American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Evaluating vaccination effectiveness of group-specific fractional-dose strategies
  • DCDS-B Home
  • This Issue
  • Next Article
    Periodicity and stability analysis of impulsive neural network models with generalized piecewise constant delays
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

Zhichao Jiang 1,, , Zexian Zhang 2, and  Maoyan Jie 3,

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.

Keywords: Double delays, Monod-Haldane functional response, stability, Hopf bifurcation, crossing curve.
Mathematics Subject Classification: 34K18, 34K20, 92D25.
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.

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

Figure 11.  Crossing curves and crossing directions
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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} $
Table Options

Download as excel

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} $
[1]

Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121
[2]

Xianlong Fu, Zhihua Liu, Pierre Magal. Hopf bifurcation in an age-structured population model with two delays. Communications on Pure and Applied Analysis, 2015, 14 (2) : 657-676. doi: 10.3934/cpaa.2015.14.657
[3]

Hermann Brunner, Chunhua Ou. On the asymptotic stability of Volterra functional equations with vanishing delays. Communications on Pure and Applied Analysis, 2015, 14 (2) : 397-406. doi: 10.3934/cpaa.2015.14.397
[4]

Yinshu Wu, Wenzhang Huang. Global stability of the predator-prey model with a sigmoid functional response. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1159-1167. doi: 10.3934/dcdsb.2019214
[5]

Tongtong Chen, Jixun Chu. Hopf bifurcation for a predator-prey model with age structure and ratio-dependent response function incorporating a prey refuge. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022082
[6]

Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325
[7]

Chun-Hsiung Hsia, Tian Ma, Shouhong Wang. Bifurcation and stability of two-dimensional double-diffusive convection. Communications on Pure and Applied Analysis, 2008, 7 (1) : 23-48. doi: 10.3934/cpaa.2008.7.23
[8]

Ferenc Hartung, Janos Turi. Linearized stability in functional differential equations with state-dependent delays. Conference Publications, 2001, 2001 (Special) : 416-425. doi: 10.3934/proc.2001.2001.416
[9]

Udhayakumar Kandasamy, Rakkiyappan Rajan. Hopf bifurcation of a fractional-order octonion-valued neural networks with time delays. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2537-2559. doi: 10.3934/dcdss.2020137
[10]

Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6185-6205. doi: 10.3934/dcdsb.2021013
[11]

Xiao He, Sining Zheng. Bifurcation analysis and dynamic behavior to a predator-prey model with Beddington-DeAngelis functional response and protection zone. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4641-4657. doi: 10.3934/dcdsb.2020117
[12]

Shengqin Xu, Chuncheng Wang, Dejun Fan. Stability and bifurcation in an age-structured model with stocking rate and time delays. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2535-2549. doi: 10.3934/dcdsb.2018264
[13]

Yiwen Tao, Jingli Ren. The stability and bifurcation of homogeneous diffusive predator–prey systems with spatio–temporal delays. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 229-243. doi: 10.3934/dcdsb.2021038
[14]

Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997
[15]

Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045
[16]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805
[17]

Qiumei Zhang, Daqing Jiang, Li Zu. The stability of a perturbed eco-epidemiological model with Holling type II functional response by white noise. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 295-321. doi: 10.3934/dcdsb.2015.20.295
[18]

Kexin Wang. Influence of feedback controls on the global stability of a stochastic predator-prey model with Holling type Ⅱ response and infinite delays. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1699-1714. doi: 10.3934/dcdsb.2019247
[19]

Tzung-shin Yeh. S-shaped and broken s-shaped bifurcation curves for a multiparameter diffusive logistic problem with holling type-Ⅲ functional response. Communications on Pure and Applied Analysis, 2017, 16 (2) : 645-670. doi: 10.3934/cpaa.2017032
[20]

Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (341)
  • HTML views (439)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy