Grazing promotes soil phosphorus cycling by enhancing soil microbial functional genes for phosphorus transformation in plant rhizosphere in a semi-arid natural grassland

•Grazing increases soil available P concentration and alkaline phosphatase activity.•Grazing enhances microbial functional genes promoting P cycling in rhizosphere soil.•Enhanced genes are for P-uptake and organic P mineralization.•Enhancement of C–P lyase genes is the key mechanism of grazing promoting P cycling.