Stoichiometric constraints and cooperative hunting reshape tri-trophic food-chain dynamics and trophic transfer...

Cooperative hunting by social carnivores is widespread, but its interplay with nutrient stoichiometry in tri-trophic food chains remains poorly understood. We develop a stoichiometrically explicit producer-herbivore-carnivore model in which the top predator engages in density-dependent cooperative hunting, the producer grows under a light-controlled carrying capacity, and all carbon flows are constrained by a fixed phosphorus pool. The carnivore attack rate increases with carnivore biomass and reduces to a standard non-cooperative form when cooperation vanishes, leading to a piecewise smooth system. We establish well-posedness and a positive invariant set, and derive existence and stability conditions for boundary and coexistence equilibria. Cooperative hunting intensity together with herbivore and carnivore loss rates can generate up to four coexistence equilibria and large parameter domains with producer-only or carnivore-free states. Bifurcation analysis with respect to cooperation and light reveals rich dynamics, including multiple types of bistability between equilibria and limit cycles, multi-peak oscillations, and numerically detected chaotic attractors. We quantify herbivore, carnivore and overall food-chain efficiencies at equilibria and as long-time averages of oscillations, and show that intermediate cooperation and moderate light typically promote three-level coexistence and high transfer efficiency, whereas very weak or very strong cooperation and extreme light favor partial food chains or low transfer. In a rate-dependent extension with slowly varying cooperation or light, results reveal tracking of unstable sets and delayed switches between coexisting attractors, including delayed carnivore extinction relative to frozen-parameter predictions. Sensitivity and bifurcation analyses for additional parameters confirm that coupling resource stoichiometry with cooperative hunting yields new mechanisms for coexistence, abrupt regime shifts and efficiency loss in tri-trophic systems.