ANALYSIS OF PROPAGATION FOR IMPULSIVE REACTION-DIFFUSION MODELS

We study a hybrid impulsive reaction-advection-diffusion model given by a reactiona-dvection-diffusion equation composed with a discrete-time map in space dimension n is an element of N. The reaction-advection-diffusion equation takes the form u(t)((m)) =div(A del u((m))-qu((m)))+f (u((m))) for (x, t) is an element of R-n x (0, 1], for some function f, a drift q, and a diffusion matrix A. When the discrete-time map is local in space we use N-m(x) to denote the density of population at a point x at the beginning of reproductive season in the mth year, and when the map is nonlocal we use u(m)(x). The local discrete-time map is {u((m))(x, 0) = g(N-m(x)) for x is an element of R-n, Nm+1(x) := u((m))(x,1) for x is an element of R-n} for some function g. The nonlocal discrete time map is {u((m))(x, 0) = u(m)(x) for x is an element of R-n, u(m+1) :g(integral(Rn) K(x - y)u((m))(y, 1)dy) for x is an element of R-n}, when K is a nonnegative normalized kernel. Here, we analyze the above model from a variety of perspectives so as to understand the phenomenon of propagation. We provide explicit formulas for the spreading speed of propagation in any direction e is an element of R-n. Due to the structure of the model, we apply a simultaneous analysis of the differential equation and the recurrence relation to establish the existence of traveling wave solutions. The remarkable point is that the roots of spreading speed formulas, as a function of drift, are exactly the values that yield blow-up for the critical domain dimensions, just as with the classical Fisher's equation with advection. We provide applications of our main results to impulsive reaction-advection-diffusion models describing periodically reproducing populations subject to climate change, insect populations in a stream environment with yearly reproduction, and grass growing logistically in the savannah with asymmetric seed dispersal and impacted by periodic fires.