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<p>We develop a novel discrete-time model for plant–herbivore dynamics that incorporates three key ecological mechanisms: induced plant defenses, an Allee-type limitation on plant performance at low density, and plant augmentation through immigration. The system is represented by a two-dimensional nonlinear map for which we establish positivity and boundedness and determine the fixed points. Stability analysis shows that the system undergoes transcritical, period-doubling, and Neimark–Sacker bifurcations. We derive conditions for each bifurcation and characterize the resulting dynamics. Results indicate that variation in immigration rate, intrinsic growth rate, and defense intensity can shift the system from stable coexistence to oscillatory, quasi-periodic, or chaotic behavior. To address the potential destabilizing effects of these nonlinear interactions, we design a state-feedback control scheme that successfully suppresses chaotic fluctuations and stabilizes the coexistence equilibrium. The resulting control framework provides a quantitative basis for state-dependent management actions (e.g., planting or regulated removal) aimed at maintaining sustainable plant–herbivore coexistence.</p>

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Integrated effects of induced plant defenses, Allee thresholds, and immigration in a discrete-time ecological model: bifurcations, chaos, and stabilization

  • 期刊论文

  • 发表日期：

    2026年4月16日

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  • Hajar Mouhsine,
  • Karima Mokni,
  • Amina Eladdadi,
  • Mohamed Ch-Chaoui

摘要

We develop a novel discrete-time model for plant–herbivore dynamics that incorporates three key ecological mechanisms: induced plant defenses, an Allee-type limitation on plant performance at low density, and plant augmentation through immigration. The system is represented by a two-dimensional nonlinear map for which we establish positivity and boundedness and determine the fixed points. Stability analysis shows that the system undergoes transcritical, period-doubling, and Neimark–Sacker bifurcations. We derive conditions for each bifurcation and characterize the resulting dynamics. Results indicate that variation in immigration rate, intrinsic growth rate, and defense intensity can shift the system from stable coexistence to oscillatory, quasi-periodic, or chaotic behavior. To address the potential destabilizing effects of these nonlinear interactions, we design a state-feedback control scheme that successfully suppresses chaotic fluctuations and stabilizes the coexistence equilibrium. The resulting control framework provides a quantitative basis for state-dependent management actions (e.g., planting or regulated removal) aimed at maintaining sustainable plant–herbivore coexistence.