Strongly coupled elliptic systems and applications to Lotka-Volterra models with cross-diffusion

The aim of this paper is to investigate the existence and method of construction of solutions for a general class of strongly coupled elliptic systems by the method of upper and lower solutions and its associated monotone iterations. The existence problem is for nonquasimonotone functions arising in the system, while the monotone iterations require some mixed monotone property of these functions. Applications are given to three Lotka-Volterra model problems with cross-diffusion and self-diffusion which are some extensions of the classical competition, prey-predator, and cooperating ecological systems. The monotone iterative schemes lead to some true positive solutions of the competition system, and to quasisolutions of the prey-predator and cooperating systems. Also given are some sufficient conditions for the existence of a unique positive solution to each of the three model problems.