Stability and Bifurcation of a Multi-Trait Eco-Evolutionary Logistic-Type Model
Rafael Luís
Source abstract
We extend a two-dimensional discrete-time eco-evolutionary logistic model to the case of finitely many quantitative traits. The original planar model couples population density to a single mean trait under stabilizing selection; the extension retains this mechanism while allowing trait-specific evolutionary response rates and feedback coefficients. We establish a positively invariant ecological core and construct compact absorbing boxes, which yield a compact global attractor for the restricted dynamics. All equilibria are characterized through a scalar equation, and the positive equilibrium is shown to exist uniquely exactly when the intrinsic growth factor exceeds one. The Jacobian at this equilibrium has an arrowhead structure, leading to an explicit characteristic polynomial. We prove that the persistence threshold is a transcritical boundary bifurcation and formulate complete spectral and nondegeneracy conditions for flip and Neimark–Sacker bifurcations. Numerical bifurcation diagrams illustrate the transcritical, period-doubling, and Neimark–Sacker bifurcations predicted by the analytical results.
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