title: A delay reaction-diffusion model of the dynamics of botulinum in fish.
abstract: A model of interaction between fish and a bacterium (Clostridium botulinum) responsible for avian botulism is introduced, considering diffusion of both fish and bacterium in water. The fish population moves randomly in water. Death fish disintegrate in water, at different locations, causing bacteria to diffuse through water and infect other fish. Existence of uniform steady states is investigated and the linearized stability of the positive uniform steady state is analyzed. A Hopf bifurcation is proved to occur from the uniform steady state when the bifurcation parameter, here the time delay, passes through a critical value and diffusion coefficients satisfy some conditions, that induces time oscillations of the populations. Comments on diffusion-driven instability are provided, and numerical simulations are carried out to illustrate the results.
authors.affiliation: Université de Lyon, Université Lyon I, CNRS, UMR 5208, Institut Camille Jordan, Batiment du Doyen Jean Braconnier, 43, blvd du 11 novembre 1918, F-69622 Villeurbanne Cedex, France. crauste@math.univ-lyon1.fr
authors.forename: Fabien
authors.initials: F
authors.lastname: Crauste
authors.forename: M Lhassan
authors.initials: ML
authors.lastname: Hbid
authors.forename: Abdelaziz
authors.initials: A
authors.lastname: Kacha