Gause's law of competitive exclusion holds that the coexistence of competing species is extremely unlikely when niches are not differentiated. This law is supported by many mathematical studies, yet the coexistence of competing species is nearly ubiquitous in real ecosystems. We pay attention to the fact that plants and animals usually contact with microbial species as mutualistic partners. The activity spaces of host species are different from those of micro-organisms. In the present study, we apply double-layer model to the association of two competing hosts and a microorganism. Two lattices are prepared: one is for hosts, and the other is for microorganism. The basic equation obtained by mean-field theory is an extension of Lotka-Volterra competition model. Both mathematical analysis and numerical simulations reveal that a shared microbial mutualist can permit the coexistence of competing hosts. From the derived condition of coexistence, we believe the microbial mutualism promotes biodiversity in many ecological systems.
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