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Bifurcation Patterns and Chaos Control in Discrete-Time Coral Reef Model

M. Priyanka, P. Muthukumar

TL;DR

This study develops a discrete-time model for coral–macroalgae–turfs interactions and shows that the system exhibits codimension-one bifurcations, notably flip and Neimark-Sacker bifurcations at an interior equilibrium $E^*(M^*,C^*)$. Using center-manifold theory and bifurcation analysis, the authors derive conditions under which these bifurcations occur and characterize the resulting dynamics, including invariant closed curves and chaotic regimes. They implement OGY and a hybrid feedback-control approach to stabilize chaotic trajectories arising from Neimark-Sacker bifurcations, and validate the theory with numerical simulations that reveal reverse period-doubling, bubble bifurcations (period-4, 8, 24), and chaos. The results highlight the richer dynamics of discrete-time reef models and offer practical chaos-control strategies for ecological management under varying macroalgal growth rates. The work lays groundwork for higher-dimensional extensions incorporating additional ecological interactions.

Abstract

The reduction in coral reef densities, characterized by the proliferation of macroalgae, has emerged as a global threat. In this paper, we present a discrete-time coral reef dynamical model that incorporates macroalgae. We explore all ecologically possible equilibrium points for the proposed model. The conditions for the local stability of the interior equilibrium point are analyzed, which represents the coexistence of both coral and macroalgae. Furthermore, we investigate the model's behavior using the center manifold theorem and bifurcation theory. Our analysis reveals that the model undergoes codimension-one bifurcations, specifically period-doubling and Neimark-Sacker bifurcations. To address the chaos resulting from the emergence of the Neimark-Sacker bifurcation, we apply the OGY feedback control method and a hybrid control methodology. Finally, we provide numerical simulations not only to validate the obtained results but also to demonstrate the complex dynamic behaviors that arise. These behaviors include reversal period-doubling bifurcation, period-4, 8, and 24 bubble bifurcations, as well as chaotic behavior.

Bifurcation Patterns and Chaos Control in Discrete-Time Coral Reef Model

TL;DR

This study develops a discrete-time model for coral–macroalgae–turfs interactions and shows that the system exhibits codimension-one bifurcations, notably flip and Neimark-Sacker bifurcations at an interior equilibrium . Using center-manifold theory and bifurcation analysis, the authors derive conditions under which these bifurcations occur and characterize the resulting dynamics, including invariant closed curves and chaotic regimes. They implement OGY and a hybrid feedback-control approach to stabilize chaotic trajectories arising from Neimark-Sacker bifurcations, and validate the theory with numerical simulations that reveal reverse period-doubling, bubble bifurcations (period-4, 8, 24), and chaos. The results highlight the richer dynamics of discrete-time reef models and offer practical chaos-control strategies for ecological management under varying macroalgal growth rates. The work lays groundwork for higher-dimensional extensions incorporating additional ecological interactions.

Abstract

The reduction in coral reef densities, characterized by the proliferation of macroalgae, has emerged as a global threat. In this paper, we present a discrete-time coral reef dynamical model that incorporates macroalgae. We explore all ecologically possible equilibrium points for the proposed model. The conditions for the local stability of the interior equilibrium point are analyzed, which represents the coexistence of both coral and macroalgae. Furthermore, we investigate the model's behavior using the center manifold theorem and bifurcation theory. Our analysis reveals that the model undergoes codimension-one bifurcations, specifically period-doubling and Neimark-Sacker bifurcations. To address the chaos resulting from the emergence of the Neimark-Sacker bifurcation, we apply the OGY feedback control method and a hybrid control methodology. Finally, we provide numerical simulations not only to validate the obtained results but also to demonstrate the complex dynamic behaviors that arise. These behaviors include reversal period-doubling bifurcation, period-4, 8, and 24 bubble bifurcations, as well as chaotic behavior.
Paper Structure (10 sections, 6 theorems, 77 equations, 25 figures, 1 table)

This paper contains 10 sections, 6 theorems, 77 equations, 25 figures, 1 table.

Table of Contents

  1. Introduction
  2. Formulation of model with key assumptions
  3. Stability analysis of two-component coral reef model
  4. Bifurcation analysis
  5. Flip bifurcation analysis around $E^*(M^*, C^*)$
  6. Neimark-Sacker bifurcation analysis around $E^*(M^*, C^*)$
  7. Chaos control
  8. OGY chaos control method
  9. Feedback control method
  10. Results and discussions

Key Result

Lemma 1

paper6-LuoAC Suppose that $F(1)>0$, $\lambda_1$ and $\lambda_2$ are two characteristic roots of $F(\lambda)=0.$ Then

Figures (25)

  • Figure 1: The schematic representation of coral reef model.
  • Figure 2: Nullclines indicates the occurrence of two equilibrium points of model \ref{['paper6-mod2']}.
  • Figure 3: Nullclines indicates the occurrence of unique equilibrium point of model \ref{['paper6-mod2']}.
  • Figure 4: Nullclines indicates the occurrence of no equilibrium point of model \ref{['paper6-mod2']}.
  • Figure 5: Reverse period-doubling bifurcation for $\delta$ versus $M$ and $C$.
  • ...and 20 more figures

Theorems & Definitions (17)

  • Lemma 1
  • Lemma 2
  • Remark 1
  • Theorem 1
  • Theorem 2
  • proof
  • Remark 2
  • Theorem 3
  • proof
  • Theorem 4
  • ...and 7 more