Biological species routinely collaborate for their mutual benefit or compete for available resources, thereby displaying dynamic behavior that is challenging to replicate in synthetic systems. Here we use computational modeling to design microscopic, chemically active sheets and self-propelled particles encompassing the appropriate synergistic interactions to exhibit bioinspired feeding, fleeing, and fighting. This design couples two different mechanisms for chemically generating motion in fluid-filled microchambers: solutal buoyancy and diffusiophoresis. Catalyst-coated sheets, which resemble crabs with four distinct claws, convert reactants in solution into products and thereby create local variations in the density and chemical composition of the fluid. Via the solutal buoyancy mechanism, the density variations generate fluid flows, which modify the shape and motility of the crabs. Concomitantly, the chemical variations propel the motion of the particles via diffusiophoresis, and thus, the crabs' and particles' motion becomes highly interconnected. For crabs with restricted lateral mobility, these two mechanisms can be modulated to either drive a crab to catch and appear to feed on all of the particles or enable the particles to flee from this sheet. Moreover, by adjusting the sheet's size and the catalytic coating, two crabs can compete and fight over the motile, diffusiophoretic particles. Alternatively, the crabs can temporally share resources by shuttling the particles back and forth between themselves. With completely mobile sheets, four crabs can collaborate to perform a function that one alone cannot accomplish. These findings provide design rules for creating chemically driven soft robotic sheets that significantly expand the functionality of microfluidic devices.