Mathematical modeling and analysis of Phytoplankton–Zooplankton–Nanoparticle dynamics

In this paper, we investigate the population dynamics of phytoplankton–zooplankton–nanoparticle model with diffusion and density dependent death rate of predator.  The functional response of predator in this model is considered as Beddington–DeAngelis type.  The stability analysis of the equilibrium points is observed by applying the Routh–Hurwitz criterion.  Numerical simulations are given to illustrate the theoretical results.

prey–predator model
наночастинки
дифузія
Beddington–DeAngelis functional response
стабільність
чисельне моделювання
  1. Rana S., Samanta S., Bhattacharya S., Al-Khaled K., Goswami A., Chattopadhyay J.  The effect of nanoparticles on plankton dynamics: A mathematical model.  Biosystems. 127, 28–41 (2015).
  2. Handy R. D., Owen R., Valsami-Jones E.  The ecotoxicology of nanoparticles and nanomaterials: current status, knowledge gaps, challenges, and future needs. Ecotoxicology. 17 (5), 315–325 (2008).
  3. Beretta E., Kuang Y.  Modeling and analysis of a marine bacteriophage infection.  Mathematical Biosciences. 149 (1), 57–76 (1998).
  4. Navarro E., Piccapietra F., Wagner B., Marconi F., Kaegi R., Odzak N., Sigg L., Behra R.  Toxicity of silver nanoparticles to Chlamydomonas reinhardtii.  Environmental science & technology. 42 (23), 8959–8964 (2008).
  5. Miao A. J., Schwehr K. A., Xu C., Zhang S. J., Luo Z., Quigg A., Santschi P. H.  The algal toxicity of silver engineered nanoparticles and detoxification by exopolymeric substances.  Environmental pollution. 157 (11), 3034–3041 (2009).
  6. Miller R. J., Bennett S., Keller A. A., Pease S., Lenihan H. S.  TiO$_2$ Nanoparticles Are Phototoxic to Marine Phytoplankton.  PLoS ONE. 7 (1), e30321 (2012).
  7. Beretta E., Kuang Y.  Modeling and analysis of a marine bacteriophage infection with latency period.  Nonlinear Analysis: Real World Applications. 2 (1), 35–74 (2001).
  8. Chattopadhyay J., Pal S.  Viral infection on phytoplankton–zooplankton system – a mathematical model.  Ecological Modelling. 151 (1), 15–28 (2002).
  9. Garain K., Kumar U., Mandal P. S.  Global Dynamics in a Beddington-DeAngelis Prey–Predator Model with Density Dependent Death Rate of Predator.  Differential Equations and Dynamical Systems. 29, 265–283 (2021).
  10. Zhang X., Huang Y., Weng P.  Permanence and stability of a diffusive predator–prey model with disease in the prey.  Computers & Mathematics with Applications. 68 (10), 1431–1445 (2014).
  11. Kinoshita S.  1 – Introduction to Nonequilibrium Phenomena.  Pattern Formations and Oscillatory Phenomena. 1–59 (2013).
  12. Cantrell R. S., Cosner C.  On the dynamics of predator–prey models with the Beddington–DeAngelis functional response.  Journal of Mathematical Analysis and Applications. 257 (1), 206–222 (2001).
  13. Li H., Takeuchi Y.  Dynamics of the density dependent predator–prey system with Beddington–DeAngelis functional response.  Journal of Mathematical Analysis and Applications. 374 (2), 644–654 (2011).
  14. Tripathi J. P., Abbas S., Thakur M.  Dynamical analysis of a prey–predator model with Beddington–DeAngelis type function response incorporating a prey refuge.  Nonlinear Dynamics. 80, 177–196 (2015).
  15. Mandal A., Tiwari P. K., Samanta S., Venturino E., Pal S.  A nonautonomous model for the effect of environmental toxins on plankton dynamics.  Nonlinear Dynamics. 99 (4), 3373–3405 (2020).
  16. Mandal A., Tiwari P. K., Pal S.  Impact of awareness on environmental toxins affecting plankton dynamics: a mathematical implication.  Journal of Applied Mathematics and Computing. 66, 369–395 (2020).
  17. Friedman A.  Partial Differential Equations of Parabolic Type.  Prentice-Hall, Englewood Cliffs, New York (1964).
  18. Pao C. V.  Nonlinear Parabolic and Elliptic Equations. Plenum, New York (1992).
  19. Murray J. D.  Mathematical biology: I. An introduction. Springer Science & Business Media (2007).
  20. Gupta R. P., Chandra P., Banerjee M.  Dynamical complexity of a prey-predator model with nonlinear predator harvesting.  Discrete & Continuous Dynamical Systems – B. 20 (2), 423–443 (2015).
  21. Xiao D., Ruan S.  Global dynamics of a ratio-dependent predator-prey system.  Journal of Mathematical Biology. 43, 268–290 (2001).
  22. Liao M., Tang X., Xu C.  Stability and instability analysis for a ratio-dependent predator–prey system with diffusion effect.  Nonlinear Analysis: Real World Applications. 12 (3), 1616–1626 (2011).
  23. Xiang T.  Global dynamics for a diffusive predator–prey model with prey-taxis and classical Lotka–Volterra kinetics.  Nonlinear Analysis: Real World Applications. 39, 278–299 (2018).
  24. Yao S. W., Ma Z. P., Cheng Z. B.  Pattern formation of a diffusive predator–prey model with strong Allee effect and nonconstant death rate.  Physica A: Statistical Mechanics and its Applications. 527, 121350 (2019).
  25. Chakraborty B., Ghorai S., Bairagi N.  Reaction–diffusion predator–prey–parasite system and spatiotemporal complexity.  Applied Mathematics and Computation. 386, 125518 (2020).
  26. Liu G., Chang Z., Meng X., Liu S.  Optimality for a diffusive predator–prey system in a spatially heterogeneous environment incorporating a prey refuge.  Applied Mathematics and Computation. 384, 125385 (2020).
  27. Nivethitha M., Senthamarai R.  Analytical approach to a steady-state predator–prey system of Lotka–Volterra model.  AIP Conference Proceedings. 2277, 210005 (2020).
  28. Vijayalakshmi T., Senthamarai R.  Analytical approach to a three species food chain model by applying Homotopy perturbation method.  International Journal of Advanced Science and Technology. 29 (6), 2853–2867 (2020).
  29. Senthamarai R., Vijayalakshmi T.  An analytical approach to top predator interference on the dynamics of a food chain model.  Journal of Physics: Conference Series. 1000, 012139 (2018).
  30. Vijayalakshmi T., Senthamarai R.  An analytical approach to the density dependent prey–predator system with Beddington–Deangelies functional response.  AIP Conference Proceedings. 2112, 020077 (2019).