Article Contents
Article Contents

# Bifurcation analysis in a delayed toxic-phytoplankton and zooplankton ecosystem with Monod-Haldane functional response

This research is supported by National Natural Science Foundation of China (No. 11801014), Natural Science Foundation of Hebei Province from China (No. A2018409004), University Discipline Top Talent Selection and Training Program of Hebei Province from China (No. SLRC2019020) and Graduate Student Demonstration Course Construction of Hebei Province from China (No. KCJSX2020093)

• We structure a phytoplankton zooplankton interaction system by incorporating (i) Monod-Haldane type functional response function; (ii) two delays accounting, respectively, for the gestation delay $\tau$ of the zooplankton and the time $\tau_1$ required for the maturity of TPP. Firstly, we give the existence of equilibrium and property of solutions. The global convergence to the boundary equilibrium is also derived under a certain criterion. Secondly, in the case without the maturity delay $\tau_1$, the gestation delay $\tau$ may lead to stability switches of the positive equilibrium. Then fixed $\tau$ in stable interval, the effect of $\tau_1$ is investigated and find $\tau_1$ can also cause the oscillation of system. Specially, when $\tau = \tau_1$, under certain conditions, the periodic solution will exist with the wide range as delay away from critical value. To deal with the local stability of the positive equilibrium under a general case with all delays being positive, we use the crossing curve methods, it can obtain the stable changes of positive equilibrium in $(\tau, \tau_1)$ plane. When choosing $\tau$ in the unstable interval, the system still can occur Hopf bifurcation, which extends the crossing curve methods to the system exponentially decayed delay-dependent coefficients. Some numerical simulations are given to indicate the correction of the theoretical analyses.

Mathematics Subject Classification: 34K18, 34K20, 92D25.

 Citation:

• Figure 1.  The plots of TPP and zooplankton at equilibrium versus $R_\tau$ when $r = 0.8, m = 10, \alpha = 5$ and $L = 6$. There is a forward bifurcation from the zooplankton free equilibrium at $R_\tau = 1.1333$

Figure 2.  The figures of TPP and zooplankton at equilibrium versus $R_\tau$ when $r = 0.3, m = 3, \alpha = 0.8,$ and $L = 10$. There is a backward bifurcation at $R_\tau = 1.8167$, which leads to the existence of multiple positive equilibria

Figure 3.  $(\tau, \mathcal{S}_{n}(\tau))\; (n = 0, 1)$ plots

Figure 4.  $E^*$ is stable when $\tau = 30$

Figure 5.  $E^*$ is unstable when $\tau = 54.4$ and there exists a stable periodic solution

Figure 6.  $E^*$ is still stable when $\tau = 200$

Figure 7.  $E^*$ is stable when $\tau = 30$ and $\tau_1 = 5$

Figure 8.  $(\nu, \mathbf{T}_n(\nu))$ plots $(n = 0, 1, 2)$

Figure 9.  $E^*$ is stable for (A) and (C). $E^*$ is unstable and there exists a stable periodic solution for (B). The bifurcation diagram showing stability switches at $E^*$ and all global Hopf bifurcations shown in (D)

Figure 10.  Feasible region and curve $C$ in $(\tau, \omega)$ plane

Figure 11.  Crossing curves and crossing directions

Figure 12.  $E^*$ is stable when $\tau = 20$ and $\tau_1 = 10$

Figure 13.  $E^*$ is unstable and there exists a stable periodic solution when $\tau = 80$ and $\tau_1 = 10$

Figure 14.  $E^*$ is stable when $\tau = 180$ and $\tau_1 = 10$

Table 1.  Descriptions and units of parameters of system (2)

 Symbol Parameter Definition Unit $r$ Intrinsic growth rate of TPP day$^{-1}$ $L$ Environmental carrying capacity $g C m^{-3}$ $\alpha$ Grazing efficiency of zooplankton day$^{-1}g C m^{-3}$ $\beta$ Growth efficiency of zooplankton day$^{-1}g C m^{-3}$ $\mu$ Natural death rate of zooplankton day$^{-1}$ $m$ Half-saturation constant $[g C m^{-3}]^2$ $\rho$ Toxin-producing rate of TPP $g C m^{-3}$ day$^{-1}$ $\tau$ Gestation delay of zooplankton day$^{-1}$ $\tau_1$ Delay required for the maturity of TPP day$^{-1}$
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