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Stability, Bifurcation, and Chaos Control in a Discrete-Time Phytoplankton-Zooplankton Model with Holling Type II and Type III Functional Responses

Sobirjon Shoyimardonov

TL;DR

The paper analyzes a discrete-time phytoplankton-zooplankton model with unified Holling type II/III functional responses and toxin dispersion. By nondimensionalizing and applying Euler discretization, it characterizes fixed points, global stability, and proves a Neimark-Sacker bifurcation at the unique positive equilibrium as the bifurcation parameter $\gamma=\beta-\theta$ crosses a critical value; the type of emergent invariant curve is determined by the first Lyapunov coefficient $\mathcal{L}$. Through numerical simulations, the authors demonstrate NS-induced quasi-periodic dynamics and obtain chaos-control via linear state-feedback, yielding stable behavior under appropriate gains. Ecologically, these results illuminate transitions from steady-state to oscillatory regimes in plankton communities and offer a potential mechanism for managing bloom dynamics in discrete-time ecological contexts.

Abstract

In this paper, we investigate the dynamics of a discrete-time phytoplankton-zooplankton model where the predator functional response and toxin distribution functions follow both Holling Type II and Holling Type III forms simultaneously. We analyze the types of fixed points and the global stability of the system. Additionally, we prove the occurrence of a Neimark-Sacker bifurcation at the positive fixed point. The theoretical findings are validated through numerical simulations

Stability, Bifurcation, and Chaos Control in a Discrete-Time Phytoplankton-Zooplankton Model with Holling Type II and Type III Functional Responses

TL;DR

The paper analyzes a discrete-time phytoplankton-zooplankton model with unified Holling type II/III functional responses and toxin dispersion. By nondimensionalizing and applying Euler discretization, it characterizes fixed points, global stability, and proves a Neimark-Sacker bifurcation at the unique positive equilibrium as the bifurcation parameter crosses a critical value; the type of emergent invariant curve is determined by the first Lyapunov coefficient . Through numerical simulations, the authors demonstrate NS-induced quasi-periodic dynamics and obtain chaos-control via linear state-feedback, yielding stable behavior under appropriate gains. Ecologically, these results illuminate transitions from steady-state to oscillatory regimes in plankton communities and offer a potential mechanism for managing bloom dynamics in discrete-time ecological contexts.

Abstract

In this paper, we investigate the dynamics of a discrete-time phytoplankton-zooplankton model where the predator functional response and toxin distribution functions follow both Holling Type II and Holling Type III forms simultaneously. We analyze the types of fixed points and the global stability of the system. Additionally, we prove the occurrence of a Neimark-Sacker bifurcation at the positive fixed point. The theoretical findings are validated through numerical simulations
Paper Structure (10 sections, 9 theorems, 85 equations, 8 figures)

This paper contains 10 sections, 9 theorems, 85 equations, 8 figures.

Table of Contents

  1. Introduction
  2. stability of fixed points
  3. bifurcation analysis and chaos control
  4. Neimark-Sacker bifurcation
  5. Chaos control
  6. Global dynamics of (\ref{['h12']})
  7. numerical simulations
  8. Neimark-Sacker bifurcation of the system (\ref{['h1']})
  9. Neimark-Sacker bifurcation of the system (\ref{['h2']})
  10. Conclusion

Key Result

Lemma 1

Let $F(\lambda) = \lambda^2 + B\lambda + C$, where $B$ and $C$ are two real constants. Suppose $\lambda_1$ and $\lambda_2$ are the roots of $F(\lambda) = 0$. If $F(1) > 0$, then the following statements hold:

Figures (8)

  • Figure 1: Triangular regions of stability of the system (\ref{['h12c']}) with parameters $r=0.5, \ \ c=1$ and $\gamma=2.$
  • Figure 2: Stability region between two surfaces (red and green) of the positive fixed point $\overline{E}$ for the operator (\ref{['h1']}), shown in the parameter space $(r, c,\gamma)$ with $0 < r \leq 1$ and $0 < c \leq 2$.
  • Figure 3: Three invariant sets
  • Figure 4: In Figures (a) and (b), the trajectories are shown for system (\ref{['h1']}) (i.e., $h = 1$), while in Figures (c) and (d), the trajectories are presented for system (\ref{['h2']}) (i.e., $h = 2$).
  • Figure 5: Bifurcation diagrams for the system \ref{['h1']} with the parameters $r = 0.5, c = 1,$ and initial values $u^0 = 0.35, v^0 = 0.6$ when the bifurcation parameter $\gamma$ varying on the interval $0.5\leq\gamma\leq3$. In (c), the Maximum Lyapunov exponent corresponding to (a) and (b) are presented. The maximum Lyapunov exponent indicates that chaotic dynamics are observed when approximately $1.78 < \gamma < 2.357$ and $2.443 < \gamma < 2.482$.
  • ...and 3 more figures

Theorems & Definitions (19)

  • Definition 1
  • Lemma 1: Lemma 2.1, Cheng
  • Theorem 1
  • proof
  • Theorem 2
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • Lemma 3
  • ...and 9 more