[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Journal of Applied Nonlinear Dynamics
ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Impact of Refuge Prey: A bottom-up top-down Phytoplankton-Zooplankton Interaction Model

Journal of Applied Nonlinear Dynamics 11(1) (2022) 179--194 | DOI:10.5890/JAND.2022.03.011

Anal Chatterjee$^1$, Samares Pal$^2$

$^1$ Department of Mathematics, Barrackpore Rastraguru Surendranath College, North 24 Parganas 700120, India

$^2$ Department of Mathematics, University of Kalyani, Kalyani- 741235, India

Download Full Text PDF

 

Abstract

A detailed study on the effect of refuge on the phytoplankton-zooplankton ecosystem is explored. At first, the coexistence and stability conditions of different equilibria of the plankton system are analyzed. Our observations established that refuges have a strong impact on plankton dynamics. When the strength of phytoplankton crosses a certain critical value, the coexistence equilibrium loses its stability and enters into Hopf bifurcation that leads to oscillations of all species. The direction of the Hopf bifurcation is also established. Next, we used Pontryagins maximum principle to study a path of optimal harvesting policy. Also, we observed that the bottom-up and top-down effects like constant nutrient input, rate of zooplankton decay due to the toxic effect of phytoplankton play important roles for switching from one steady state to another that relates to the transcritical bifurcation. We derived the bifurcation scenarios when two different parameters vary together at the same time. At last, numerical simulations are implemented to support our results.

Acknowledgments

The research of Samares Pal is partially supported by Science and Engineering Research Board, Government of India, Grant No. CRG/2019/003248.

References

  1. [1]  Hallegraeff, G.M. (1993), A review of harmful algal blooms and their apparent global increase, Phycologia, 32, 79-99.
  2. [2]  Smayda, T.J. (1989), Primary production and the global epidemic of phytoplankton blooms in the sea: A Linkage?, in Noval Phytoplankton Blooms: Causes and Impact of Recurrent Brown Tides and Other Unusual Blooms, E.M. Cosper et al., eds., Springer Verlag, Berlin, 449-483.
  3. [3]  Anderson, D.M., Kaoru, Y., and White, A.W. (2000), Estimated Annual Economic Impacts form Harmful Algal Blooms (HABs) in the United States Sea, Woods Hole Sea Grant, Woods Hole, MA.
  4. [4]  Edwards, A.M. and Brindley, J. (1999), Zooplankton mortality and the dynamical behaviour of plankton population models, Bulletin of Mathematical Biology, 61, 303-339.
  5. [5]  Truscott, J.E. and Brindley, J. (1994), Ocean plankton populations as excitable media, Bulletin of Mathematical Biology, 56, 981-998.
  6. [6]  Clother, D.R. and Brindley, J. (1999), Excitability of an age-structured plankton ecosystem, Journal of Mathematical Biology, 39, 377-420.
  7. [7]  Pitchford, J.W. and Brindley, J. (1999), Iron limitation, grazing pressure and oceanic high nutrient-low chlorophyll (HNLC) regions, Journal of Plankton Research, 21(3), 525-547.
  8. [8]  Robson, B.J. and Hamilton, D.P. (2004), Three-dimensional modelling of a Microcystis bloom event in the Swan River estuary, Western Australia, Ecological Modelling, 174/1-2, 203-222.
  9. [9]  Chan, T.U., Robson, B.J., and Hamilton, D.P. (2003), Modelling phytoplankton succession and biomass in seasonal West Australian estuary, Verh. Int. Ver. Limnol., 28(2), 1086-1088.
  10. [10]  Huppert, A., Blasius, B., and Stone, L. (2002), A model of phytoplankton blooms, The American Naturalist, 159, 156-171.
  11. [11]  Segal, R., Waite, A.M., and Hamilton, D.P. (2006), Transition from planktonic to benthic algal dominance along a salinity gradient, Hydrobiologia, 556, 119-135.
  12. [12]  Chatterjee, A., Pal, S., and Chatterjee, S. (2011), Bottom up and top down effect on toxin producing phytoplankton and its consequence on the formation of plankton bloom, Applied Mathematics and Computation, 218(7), 3387-398.
  13. [13]  Pal, S., Chatterjee, S., and Chattopadhyay, J, (2007), Role of toxin and nutrient for the occurrence and termination of plankton bloom. Results drawn from field observations and a mathematical model, Biosystems, 90(1), 87-100.
  14. [14]  Banerjee, M. and Venturino, E. (2011), A phytoplankton-toxic phytoplankton-zooplankton model, Ecological Complexity, 8(3), 239-248.
  15. [15]  Jang, S.R.J., Baglama, J., and Rick, J. (2006), Nutrient phytoplankton-zooplankton models with a toxin, Mathematical and Computer Modelling, 43, 105-118.
  16. [16]  Chattopadhyay, J., Sarkar, R.R., and Pal, S. (2004), Mathematical modelling of harmful algal blooms supported by experimental findings, Ecological Complexity, 1, 225-235.
  17. [17]  Khajanchi, S. and Banerjee, S. (2017), Role of constant prey refuge on stage structure predator-prey model with ratio dependent functional response, Applied Mathematics and Computation, 314, 193-198.
  18. [18]  Haque, M., Rahman, S., Venturino, E., and Li, B.L. (2014), Effect of a functional response-dependent prey refuge in a predator-prey model, Ecological Complexity, 20, 248-256.
  19. [19]  Jana, D. and Ray, S. (2016), Impact of physical and behavioral prey refuge on the stability and bifurcation of Gause type Filippov prey-predator system, Modeling Earth Systems and Environment, 2(1), 24. https://doi.org/10.1007/s40808-016-0077-y.
  20. [20]  Das, A and Samanta, G.P. (2020), A prey-predator model with refuge for prey and additional food for predator in a fluctuating environment, Physica A: Statistical Mechanics and its Applications, 538, 122844.
  21. [21]  Guin, L.N. and Acharya, S. (2017), Dynamic behaviour of a reaction-diffusion predator-prey model with both refuge and harvesting, Nonlinear Dynamics, 88, 501-1533.
  22. [22]  Zhang, W. and Zhao, M. (2013), Dynamical Complexity of a Spatial Phytoplankton-Zooplankton Model with an Alternative Prey and Refuge Effect, Hindawi Publishing Corporation Journal of Applied Mathematics, Volume 2013, Article ID 608073.
  23. [23]  Li, J., Song, Y., Wan, H., and Zhu, H. (2017), Dynamical analysis of a toxin-producing phytoplankton-zooplankton model with refuge, Mathematical Biosciences and Engineering, 14(2), 529-557.
  24. [24]  Paul, S., Mandal, S.P., and Bhattacharya, P. (2016), Discussion on fuzzy quota harvesting model in fuzzy environment: fuzzy differential equation approach, Modeling Earth Systems and Environment, 2(70), https://doi.org/10.1007/s40808-016-0113-y.
  25. [25]  Fredrickson, A.G. and Stephanopoulos, G. (1981), Microbial competition, Science, 213(4511), 972-979.
  26. [26]  Smith, H.L. (1981), Competitive coexistence in an oscillating chemostat, SIAM Journal on Applied Mathematics, 40(3), 498-522.
  27. [27]  Mukhopadhyay, B. and Bhattacharyya, R. (2011), A stage structured food chain model with stage dependent predation: Existence of codimension one and codimension two bifurcations, Nonlinear Analysis: Real World Applications, 12, 3056-3072.
  28. [28] Toaha, S., Kusuma, J.K., and Bahri, M. (2014), Stability Analysis and Optimal Harvesting Policy of Prey-Predator Model with Stage Structure for Predator, Applied Mathematical Sciences, 8(159), 7923-7934.
  29. [29] Toaha, S. and Rustam, R. (2017), Optimal harvesting policy of predator-prey model with free fishing and reserve zones, AIP Conference Proceedings, 1825, 020023, DOI: 10.1063/1.4978992.
  30. [30]  Toaha, S. and Azis, M.I. (2018), Stability and Optimal Harvesting of Modified Leslie-Gower Predator-Prey Model, Journal of Physics: Conf. Series, 979, 012069, DOI:10.1088/1742-6596/979/1/012069.

Copyright © 2011-2024 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates