Modeling the dynamics of a marine system using the fractional order approach to assess its susceptibility to global warming


Abstract views: 227 / PDF downloads: 158

Authors

Keywords:

Caputo fractional derivatives, simulation, oxygen, plankton, fish, global warming, numerical results

Abstract

In the marine ecosystem, phytoplankton plays a vital role as the primary supplier of oxygen, contributing to 50\% of the total oxygen production. Not only does it serve as a significant source of food for other species, but it also sustains life in the ocean. However, the rising ocean temperatures caused by global warming have severely hindered the ability of phytoplankton to generate oxygen. Furthermore, fishes are crucial consumers of oxygen within the marine ecosystem. This research paper presents a model that intricately connects the dynamics of oxygen, phytoplankton, zooplankton, and fish using the Caputo fractional derivative. The model aims to examine the impact of global warming on the collective dynamics by considering the relationship between the rate of oxygen generation, temperature, and time. The paper establishes the existence and uniqueness of solutions and also analyzes the stability of equilibrium points. Numerical simulations are conducted to demonstrate the impact of fractional derivatives and global warming on oxygen depletion and species extinction.

Author Biography

Mohammad Sajid, Qassim University

Professor

Department of Mechanical Engineering, College of Engineering, Qassim University, Buraydah-51452, Saudi Arabia

References

J. H. Steele, Spatial pattern in plankton communities, vol. 3. Springer Science & Business

Media, 1978.

O. Akira, “Diffusion and ecological problems:(mathematical models),” Biomathematics, 1980.

M. John P, “Ecological consequences of recent climate change,” Conserv Biol, vol. 15, pp. 320–

, 2001.

M. Horst, Spatiotemporal patterns in ecology and epidemiology: theory, models, and simulation.

Chapman and Hall/CRC, 2007.

H. Graham, Phytoplankton ecology: structure, function and fluctuation. Springer Science &

Business Media, 2012.

J. Lukas, H. Helmut, and K. Michal, “Global change drives modern plankton communities away

from the pre-industrial state,” Nature, vol. 570, pp. 372–375, 2019.

Y. Sekerci and S. Petrovskii, “Mathematical modelling of plankton-oxygen dynamics under the

climate change,” Bulletin of mathematical biology, vol. 77, 2015.

P. Yongzhen, L. Yunfei, and L. Changguo, “Evolutionary consequences of harvesting for a

two-zooplankton one-phytoplankton system,” Applied Mathematical Modelling, vol. 36, no. 4,

pp. 1752–1765, 2012.

M. Benfort, U. Feudel, M. Hilker, F, and H. Malchow, “Plankton blooms and patchiness generated

by heterogeneous physical environment,” Ecol. Complexity, no. 20, pp. 185 – 194, 2014.

Lewis, N. D., Morozov, A., Breckels, M. N., Steinke, M., and Codling, E. A., “Multitrophic

interactions in the sea: Assessing the effect of infochemical-mediated foraging in a 1-d spatial

model,” Math. Model. Nat. Phenom., vol. 8, no. 6, pp. 25–44, 2013.

S. V. Petrovskii and M. Horst, Mathematical Models of Marine Ecosystem, Mathematical Models,

vol. 3. Eolss, 2009.

T. Zhang and W. Wang, “Hopf bifurcation and bistability of a nutrient-phytoplanktonzooplankton

model,” Applied Mathematical Modelling, vol. 36, no. 12, pp. 6225–6235, 2012.

A. K. Mishra, “Modeling the depletion of dissolved oxygen in a lake due to submerged macrophytes,”

Nonlinear Analysis: Modelling and Control, vol. 15, no. 2, pp. 185–198, 2010.

R. Ozarsian and Y. Sekerci, “Fractional order oxygen-plankton system under climate change,”

Chaos, vol. 30, 2020.

M. Caputo and M. Fabrizio, “A new definition of fractional derivatives without singular kernal,”

Progr.Fract Differ Appl, vol. 1, pp. 1–13, 2015.

A. Atangana and D. Baleanu, “New fractional derivatives with non-local and non-singular kernel

theory and application to heat transfer model,” Thermal Science, vol. 20, pp. 763–769, 2016.

A. Kumar, A. Prakash, and H. M. Baskonus, “The epidemic covid-19 model via caputo–fabrizio

fractional operator,” Waves in Random and Complex Media, pp. 1–15, 05 2022.

W. Gao and H. M. Baskonus, “Deeper investigation of modified epidemiological computer virus

model containing the caputo operator,” Chaos, Solitons & Fractals, vol. 158, 05 2022.

S. J. Achar, C. Baishya, and M. K. Kaabar, “Dynamics of the worm transmission in wireless

sensor network in the framework of fractional derivatives,” Mathematical Methods in the Applied

Sciences, 2021.

P. Veeresha, N. Malagi, D. Prakasha, and H. M. Baskonus, “An efficient technique to analyze

the fractional model of vector-borne diseases,” Physica Scripta, vol. 97, 05 2022.

C. Baishya, S. J. Achar, P. Veeresha, and D. G. Prakasha, “Dynamics of a fractional epidemiological

model with disease infection in both the populations,” Chaos: An Interdisciplinary Journal

of Nonlinear Science, vol. 31, no. 4, 2021.

Z. Hammouch, M. Yavuz, and N. Ozdemir, “Numerical solutions and synchronization of a

variable-order fractional chaotic system,” Mathematical Modelling and Numerical Simulation

with Applications, vol. 1, pp. 11–23, 08 2021.

C. Baishya and P. Veeresha, “Laguerre polynomial-based operational matrix of integration for

solving fractional differential equations with non-singular kernel,” Proceedings of the Royal

Society A, vol. 477, 2021.

C. Baishya, “An operational matrix based on the independence polynomial of a complete bipartite

graph for the Caputo fractional derivative,” SeMA Journal, pp. 1–19, 2021.

R. S. Dubey, P. Goswami, H. M. Baskonus, and A. Tailor, “On the existence and uniqueness

analysis of fractional blood glucose-insulin minimal model,” International Journal of Modeling,

Simulation, and Scientific Computing, 05 2022.

S. Mohammad, C. Harindri, A. Ali, and K. Santosh, “Chaos controllability in fractional-order

systems via active dual combination–combination hybrid synchronization strategy,” Fractal and

Fractional, vol. 6, no. 11, 717, 2022.

M. Yavuz, “European option pricing models described by fractional operators with classical

and generalized Mittag-Leffler kernels,” Numerical Methods for Partial Differential Equations,

vol. 38, pp. 434–456, 05 2022.

S. Djillali, A. Atangana, A. Zeb, and C. Park, “Mathematical analysis of a fractional-order

epidemic model with nonlinear incidence function,” AIMS Mathematics, vol. 7, no. 2, pp. 2160–

, 2022.

G. Biplab, Dharand Praveen Kumar and S. Mohammad, “Solution of a dynamical memory effect

covid-19 infection system with leaky vaccination efficacy by non-singular kernel fractional

derivatives,” Mathematical Biosciences and Engineering, vol. 19, no. 5, pp. 4341–4367, 2022.

B. Ghanbari and J. Gomez-Aguilar, “Modeling the dynamics of nutrient-phytoplanktonzooplankton

system with variable-order fractional derivatives,” Chaos, Solitons & Fractals,

vol. 116, pp. 114–120, 2018.

B. Ghanbari and S. Djilali, “Mathematical analysis of a fractional-order predator-prey model

with prey social behavior and infection developed in predator population,” Chaos, Solitons &

Fractals, vol. 138, 2020.

R. Shi, J. Ren, and C. Wang, “Stability analysis and hopf bifurcation of a fractional order mathematical

model with time delay for nutrient-phytoplankton-zooplankton,” Mathematical Biosciences

and Engineering, vol. 17, no. 4, pp. 3836–3868, 2020.

P. Veeresha and L. Akinyemi, “Fractional approach for mathematical model of phytoplankton–

toxic phytoplankton–zooplankton system with mittag-leffler kernel,” International Journal

of Biomathematics, vol. 16, no. 03, 2023.

M. Javidi and B. Ahmad, “Dynamic analysis of time fractional order phytoplankton–toxic phytoplankton–

zooplankton system,” Ecological Modelling, vol. 318, pp. 8–18, 2015. Ecological

management for human-dominated urban and regional ecosystems.

M. El-shahed, A. Ahmed, and I. Elsony, “Fractional order model of phytoplankton-toxic

phytoplankton-zooplankton system,” Advances in Analysis, vol. 3, pp. 37–51, 01 2018.

A. Yusuf, B. Acay, and M. Inc, “Analysis of fractional-order nonlinear dynamic systems under

caputo differential operator,” Mathematical Methods in the Applied Sciences, vol. 44, no. 13,

pp. 10861–10880, 2021.

S. Eze and M. Oyesanya, “Fractional order climate change model in a pacific ocean,” Journal of

Fractional Calculus and Applications, vol. 10, pp. 10–23, 01 2020.

S. Eze and M. Oyesanya, “Fractional order on the impact of climate change with dominant

earth’s fluctuations,” Mathematics of Climate and Weather Forecasting, vol. 5, no. 1, pp. 1–11,

A. Din, F. M. Khan, Z. U. Khan, A. Yusuf, and T. Munir, “The mathematical study of climate

change model under nonlocal fractional derivative,” Partial Differential Equations in Applied

Mathematics, vol. 5, 2022.

I. Podlubny, Fractional Differential Equations. Academic Press, 1999.

Y. Li, Y. Chen, and I. Podlubny, “Stability of fractional order nonlinear dynamic systems: Lyapunov

direct method and generalized mittag-leffler stability,” Computers & Mathematics with

Applications, vol. 59, pp. 1810–1821, 2010.

K. Diethelm, The Analysis of Fractional Differential Equations. Springer, 2010.

K. Diethelm, “An algorithm for the numerical solution of differential equations of fractional

order,” Electronic Transactions on Numerical Analysis, vol. 5, 08 1998.

K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of Mathematical

Analysis and Applications, vol. 265, no. 2, pp. 229–248, 2002.

D. Kumarathunge, B. Medlyn, J. Drake, M. Tjoelker, M. Aspinwall, M. Battaglia, F. Cano,

C. Kelsey, M. Cavaleri, L. Cernusak, J. Chambers, K. Crous, M. De Kauwe, D. Dillaway,

E. Dreyer, D. Ellsworth, O. Ghannoum, Q. Han, K. Hikosaka, and D. Way, “Acclimation and

adaptation components of the temperature dependence of plant photosynthesis at the global

scale,” New Phytologist, vol. 222, 12 2018.

C. Robinson, “Plankton gross production and respiration in the shallow water hydrothermal

systems of Milos, Aegean Sea,” Journal of Plankton Research, vol. 22, no. 5, pp. 887–906,

K. Hancke and R. Glud, “Temperature effects on respiration and photosynthesis in three diatomdominated

benthic communities,” Aquatic Microbial Ecology, vol. 37, no. 3, pp. 265–281, 2004.

J. Hansen, R. Ruedy, M. Sato, and K. Lo, “Global surface temperature change,” Reviews of

Geophysics, vol. 48, no. 4, 2010.

Downloads

Published

2023-12-30

How to Cite

Premakumari, R., Baishya, C., Sajid, M., & Manisha Krishna Naik. (2023). Modeling the dynamics of a marine system using the fractional order approach to assess its susceptibility to global warming. Results in Nonlinear Analysis, 7(1), 89–109. Retrieved from https://nonlinear-analysis.com/index.php/pub/article/view/297