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February  2022, 27(2): 691-715. doi: 10.3934/dcdsb.2021061

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

 1 School of Liberal Arts and Sciences, North China Institute of Aerospace Engineering, Langfang, 065000, China 2 283 Company, Second Research Institute of the CASIC, Beijing, 100089, China 3 Aerospace Science and Technology, North China Institute of Aerospace Engineering, Langfang, 065000, China

* Corresponding author: jzhsuper@163.com

Received  September 2020 Revised  January 2021 Published  February 2022 Early access  February 2021

Fund Project: 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.

Citation: Zhichao Jiang, Zexian Zhang, Maoyan Jie. Bifurcation analysis in a delayed toxic-phytoplankton and zooplankton ecosystem with Monod-Haldane functional response. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 691-715. doi: 10.3934/dcdsb.2021061
##### References:
 [1] Q. An, E. Beretta, Y. Kuang, C. Wang and H. Wang, Geometric stability switch criteria in delay differential equations with two delays and delay dependent parameters, J. Differential Equations, 266 (2019), 7073-7100.  doi: 10.1016/j.jde.2018.11.025. [2] M. Banerjee and E. Venturino, A phytoplankton-toxic phytoplankton-zooplankton model, Ecol. Complex., 8 (2011), 239-248.  doi: 10.1016/j.ecocom.2011.04.001. [3] E. Beretta and Y. Kuang, Geometric, stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165.  doi: 10.1137/S0036141000376086. [4] P. Bi and S. Ruan, Bifurcations in delay differential equations and applications to tumor and immune system interaction models, SIAM J. Applied Dynamical Systems, 12 (2013), 1847-1888.  doi: 10.1137/120887898. [5] J. Chattopadhyay, R. Sarkar and S. Mandal, Toxin producing plankton may act as a biological control for planktonic blooms: A field study and mathematical modelling, J. Theor. Biol., 215 (2002), 333-344.  doi: 10.1006/jtbi.2001.2510. [6] J. Chattopadhyay, R. Sarkar and AE Abdllaoui, A delay differential equation model on harmful algal blooms in the presence of toxic substances, IMA J. Appl. Math., 19 (2002), 137-161.  doi: 10.1093/imammb/19.2.137. [7] Y. Ding, W. Jiang and P. Yu, Double Hopf bifurcation in delayed vander pol-duffing equation, Internat. J. Bifur. Chaos, 23 (2013), 1350014, 15 pages. doi: 10.1142/S0218127413500144. [8] K. Gu, S. Niculescu and J. Chen, On stability crossing curves for general systems with two delays, J. Math. Anal. Appl., 311 (2005), 231-253.  doi: 10.1016/j.jmaa.2005.02.034. [9] R. Han and B. Dai, Cross-diffusion induced Turing instability and amplitude equation for a toxic-phytoplankton-zooplankton model with nonmonotonic functional response, Internat. J. Bifur. Chaos, 27 (2017), 1750088, 24 pages. doi: 10.1142/S0218127417500882. [10] J. Hale and S. Lunel, Introduction to functional differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [11] Z. Jiang and T. Zhang, Dynamical analysis of a reaction-diffusion phytoplankton-zooplankton system with delay, Chaos, Solitons Fractals, 104 (2017), 693-704.  doi: 10.1016/j.chaos.2017.09.030. [12] Z. Jiang, W. Zhang, J. Zhang and T. Zhang, Dynamical analysis of a phytoplankton-zooplankton system with harvesting term and Holling III functional response, Internat. J. Bifur. Chaos., 28 (2018), 1850162, 23 pages. doi: 10.1142/S0218127418501626. [13] Z. Jiang, J. Dai and T. Zhang, Bifurcation analysis of phytoplankton and zooplanktoninteraction system with two delays, Internat. J. Bifur. Chaos, 30 (2020), 2050039, 21 pages. doi: 10.1142/S021812742050039X. [14] H. Jiang and Y. Song, Normal forms of non-resonance and weak resonance double Hopf bifurcation in the retarded functional differential equations and applications, Appl. Math. Comput., 266 (2015), 1102-1126.  doi: 10.1016/j.amc.2015.06.015. [15] Z. Jiang and L. Wang, Global Hopf bifurcation for a predator-prey system with three delays, Internat. J. Bifur. Chaos, 27 (2017), 1750108, 15 pages. doi: 10.1142/S0218127417501085. [16] Z. Jiang and Y. Guo, Hopf bifurcation and stability crossing curvein a planktonic resource-consumer system with double delays, Internat. J. Bifur. Chaos, 30 (2020), 2050190, 20 pages. doi: 10.1142/S0218127420501904. [17] S. Ma, Q. Lu and Z. Feng, Double Hopf bifurcation for van der pol-duffing oscillator with parametric delay feedback control, J. Math. Anal. Appl., 338 (2008), 993-1007.  doi: 10.1016/j.jmaa.2007.05.072. [18] R. Pal, D. Basu and M. Banerjee, Modelling of phytoplankton allelopathy with Monod-Haldanetype functional response–A mathematical study, Biosystems, 95 (2009), 243-253.  doi: 10.1016/j.biosystems.2008.11.002. [19] Y. Qu, J. Wei and S. Ruan, Stability and bifurcation analysis in hematopoietic stem cell dynamics with multiple delays, Physica D., 239 (2010), 2011-2024.  doi: 10.1016/j.physd.2010.07.013. [20] S. Roy, S. Bhattacharya, P. Das and J. Chattopadhyay, Interaction among non-toxic phytoplankton, toxic phytoplankton and zooplankton: Inferences from field observations, J. Biol. Phys., 33 (2007), 1-17.  doi: 10.1007/s10867-007-9038-z. [21] J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc., 350 (1998), 4799-4838.  doi: 10.1090/S0002-9947-98-02083-2.

show all references

##### References:
 [1] Q. An, E. Beretta, Y. Kuang, C. Wang and H. Wang, Geometric stability switch criteria in delay differential equations with two delays and delay dependent parameters, J. Differential Equations, 266 (2019), 7073-7100.  doi: 10.1016/j.jde.2018.11.025. [2] M. Banerjee and E. Venturino, A phytoplankton-toxic phytoplankton-zooplankton model, Ecol. Complex., 8 (2011), 239-248.  doi: 10.1016/j.ecocom.2011.04.001. [3] E. Beretta and Y. Kuang, Geometric, stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165.  doi: 10.1137/S0036141000376086. [4] P. Bi and S. Ruan, Bifurcations in delay differential equations and applications to tumor and immune system interaction models, SIAM J. Applied Dynamical Systems, 12 (2013), 1847-1888.  doi: 10.1137/120887898. [5] J. Chattopadhyay, R. Sarkar and S. Mandal, Toxin producing plankton may act as a biological control for planktonic blooms: A field study and mathematical modelling, J. Theor. Biol., 215 (2002), 333-344.  doi: 10.1006/jtbi.2001.2510. [6] J. Chattopadhyay, R. Sarkar and AE Abdllaoui, A delay differential equation model on harmful algal blooms in the presence of toxic substances, IMA J. Appl. Math., 19 (2002), 137-161.  doi: 10.1093/imammb/19.2.137. [7] Y. Ding, W. Jiang and P. Yu, Double Hopf bifurcation in delayed vander pol-duffing equation, Internat. J. Bifur. Chaos, 23 (2013), 1350014, 15 pages. doi: 10.1142/S0218127413500144. [8] K. Gu, S. Niculescu and J. Chen, On stability crossing curves for general systems with two delays, J. Math. Anal. Appl., 311 (2005), 231-253.  doi: 10.1016/j.jmaa.2005.02.034. [9] R. Han and B. Dai, Cross-diffusion induced Turing instability and amplitude equation for a toxic-phytoplankton-zooplankton model with nonmonotonic functional response, Internat. J. Bifur. Chaos, 27 (2017), 1750088, 24 pages. doi: 10.1142/S0218127417500882. [10] J. Hale and S. Lunel, Introduction to functional differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [11] Z. Jiang and T. Zhang, Dynamical analysis of a reaction-diffusion phytoplankton-zooplankton system with delay, Chaos, Solitons Fractals, 104 (2017), 693-704.  doi: 10.1016/j.chaos.2017.09.030. [12] Z. Jiang, W. Zhang, J. Zhang and T. Zhang, Dynamical analysis of a phytoplankton-zooplankton system with harvesting term and Holling III functional response, Internat. J. Bifur. Chaos., 28 (2018), 1850162, 23 pages. doi: 10.1142/S0218127418501626. [13] Z. Jiang, J. Dai and T. Zhang, Bifurcation analysis of phytoplankton and zooplanktoninteraction system with two delays, Internat. J. Bifur. Chaos, 30 (2020), 2050039, 21 pages. doi: 10.1142/S021812742050039X. [14] H. Jiang and Y. Song, Normal forms of non-resonance and weak resonance double Hopf bifurcation in the retarded functional differential equations and applications, Appl. Math. Comput., 266 (2015), 1102-1126.  doi: 10.1016/j.amc.2015.06.015. [15] Z. Jiang and L. Wang, Global Hopf bifurcation for a predator-prey system with three delays, Internat. J. Bifur. Chaos, 27 (2017), 1750108, 15 pages. doi: 10.1142/S0218127417501085. [16] Z. Jiang and Y. Guo, Hopf bifurcation and stability crossing curvein a planktonic resource-consumer system with double delays, Internat. J. Bifur. Chaos, 30 (2020), 2050190, 20 pages. doi: 10.1142/S0218127420501904. [17] S. Ma, Q. Lu and Z. Feng, Double Hopf bifurcation for van der pol-duffing oscillator with parametric delay feedback control, J. Math. Anal. Appl., 338 (2008), 993-1007.  doi: 10.1016/j.jmaa.2007.05.072. [18] R. Pal, D. Basu and M. Banerjee, Modelling of phytoplankton allelopathy with Monod-Haldanetype functional response–A mathematical study, Biosystems, 95 (2009), 243-253.  doi: 10.1016/j.biosystems.2008.11.002. [19] Y. Qu, J. Wei and S. Ruan, Stability and bifurcation analysis in hematopoietic stem cell dynamics with multiple delays, Physica D., 239 (2010), 2011-2024.  doi: 10.1016/j.physd.2010.07.013. [20] S. Roy, S. Bhattacharya, P. Das and J. Chattopadhyay, Interaction among non-toxic phytoplankton, toxic phytoplankton and zooplankton: Inferences from field observations, J. Biol. Phys., 33 (2007), 1-17.  doi: 10.1007/s10867-007-9038-z. [21] J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc., 350 (1998), 4799-4838.  doi: 10.1090/S0002-9947-98-02083-2.
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 $R_\tau = 1.1333$
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
$(\tau, \mathcal{S}_{n}(\tau))\; (n = 0, 1)$ plots
$E^*$ is stable when $\tau = 30$
$E^*$ is unstable when $\tau = 54.4$ and there exists a stable periodic solution
$E^*$ is still stable when $\tau = 200$
$E^*$ is stable when $\tau = 30$ and $\tau_1 = 5$
$(\nu, \mathbf{T}_n(\nu))$ plots $(n = 0, 1, 2)$
$E^*$ 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)
Feasible region and curve $C$ in $(\tau, \omega)$ plane
Crossing curves and crossing directions
$E^*$ is stable when $\tau = 20$ and $\tau_1 = 10$
$E^*$ is unstable and there exists a stable periodic solution when $\tau = 80$ and $\tau_1 = 10$
$E^*$ is stable when $\tau = 180$ and $\tau_1 = 10$
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$^{-1}$
 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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