In　the　mutualism　system　with　three　species　if　the　effects　of　dispersion　and　time　delays　are　both　taken　into　consideration,　then　the　densities　of　the　cooperating　species　are　governed　by　a　coupled　system　of　reaction-diffusion　equations　with　time　delays.　The　aim　of　this　paper　is　to　investigate　the　asymptotic　behavior　of　the　time-dependent　solution　in　relation　to　a　positive　uniform　solution　of　the　corresponding　steady-state　problem　in　a　bounded　domain　with　Neumann　boundary　condition,　including　the　existence　and　uniqueness　of　a　positive　steady-state　solution.　A　simple　and　easily　verifiable　condition　is　given　to　ensure　the　global　asymptotic　stability　of　the　positive　steady-state　solution.　This　result　leads　to　the　permanence　of　the　mutualism　system,　the　instability　of　the　trivial　and　all　forms　of　semitrivial　solutions,　and　the　nonexistence　of　nonuniform　steady-state　solution.　The　condition　for　the　global　asymptotic　stability　is　independent　of　diffusion　and　time-delays　as　well　as　the　net　birth　rate　of　species,　and　the　conclusions　for　the　reaction-diffusion　system　are　directly　applicable　to　the　corresponding　ordinary　differential　system　and　2-species　cooperating　reaction-diffusion　systems.　Our　approach　to　the　problem　is　based　on　inequality　skill　and　the　method　of　upper　and　lower　solutions　for　a　more　general　reaction-diffusion　system.　Finally,　the　numerical　simulation　is　given　to　illustrate　our　results.　(C)　2010　Elsevier　Inc.　All　rights　reserved.