Mathematical modeling of commensalism provides insights into ecological interactions in which one species benefits while the other remains unaffected. In this study, a discrete-time commensalism model is proposed by incorporating a saturating interaction benefit and an interaction efficiency parameter. In contrast to related models, the proposed formulation accounts for both the saturation of the commensal benefit and the fraction of the host population that is either ineffective or unavailable for interaction. The existence and stability of fixed points are analyzed, and conditions for period-doubling bifurcation are derived using the center manifold and normal form theories. Numerical simulations, including bifurcation diagrams, maximum Lyapunov exponents, phase portraits, and basins of attraction, confirm the theoretical results and reveal periodic, chaotic, and multistable dynamics. The findings demonstrate the important role of interaction efficiency and saturation in shaping commensal dynamics.
Citation: Saad Jamhan Aldosari. Stability and bifurcation analysis in a commensalism model with saturation and efficiency constraints[J]. AIMS Mathematics, 2026, 11(8): 27508-27528. doi: 10.3934/math.20261101
Mathematical modeling of commensalism provides insights into ecological interactions in which one species benefits while the other remains unaffected. In this study, a discrete-time commensalism model is proposed by incorporating a saturating interaction benefit and an interaction efficiency parameter. In contrast to related models, the proposed formulation accounts for both the saturation of the commensal benefit and the fraction of the host population that is either ineffective or unavailable for interaction. The existence and stability of fixed points are analyzed, and conditions for period-doubling bifurcation are derived using the center manifold and normal form theories. Numerical simulations, including bifurcation diagrams, maximum Lyapunov exponents, phase portraits, and basins of attraction, confirm the theoretical results and reveal periodic, chaotic, and multistable dynamics. The findings demonstrate the important role of interaction efficiency and saturation in shaping commensal dynamics.
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Saad Jamhan Aldosari. Stability and bifurcation analysis in a commensalism model with saturation and efficiency constraints[J]. AIMS Mathematics, 2026, 11(8): 27508-27528. doi: 10.3934/math.20261101
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