Algebraic–spectral thresholds and discrete–continuous stability transfer in Leslie–Gower systems

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.