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Stability and Bifurcation Analysis of a Phytoplankton-Zooplankton Model with Linear Functional Responses

S. K. Shoyimardonov

TL;DR

The paper addresses stability and bifurcation in a phytoplankton-zooplankton model with linear functional responses, presenting a continuous-time system with logistic prey growth and a discrete-time analogue. Global stability is proved for the continuous model using Lyapunov functions, yielding $E_1=(1,0)$ as globally stable when $0<\gamma\le r$ and $E_2=(\frac{r}{\gamma},\frac{\gamma-r}{\gamma})$ when $\gamma>r$. For the discrete model, invariant sets and LaSalle-type arguments establish global attractivity of equilibria under certain parameter regimes, and a Neimark-Sacker bifurcation occurs at the positive fixed point $E_2$ near $\gamma_0=r+1$, with an attracting invariant closed curve for nearby parameter values; the negativity of the discriminating quantity $\mathcal{L}$ guarantees this attraction. Collectively, these results reveal rich dynamics, including bistability in certain regimes and oscillatory behavior arising from bifurcation, with implications for ecological stability and regime shifts.

Abstract

In this paper, the dynamics of a phytoplankton-zooplankton system with linear functional responses are examined. For the continuous-time model, the global asymptotic stability of the fixed points is demonstrated by constructing Lyapunov functions. For the discrete version of the model, both local and global dynamics are investigated using LaSalle's Invariance Principle. Furthermore, the occurrence of a Neimark-Sacker bifurcation at the positive fixed point is established, and it is proved that the resulting invariant closed curve is attracting.

Stability and Bifurcation Analysis of a Phytoplankton-Zooplankton Model with Linear Functional Responses

TL;DR

The paper addresses stability and bifurcation in a phytoplankton-zooplankton model with linear functional responses, presenting a continuous-time system with logistic prey growth and a discrete-time analogue. Global stability is proved for the continuous model using Lyapunov functions, yielding as globally stable when and when . For the discrete model, invariant sets and LaSalle-type arguments establish global attractivity of equilibria under certain parameter regimes, and a Neimark-Sacker bifurcation occurs at the positive fixed point near , with an attracting invariant closed curve for nearby parameter values; the negativity of the discriminating quantity guarantees this attraction. Collectively, these results reveal rich dynamics, including bistability in certain regimes and oscillatory behavior arising from bifurcation, with implications for ecological stability and regime shifts.

Abstract

In this paper, the dynamics of a phytoplankton-zooplankton system with linear functional responses are examined. For the continuous-time model, the global asymptotic stability of the fixed points is demonstrated by constructing Lyapunov functions. For the discrete version of the model, both local and global dynamics are investigated using LaSalle's Invariance Principle. Furthermore, the occurrence of a Neimark-Sacker bifurcation at the positive fixed point is established, and it is proved that the resulting invariant closed curve is attracting.
Paper Structure (5 sections, 8 theorems, 70 equations, 4 figures)

This paper contains 5 sections, 8 theorems, 70 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Stability Analysis of Continuous Model (\ref{['contin']})
  3. Stability Analysis of Discrete Model (\ref{['h12']})
  4. Dynamics on invariant sets $\mathcal{X}$ and $\mathcal{Y}$
  5. Bifurcation analysis

Key Result

Theorem 1

If $0<\gamma\leq r,$ then the equilibrium point $E_1=(1,0)$ of the continuous model (contin) is globally asymptotically stable.

Figures (4)

  • Figure 1: The parameter set $S$
  • Figure 2: The green and red lines do not intersect, ensuring the invariance of $\mathcal{M}$
  • Figure 3: Two trajectories starting from $x^0=0.8, y^0=0.7.$
  • Figure 4: Phase portraits for the system (\ref{['h12']}) with $r=0.5, \gamma=1.51$ i.e., $\gamma^*=0.01.$

Theorems & Definitions (20)

  • Remark 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Definition 1
  • Lemma 1: Lemma 2.1, Cheng
  • Proposition 1
  • proof
  • Proposition 2
  • ...and 10 more