Material for the manuscript “ Spatial dynamics of seasonal bird migration in the context of H 5 N 1 ”

which is valid for any arbitrary volume, so the integrand is zero and we recover a classical continuity equation similar to that derived for mass conservation in hydrodynamics. (1) is the equivalent to a Lagrangian conservation of bird density with dρ/dt = 0, hence, a flock of birds traveling at speed U with an initial density ρn−1 = ρ(xn−1 + ǫ, tn−1) at the interface of patch Pn−1 at time tn−1 conserves its density along its trajectory and arrives at the interface of patch Pn at xn − ǫ at time tn with a density ρ(xn − ǫ, tn) = ρ(xn−1 + ǫ, tn−1). Considering a constant death rate μ along the flight-way leads simply to the following modified equation