Weakly coupled parabolic systems describing populations undergoing diffusion and interaction dynamics are discussed. The systems are of Lotka–Volterra type and model complex food webs in which there can be several trophic levels involving predator-prey, competition, commensal and amensal interactions. Sufficient conditions on the interaction terms are given which imply the existence and global asymptotic stability of a unique equilibrium point with all species coexisting. The theory of contracting rectangles is the main tool used in proving the stability result.
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Peter N. Brown (1986) studied this question.