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

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Abstract

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 α, crowding β, enrichment γ, and depletion couplings δ, ε. 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, β); as γ varies, a two- equilibria window appears and terminates at an explicit saddle–node value γ⁺ 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 + ∆τλ| = 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.

Keywords

algebraic elimination / saddle–node and Hopf bifurcation / Euler discretization / intraguild predation

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