu, 35042 Rennes Cedex, France. E-mail: yvan.lagadeuc@univ-rennes1.fr Received 22 October 2005; accepted in final form 22 October 2005 ABSTRACT A study of stage structured model of fish population is presented. This model focuse on the anchovy population in the Bay of Biscay (Engraulis encrasicolus L.) is presented. The method of study is based on an intermediate complexity mathematical model, taking into account the spatialisation, the environmental conditions and the stage-structure of the fishes. First, to test the model, we show mathematical properties, such as unicity of the solution of structural stability. Then we provide numerical simulations, to validate the model and to test the dynamics according to the variations of the parameters. Key Words: advection-diffusion-reaction equation, structural stability, stage-structured population, biophysical coupling, anchovy, Bay of Biscay 1. INTRODUCTION The fluctuations of the hydrodynamics and food abundance are crucial for the vari- ations of fish populations. Actually, dynamics of marine fish populations, as most of the marine species, are totally or partially controlled by hydrodynamics (Sinclair, 1988; Curry et al., 1989). The second main factors which controls dynamics of fishes are space and time distribution of food (Cushing, 1975), which are also controlled by hydrody- namical variability (Franks et al., 1997). Modeling provides a tool to understand and to predict the behavior of species. How- ever, most of the models recently developed are focused either on the biological part – including dynamics populations (Xiao, 2000; Mich et al., 2002) or ecosystem view- point (L´ evy et al., 1998) – or on the hydrodynamical part (Hinrichsen et al., 2003). Indeed, it appears that studies of models of such species should either become quickly too complex to be mathematically studied or are too simple to be biologically realistic. Moreover, parameters are generally not well known due to the difficulty of quantifica- tion in the field. For prediction purpose or on the theoretical ecology point of view, it is a central question to consider with a relevant minimal number of parameters. In this Acta Biotheoretica (2005) 53: 341–358 C  Springer 2006