GLOBAL SOLUTIONS NEAR HOMOGENEOUS STEADY STATES IN A MULTIDIMENSIONAL POPULATION MODEL WITH BOTH PREDATOR- AND PREY-TAXIS

We study the system (*) { u(t) = D-1 Delta - chi(1)del . (u del v) + u(lambda(1) - mu(1)u + a(1)v) v(t) = D-2 Delta v + chi(2)del . (v del u) + v(lambda(2) - mu(2)v -a(2)u) (inter alia) for D-1, D-2, chi(1), chi(2), lambda(1), lambda(2), mu(1), mu(2), a(1), a(2) > 0 in smooth, bounded domains Omega subset of R-n, is an element of {1, 2,3}. Without any further restrictions on these parameters, we prove that there exists a constant stable steady state (u(*), v(*) ) is an element of [0, infinity)(2), meaning that there is epsilon > 0 such that if u(0), v(0) is an element of W-2,W-2 (Omega) are nonnegative with partial derivative(nu)u(0) = partial derivative(nu)v(0) = 0 in the sense of traces and parallel to u(0) - u(*)parallel to(W2,2(Omega)) + parallel to v(0) - v(*)parallel to(W2,2(Omega)) < epsilon, then there exists a global classical solution (u, v) of ((*)) with initial data u(0), v(0) converging to (u(*), v(*)) in w(2,2)(Omega). Moreover, the convergence rate is exponential, except for the case lambda(2)mu(1) = lambda(1)a(2), where it is is only algebraical. To the best of our knowledge, this constitutes the first global existence result for ((*)) in the biologically most relevant two- and three-dimensional settings. In the proof, we make use of the special structure in ((*)) and carefully balance the doubly cross-diffusive interaction therein. Indeed, we introduce certain functionals and combine them in a way allowing for cancellations of the most worrisome terms.