Nonlocal problems involving spatial structure for coupled reaction-diffusion systems

The purpose of this work is to investigate the uniqueness and existence of solutions for a system of nonlinear parabolic equations weakly coupled with ordinary differential equations in bounded domains subject to nonlocal boundary conditions that involve values of the unknown functions inside of the spatial domain. The solutions of such problems allow for considering not only physical, chemical or biological properties, but also spatial structure of the modelled system. The uniqueness of classical solutions of the nonlocal boundary problems for coupled reaction-diffusion systems is proved by means of comparison principles for differential inequalities. The existence of the unique solution is obtained via a monotone iterative method. Applications are given to some model problems in epidemiology and ecology with the nonlocal boundary conditions based on the inverse distance weighting interpolation and cokriging.