Blow-up, Quenching, Aggregation and Collapse in a Chemotaxis Model with Reproduction Term

In this paper, we consider the following chemotaxis model with ratio-dependent logistic reaction term { partial derivative u/partial derivative t = D del(del u - u del w/w) + u(a - bu/w), (x, t) epsilon Q(T), partial derivative w/partial derivative t = beta u - delta w, (x, t) epsilon Q(T), u del In(u/w)center dot(n) over right arrow = 0, x epsilon partial derivative Omega, 0 < t < T, u(x, 0) = u(0)(x) > 0, x epsilon (Omega) over bar, w(x, 0) = w(0)(x) > 0, x epsilon (Omega) over bar, It is shown that the solution to the problem exists globally if b + beta >= 0 and will blow up or quench if beta < 0 by means of function transformation and comparison method. Various asymptotic behavior related to different coefficients and initial data is also discussed.