Stability and Bifurcation in a Discrete Phytoplankton-Zooplankton Model with Holling-Type Toxic Effects

Abstract

In this paper, we investigate a discrete-time phytoplankton-zooplankton model that incorporates a linear predator functional response alongside a Holling-type toxin distribution. Both Holling type II and type III cases are considered, and we derive conditions on the model parameters that guarantee the existence of positive fixed points. We classify all fixed points and analyze their global stability. Furthermore, we establish the occurrence of a Neimark-Sacker bifurcation at the positive fixed point. Theoretical results are supported by numerical simulations, which illustrate the dynamic behavior of the system

Figures (5)

• Figure 1: Graph of $f(u) = \beta c u^3 - \beta u + 2r$ for various parameter values of $\beta$, $c$, and $r$.
• Figure 3: Phase portraits for the system (\ref{['h1']}) with parameters $c =0.25, \beta = 2$, $r = 0.5$, and $n = 10,000$. The red point represents the fixed point $E_{-}$, while the green curve indicates an invariant closed curve that is attracting. In panel (c), the initial point is taken from outside the invariant closed curve, while in panel (d), the initial point is taken from inside the invariant closed curve. In panels (e) and (f), the closed curve expands and undergoes a transformation in shape.
• Figure 4: Bifurcation diagrams for the system (\ref{['h1']}) with parameters $r = 0.5$, $c = \beta = 2$, and initial values $u^0 = 0.2$, $v^0 = 1.1$, as the bifurcation parameter $\theta$ varies in the interval $0.1 \leq \gamma \leq 5$. Panel (c) presents the maximum Lyapunov exponents corresponding to panels (a) and (b).
• Figure 5: Phase portraits for the system (\ref{['h2']}) with parameters $c = 0.25$, $\beta = 2$, $r = 0.5$, and $n = 10,000$. The red point represents the fixed point $E_{-}$, while the green curve indicates an invariant closed curve that is attracting. In panel (c), the initial point is taken from outside the invariant closed curve, while in panel (d), the initial point is taken from inside the invariant closed curve. In panels (e) and (f), the closed curve expands and undergoes a transformation in shape.
• Figure 6: Bifurcation diagrams for the system (\ref{['h2']}) with parameters $r = 0.5$, $c = 0.25$, $\beta = 2$, and initial values $u^0 = 0.2$, $v^0 = 1.1$, as the bifurcation parameter $\theta$ varies in the interval $0.1 \leq \gamma \leq 3$. Panel (c) presents the maximum Lyapunov exponents corresponding to panels (a) and (b).

Theorems & Definitions (23)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Lemma 1
• proof