Long time simulations of transport equations raise computational challenges since theyrequire both a large domain of calculation and sufficient accuracy. It is thereforeadvantageous, in terms of computational costs, to use a time varying adaptive mesh, withsmall cells in the region of interest and coarser cells where the solution is smooth.Biological models involving cell dynamics fall for instance within this framework and areoften non conservative to account for cell division. In that case the thresholdcontrolling the spatial adaptivity may have to be time-dependent in order to keep up withthe progression of the solution. In this article we tackle the difficulties arising whenapplying a Multiresolution method to a transport equation with discontinuous fluxesmodeling localized mitosis. The analysis of the numerical method is performed on asimplified model and numerical scheme. An original threshold strategy is proposed andvalidated thanks to extensive numerical tests. It is then applied to a biological model inboth cases of distributed and localized mitosis.