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<p>In this article, a phytoplankton-zooplankton (PZ) model incorporating both size and stage structure for zooplankton is investigated. We first convert the model into a system with a threshold-type and state-dependent delay, and further into a system with a fixed delay. Subsequently, we analyze the existence and local stability of the positive equilibrium, and derive the conditions for the Hopf and double Hopf bifurcations. Next, using center manifold argument and normal form theory, we compute the normal forms near the double Hopf singularity. Finally, numerical simulations are conducted to verify the theoretical findings. Our results demonstrate that the system near the double Hopf singularity exhibits rich dynamics, including stable periodic orbits, quasiperiodic orbits, and switching between the two.</p>

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Double Hopf bifurcation in a phytoplankton-zooplankton model incorporating size and stage structure

  • 期刊论文

  • 发表日期：

    2026年2月11日

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  • Nannan Lu,
  • Zhihua Liu

摘要

In this article, a phytoplankton-zooplankton (PZ) model incorporating both size and stage structure for zooplankton is investigated. We first convert the model into a system with a threshold-type and state-dependent delay, and further into a system with a fixed delay. Subsequently, we analyze the existence and local stability of the positive equilibrium, and derive the conditions for the Hopf and double Hopf bifurcations. Next, using center manifold argument and normal form theory, we compute the normal forms near the double Hopf singularity. Finally, numerical simulations are conducted to verify the theoretical findings. Our results demonstrate that the system near the double Hopf singularity exhibits rich dynamics, including stable periodic orbits, quasiperiodic orbits, and switching between the two.