Nonlinear dynamics and bifurcation analysis of a discrete plant-herbivore system: Empirical validation and ecological implications

Qamar Din and Iqra Fareed

Mathematics and Computers in Simulation (MATCOM), 2026, vol. 239, issue C, 44-82

Abstract: The present study explores the complex dynamics of plant-herbivore interactions using a discrete-time model incorporating Ricker-type growth for both plant and herbivore populations and a Holling type III functional response. The model was validated against empirical data from observed plant-herbivore interactions, specifically considering species such as Betula glandulosa (dwarf birch) as the plant species and Lepus americanus (snowshoe hare) as the herbivore species, as derived from experimental studies such as the Kluane experiment. The best-fit parametric values were obtained through rigorous statistical inference, with confidence intervals estimated to ensure the robustness of parameter selection. The biological feasibility of equilibrium points was examined, followed by a detailed stability analysis and topological classification. Various codimension-one bifurcations were analyzed, including transcritical bifurcation at boundary equilibrium points, period-doubling bifurcation at both boundary and interior fixed points, and Neimark-Sacker bifurcation at the positive equilibrium using normal form theory. Furthermore, codimension-two bifurcations were explored, revealing the presence of strong resonance cases of 1:2, 1:3, and 1:4 bifurcations. These findings provide insight into the complex interplay of nonlinearity and feedback mechanisms governing plant-herbivore interactions. The dynamical behavior of the discrete plant-herbivore system is further elucidated through comprehensive bifurcation diagrams and phase portraits, revealing intricate transitions between stable equilibria, periodic oscillations, and chaotic regimes. Numerical simulations demonstrate the occurrence of period-doubling bifurcations at boundary and interior fixed points, as well as Neimark-Sacker bifurcations leading to quasi-periodic orbits. Strong resonances (1:2, 1:3, and 1:4) are identified, highlighting the sensitivity to the parameter variations. The Kaplan-Yorke dimension and maximum Lyapunov exponents confirm the presence of chaotic attractors, while basin of attraction plots illustrate the coexistence of multiple stable states. These results underscore the rich, nonlinear dynamics inherent in plant-herbivore interactions, driven by feedback mechanisms and environmental constraints.

Keywords: Discrete-time model; Plant-herbivore interaction; Stability analysis; Transcritical bifurcation; Period-doubling bifurcation; Neimark-Sacker bifurcation; Codimension-two bifurcation; Empirical validation (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:239:y:2026:i:c:p:44-82

DOI: 10.1016/j.matcom.2025.05.026

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