Stability switches and global Hopf bifurcation in a nutrient-plankton model

Abstract

This paper is devoted to the analysis of a nutrient-plankton model with delayed nutrient cycling. Firstly, stability and Hopf bifurcation of the positive equilibrium are given, and the direction and stability of Hopf bifurcation are also studied. We show that delay, which is considered in the decomposition of dead phytoplankton, can induce stability switches, such that the positive equilibrium switches from stability to instability, to stability again and so on. One can observe that the influence of delay on the system dynamics is essential. Then, we prove that there exists at least one positive periodic solution as the time delay varies in some regions using the global Hopf bifurcation result of Wu (1998, Trans Am Math Soc 350:4799–4838) for functional differential equations. Furthermore, the impact of input rate of nutrient is discussed along with numerical results, and the role of delay in the nutrient cycling is interpreted ecologically. Finally, several groups of illustrations are performed to justify analytical findings.

Appendix

1.1 Properties of Hopf bifurcation

In this section, based on numerical evaluation of Hopf bifurcation (see Hassard et al. [22], Chap. 3), and using a similar computation process as in [6], we give the computing process of the properties of Hopf bifurcation and stability of the bifurcating periodic solutions from $S^{*}$ under the conditions of Theorem 1 $(ii)$. We obtain the coefficients determining the important quantities:

• $g_{20}=2\bar{D}(\Delta _{11}+\bar{\rho }_{1}^{*}\Delta _{21} + \bar{\rho }_{2}^{*}\Delta _{31}), g_{11}=\bar{D}(\Delta _{12} + \bar{\rho }_{1}^{*}\Delta _{22}+\bar{\rho }_{2}^{*}\Delta _{32}),$

• $g_{02}=2\bar{D}(\Delta _{13}+\bar{\rho }_{1}^{*}\Delta _{23} + \bar{\rho }_{2}^{*}\Delta _{33}), g_{21}=2\bar{D}(\Delta _{14} + \bar{\rho }_{1}^{*}\Delta _{24} + \bar{\rho }_{2}^{*}\Delta _{34}),$

where

• $\Delta _{11}=\rho _{1} b_{12}+b_{20}, \Delta _{12} = (\rho _{1}+\bar{\rho }_{1}) b_{12}+2b_{20}, \Delta _{13}=\bar{\rho }_{1} b_{12}+b_{20},$

• $\Delta _{14}=b_{12}(W_{11}^{2}(0)+\rho _{1}W_{11}^{1}(0)+\frac{1}{2}W_{20}^{2}(0) +\frac{1}{2}\bar{\rho }_{1}W_{20}^{1}(0))+b_{20}(W_{20}^{1}(0)+2W_{11}^{1}(0))$,

• $\Delta _{21}=c_{12}\rho _{1}+c_{23}\rho _{1}\rho _{2}+c_{20}\rho _{1}^{2},$

• $\Delta _{22}=c_{12}(\rho _{1}+\bar{\rho }_{1})+c_{23}(\rho _{1}\bar{\rho }_{2} + \bar{\rho }_{1}\rho _{2})+2c_{20}\rho _{1}\bar{\rho }_{1},$

• $\Delta _{23}=c_{12}\bar{\rho }_{1}+c_{23}\bar{\rho }_{1}\bar{\rho }_{2} + c_{20}\bar{\rho }_{1}^{2},$

• $\Delta _{24}\!=\!c_{12}(W_{11}^{2}(0)+\rho _{1}W_{11}^{1}(0) + \frac{1}{2}W_{20}^{2}(0)+\frac{1}{2}\bar{\rho }_{1}W_{20}^{1}(0)) +c_{23}(\frac{1}{2}\bar{\rho }_{2}W_{20}^{2}(0) +\frac{1}{2}\bar{\rho }_{1}W_{20}^{3} (0)+\rho _{2}W_{11}^{2}(0) + \rho _{1}W_{11}^{3}(0))+c_{20}(\bar{\rho }_{1} W_{20}^{2}(0)+2\rho _{1} \qquad W_{11}^{2}(0))$

• $\Delta _{31}=d_{23}\rho _{1}\rho _{2}+d_{20}\rho _{1}^{2},$

• $\Delta _{32}=d_{23}(\rho _{1}\bar{\rho }_{2}+\bar{\rho }_{1}\rho _{2}) + 2d_{20}\rho _{1}\bar{\rho }_{1},$

• $\Delta _{33}=d_{23}\bar{\rho }_{1}\bar{\rho }_{2}+d_{20}\bar{\rho }_{1}^{2},$

• $\Delta _{34}=d_{23}(\frac{1}{2}\bar{\rho }_{2}W_{20}^{2}(0) + \frac{1}{2}\bar{\rho }_{1}W_{20}^{3}(0)+\rho _{2}W_{11}^{2}(0) +\rho _{1}W_{11}^{3}(0))+d_{20}(\bar{\rho }_{1} W_{20}^{2}(0) +2\rho _{1}W_{11}^{2}(0)),$

• $D=\frac{1}{1+\rho _{1}^{*}\bar{\rho }_{1}+\rho _{2}^{*}\bar{\rho }_{2} + a_{13}\bar{\rho }_{1}\tau _{0}e^{i\omega _{k}\tau _{j}^{k}}},$

• $\rho _{1}=\frac{i\omega _{k}-a_{11}}{a_{12}+a_{13}e^{-i\omega _{k}\tau _{j}^{k}}}, \rho _{2}=\frac{a_{32}}{i\omega _{k}}\rho _{1},$

• $\rho _{1}^{*}=-\frac{i\omega _{k}+a_{11}}{a_{21}}, \rho _{2}^{*}=-\frac{a_{23}}{i\omega _{k}}\rho _{1}^{*}.$

and

• $W_{20}(0)=\frac{ig_{20}}{\omega _{k}}q(0)+\frac{i\bar{g}_{02}}{3\omega _{k}} \bar{q}(0)$,

• $W_{11}(0)=\frac{-ig_{11}}{\omega _{k}}q(0)+\frac{i\bar{g}_{11}}{\omega _{k}} \bar{q}(0),$

• $q(0)=(1,\rho _{1}$, $\rho _{2})$.

Thus, we can compute the following values:

$$\begin{aligned} c_{1}(0)&= \frac{i}{2\omega ^{*}}[g_{20}g_{11}-2| g_{11}|^{2} - \frac{1}{3}| g_{02}|^{2}]+\frac{g_{21}}{2},\\ \beta _{2}&= \text{ Re }(c_{1}(0)),\\ \mu _{2}&= \text{ Re }(\lambda ^{\prime }(\tau ^{*})), \end{aligned}$$

Then we can determine the properties of bifurcating periodic solutions at the critical value $\tau ^{*}$. According to Case 1 and Case 2 in [29] (p. 202), we have

• 1. If $\beta _2<0,~ \mu _{2}>0~ ( \beta _2<0, \mu _{2}<0)$, the bifurcating periodic solutions on the center manifold are stable. This case is referred to as a supercritical bifurcation.

• 2. If $\beta _2>0,~ \mu _{2}>0~ ( \beta _2>0, \mu _{2}<0)$, the bifurcating periodic solutions on the center manifold are unstable. This case is referred to as a subcritical bifurcation.