Stability analysis and Hopf bifurcation of a fractional order mathematical model with time delay for nutrient-phytoplankton-zooplankton

Ruiqing Shi; Jianing Ren; Cuihong Wang

doi:10.3934/mbe.2020214

Mathematical Biosciences and Engineering, 2020, 17(4): 3836-3868. doi: 10.3934/mbe.2020214. Research article

Export file:

Format

RIS(for EndNote,Reference Manager,ProCite)
BibTex
Text

Content

Citation Only
Citation and Abstract

Stability analysis and Hopf bifurcation of a fractional order mathematical model with time delay for nutrient-phytoplankton-zooplankton

Ruiqing Shi, Jianing Ren, Cuihong Wang

School of Mathematics and Computer Science, Shanxi Normal University, Linfen, 041004, China

Received: , Accepted: , Published:

Abstract Full Text(HTML) Figure/Table Related pages

In recent years, some researchers paid their attention to the interaction between toxic phytoplankton and zooplankton. Their studies showed that the mechanism of food selection in zooplankton is still immature and when different algae of the same species (toxic and nontoxic) coexist, some zooplankton may not be able to distinguish between toxic and nontoxic algae, and even show a slight preference for toxic strains. Thus, in this article, a fractional order mathematical model with time delay is constructed to describe the interaction of nutrient-phytoplankton-toxic phytoplankton-zooplankton. The main purpose of this paper is to study the influence of fractional order and time delay on the ecosystem. The sufficient conditions for the existence and local stability of each equilibrium are obtained by using fractional order stability theory. By choosing time delay as the bifurcation parameter, we find that Hopf bifurcation occurs when the time delay passes through a sequence of critical values. After that, some numerical simulations are performed to support the analytic results. At last we make some conclusion and point out some possible future work.

Figure/Table

Supplementary

Article Metrics

Keywords: fractional order; nutrient-phytoplankton-zooplankton; time delay; Hopf bifurcation; stability

Citation: Ruiqing Shi, Jianing Ren, Cuihong Wang. Stability analysis and Hopf bifurcation of a fractional order mathematical model with time delay for nutrient-phytoplankton-zooplankton. Mathematical Biosciences and Engineering, 2020, 17(4): 3836-3868. doi: 10.3934/mbe.2020214

References:

1. T. Saha, M. Bandyopadhyay, Dynamical analysis of toxin producing phytoplankton-zooplankton interactions, Nonlinear Anal. Real. World Appl., 10 (2009), 314-332.
2. O. A. Chichigina, A. A. Dubkov, D. Valenti, B. Spagnolo, Stability in a system subject to noise with regulated periodicity, Phys. Rev. E, 84 (2011), 021134.
3. A. L. Barbera, B. Spagnolo, Spatio-temporal patterns in population dynamics, Phys. A, 314 (2002), 120-124.
4. D. Valenti, L. Tranchina, M. Brai, A. Caruso, C. Cosentino, B. Spagnolo, Environmental metal pollution considered as noise: Effects on the spatial distribution of benthic foraminifera in two coastal marine areas of Sicily (Southern Italy), Ecol. Model., 213 (2008), 449-462.
5. H. Zhang, T. Zhang, The stationary distribution of a microorganism flocculation model with stochastic perturbation, Appl. Math. Lett., 103 (2020), 106217.
6. T. Zhang, N. Gao, T. Wang, H. Liu, Z. Jiang, Global dynamics of a model for treating microorganisms in sewage by periodically adding microbial flocculants, Math. Biosci. Eng., 17 (2020), 179-201.
7. M. Chen, M. Fan, R. Liu, X. Wang, X. Yuan, H. Zhu, The dynamics of temperature and light on the growth of phytoplankton, J. Theor. Biol., 385 (2015), 8-19.
8. Y. Sekerci, S. Petrovskii, Mathematical modelling of plankton-oxygen dynamics under the climate change, B Math. Biol., 77 (2015), 2325-2353.
9. J. Zhao, J. Wei, Stability and bifurcation in a two harmful phytoplankton-zooplankton system, Chaos Soliton. Fract., 39 (2009), 1395-1409.
10. S. Abdallah, Stability and persistence in plankton models with distributed delays, Chaos Soliton. Fract., 17 (2003), 879-884.
11. R. R. Sarkar, B. Mukhopadhyay, R. Bhattacharyya, S. Banerjee, Time lags can control algal bloom in two harmful phytoplankton-zooplankton system, Appl. Math. Comput., 186 (2007), 445-459.
12. G. Denaro, D. Valenti, A. L. Cognata, B. Spagnolo, A. Bonanno, G. Basilone, et al., Spatiotemporal behaviour of the deep chlorophyll maximum in Mediterranean Sea: development of a stochastic model for picophytoplankton dynamics, Ecol. Complex., 13 (2013), 21-34.
13. G. Denaro, D. Valenti, B. Spagnolo, G. Basilone, S. Mazzola, S. W. Zgozi, et al., Dynamics of two picophytoplankton groups in mediterranean sea: Analysis of the deep chlorophyll maximum by a stochastic advection-reaction-diffusion model, PLoS One, 8 (2013), e66765.
14. D. Huang, H. Wang, J. Feng, Z. Zhu, Hopf bifurcation of the stochastic model on HAB nonlinear stochastic dynamics, Chaos Soliton. Fract., 27 (2006), 1072-1079.
15. J. Chattopadhyay, R. R. Sarkar, S. Pal, Mathematical modelling of harmful algal blooms supported by experimental findings, Ecol. Complex., 1 (2004), 225-235.
16. M. Javidi, B. Ahmad, Dynamic analysis of time fractional order phytoplankton-toxic phytoplankton-zooplankton system, Ecol. Model., 138 (2015), 8-18.
17. D. Straile, Meteorological forcing of plankton dynamics in a large and deep continental European lake, Oecologia, 122 (2000), 44-50.
18. R. W. Black, L. B. Slobodkin, What is cyclomorphosis? Freshwater Biol., 18 (1987), 373-378.
19. Y. He, Z. Li, Epigenetic environmental memories in plants: Establishment, maintenance, and reprogramming, Trends Genet., 34 (2018), 1-11.
20. S. I. Dodson, T. A. Crowl, B. L. Peckarsky, L. B. Kats, A. P. Covich, J. M. Culp, Non-visual communication in freshwater benthos: an overview, J. N. Am. Benthol. Soc., 13 (1994), 268-282.
21. D. P. Chivers, R. J. F. Smith, Chemical alarm signalling in aquatic predator-prey systems: A review and prospectus, Ecoscience, 5 (1998), 338-352.
22. C. Huang, J. Cao, M. Xiao, A. Alsaedi, F. E. Alsaadi, Controlling bifurcation in a delayed fractional predator-prey system with incommensurate orders, Appl. Math. Comput., 293 (2017), 293-310.
23. E. Ahmed, A. M. A. El-Sayed, H. A. A. El-Saka, Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models, J. Math. Anal. Appl., 325 (2007), 542-553.
24. F. A. Rihan, S. Lakshmanan, A. H. Hashish, R. Rakkiyappan, E. Ahmed, Fractional-order delayed predator-prey systems with Holling type-II functional response, Nonlinear Dynam., 80 (2015), 777-789.
25. V. E. Tarasov, V. V. Tarasova, Macroeconomic models with long dynamic memory: fractional calculus approach, Appl. Math. Comput., 338 (2018), 466-486.
26. A. Khan, J. F. Gómez-Aguilar, T. Abdeljawad, H. Khan, Stability and numerical simulation of a fractional order plant-nectar-pollinator model, Alex. Eng. J., 59 (2019), 49-59.
27. C. I. Muresan, C. Ionescu, S. Folea, R. D. Keyser, Fractional order control of unstable processes: the magnetic levitation study case, Nonlinear Dynam., 80 (2014), 1761-1772.
28. M. D. Ortigueira, Fractional calculus for scientists and engineers, Springer Netherlands, Berlin, 2011.
29. D. Copot, R. D. Keyser, E. Derom, M. Ortigueira, C. M. Ionescu, Reducing bias in fractional order impedance estimation for lung function evaluation, Biomed. Signal Process, 39 (2018), 74-80.
30. G. S. F. Frederico, D. F. M. Torres, Fractional conservation laws in optimal control theory, Nonlinear Dynam., 53 (2008), 215-222.
31. R. L. Magin, O. Abdullah, D. Baleanu, X. J. Zhou, Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation, J. Magn. Reson., 190 (2008), 255-270.
32. E. H. Doha, A. H. Bhrawy, S. S. Ezz-Eldien, A new Jacobi operational matrix: an application for solving fractional differential equations, Appl. Math. Model., 36 (2012), 4931-4943.
33. M. S. Asl, M. Javidi, An improved PC scheme for nonlinear fractional differential equations: Error and stability analysis, J. Comput. Appl. Math., 324 (2017), 101-117.
34. I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press, San Diego, Calif, USA, 1999.
35. S. Gakkhar, A. Singh, Effects of delay and seasonality on toxin producing phytoplanktonzooplankton system, Int. J. Biomath., 5 (2012), 1-21.
36. N. Juneja, K. Agnihotri, H. Kaur, Effect of delay on globally stable prey-predator system, Chaos Soliton. Fract., 111 (2018), 146-156.
37. T. Zhang, J. Liu, Z. Teng, Stability of Hopf bifurcation of a delayed SIRS epidemic model with stage structure, Nonlinear Anal. Real. World Appl., 11 (2010), 293-306.
38. Z. Wang, Y. Xie, J. Lu, Y. Li, Stability and bifurcation of a delayed generalized fractional-order prey-predator model with interspecific competition, Appl. Math. Comput., 347 (2019), 360-369.
39. Z. Wang, X. Wang, Y. Li, X. Huang, Stability and Hopf bifurcation of fractional-order complexvalued single neuron model with time delay, Int. J. Bifurcat. Chaos, 27 (2017), 1-13.
40. M. S. Asl, M. Javidi, Novel algorithms to estimate nonlinear FDEs: applied to fractional order nutrient-phytoplankton-zooplankton system, J. Comput. Appl. Math., 339 (2018), 193-207.
41. Z. Chen, Z. Tian, S. Zhang, C. Wei, The stationary distribution and ergodicity of a stochastic phytoplankton-zooplankton model with toxin-producing phytoplankton under regime switching, Phys. A, 537 (2020), 122728.
42. A. Barreiro, C. Guisande, I. Maneiro, A. R. Vergara, I. Riveiro, P. Iglesias, Zooplankton interactions with toxic phytoplankton: Some implications for food web studies and algal defence strategies of feeding selectivity behaviour, toxin dilution and phytoplankton population diversity, Acta Oecol., 32 (2007), 279-290.
43. I. Petras, Fractional-Order nonlinear systems: Modeling, analysis and simulation, HEP/Springer, London, 2011.
44. E. Ahmed, A. M. A. El-Sayed, H. A. A. El-Saka, On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems, Phys. Lett. A, 358 (2006), 1-4.
45. B. Tao, M. Xiao, Q. Sun, J. Cao, Hopf bifurcation analysis of a delayed fractional-order genetic regulatory network model, Neurocomputing, 275 (2017), 677-686.
46. S. Bhalekar, V. Daftardar-Gejji, A predictor-corrector scheme for solving nonlinear delay differential equations of fractional order, Fract. Calc. Appl. Anal., 1 (2011), 1-9.
47. C. Guisande, M. Frangopulos, I. Maneiro, A. R. Vergara, I. Riveiro, Ecological advantages of toxin production by the dinoflagellate Alexandrium minutum under phosphorous limitation, Mar. Ecol. Prog. Ser., 225 (2002), 169-176.
48. N. Turriff, J. A. Runge, A. D. Cembella, Toxin accumulation and feeding behaviour of the planktonic copepod Calanus jinmarchicus exposed to the red-tide dinoflagellate Alexandrium excavatum, Mar. Biol., 123 (1995), 55-64.
49. M. Schultz, T. Kiorboe, Active prey selection in two pelagic copepods feeding on potentially toxic and non-toxic dinoflagellates, J. Plankton Res., 31 (2009), 553-561

show all references

Reader Comments

your name: * your email: *

Download full text in PDF

Associated material

Metrics