Biological growth and spread modeled by systems of recursions. I. Mathematical theory

We study the asymptotic behavior of solutions to a system of recursions u in+1 = Qi[mu n], i = 1, ..., k. The vector operator Q has the origin theta and a positive vector beta as fixed points and is defined for vector-valued functions bounded between theta and gamma where gamma greater than or equal to beta. In addition, Q is order-preserving, commutes with translation, and is continuous in the topology of uniform convergence on compact subsets. Let theta less than or equal to pi much less than beta, and suppose that for all pi much less than alpha much less than beta, Q(n) alpha]----beta as n----infinity. If u0 much greater than pi on a sufficiently large ball and has bounded support, then un propagates with a speed c*(xi) in the direction of the unit vector xi as n----infinity. In certain cases, c*(xi) can be calculated explicitly. The results generalize those of a scalar equation studied by Weinberger.