We studied an intraguild–predation system where an intermediate consumer and a top consumer exploit a shared basal resource. A compact nondimensionalization yielded five interpretable parameters—relative predator growth $\alpha$, crowding $\beta$, enrichment $\gamma$, and depletion couplings $\delta ,\epsilon$. We presented closed-form thresholds that organize the dynamics: the coexistence equilibrium exists exactly when a quadratic in the resource steady state has a root in $(0,\beta )$; as $\gamma$ varies, a two-equilibria window appears and terminates at an explicit saddle–node value ${\gamma }_1^{+}$, with transversality confirmed and transcritical/pitchfork alternatives excluded. A Hopf onset criterion was given via the characteristic polynomial coefficients along the interior branch. For the forward-Euler discretization we established positivity, an absorbing set under an explicit stepsize bound, and stability tests that reduce to $|1+\mathrm{\Delta }\tau \lambda |=1$. Extensions to diffusion and stochastic forcing suggest the incorporation of more realistic spatial and stochastic factors. The thresholds were directly calibratable, enabling reproducible, mechanistic predictions for applied systems.

Environmental enrichment can destabilize predator–prey coexistence through a Hopf bifurcation, yet real ecosystems are finite and intrinsically stochastic. We investigate how mechanistically derived demographic noise shapes near-Hopf dynamics in the Rosenzweig–MacArthur model by systematically comparing two diffusion closures that share identical deterministic drift but differ solely in their predation-induced covariance structure. Starting from a continuous-time Markov chain description, we derive a full-covariance stochastic differential equation whose diffusion tensor inherits stoichiometric coupling, generating a negative prey–predator cross-covariance. Our exact nonlinear simulations demonstrate that the dominant near-Hopf phenomenon is the profound breakdown of the linear noise approximation itself. While the linear noise approximation predicts unbounded variance and spectral amplification, the true nonlinear quasi-cycles remain strictly bounded, rapidly driving the system into absorbing extinction boundaries. We conclude that accurate early warning inference in finite ecosystems depends not on resolving fine-scale stoichiometric covariance, but on properly accounting for nonlinear saturation and noise-induced boundary hitting.