Dynamic analysis and optimal control of an ecological snowdrift game-coupled delayed information dissemination model considering information beneficiaries and retention behavior

Online information spreading depends on individual judgment, strategy choice, and network structure. Here, we used the eco-evolutionary snowdrift game of Lv et al. and assigned the benefit $ b $, cost $ c $, and environmental pressure $ Q $ interpretations suited to effective information dissemination. The selected attracting ecological densities of cooperators and defectors were mapped to the exit rates of spreaders and information holders; in a multistable region, this selection may depend on the game parameters and initial state. This provided an IHSCR delay model with information-ignorant individuals, beneficiaries, spreaders, holders, and information-immune individuals. A survival-consistent delayed transition formulation preserved nonnegativity and yielded $ R_0(\tau) = e^{-\mu\tau}R_0(0) $, with the critical delay $ \tau_p = \mu^{-1}\ln R_0(0) $ when $ R_0(0) > 1 $. The information-free equilibrium was globally asymptotically stable for $ R_0(\tau)\leq1 $, and no delay-induced imaginary-axis crossing occurred for the positive equilibrium on its existence interval. We also formulated a joint optimal-control problem for the spreader–holder transition channels. A larger cooperative ecological density raised $ \lambda_C $ and shortened the recurrent active cycle; the resulting decrease in $ R_0 $ does not imply a negative social effect of cooperation. Numerical simulations illustrated the threshold and showed that, in the tested settings, longer delays weaken finite-horizon control performance. Joint control gives the largest net objective among the tested strategies, while sensitivity tests across six scenarios support the dominance of $ u_1 $ under the stated control bounds and cost weights.