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Journal of Applied Nonlinear Dynamics 14(2) (2025) 371--398 | DOI:10.5890/JAND.2025.06.010

Rakesh Kumar$^{1}$, Amanpreet Kaur$^{2}$, Krishna Pada Das$^{3}$, Ajender Kumar Malik$^{4}$

$^{1}$ Department of Applied Science and Humanities, Shaheed Bhagat Singh State University, Ferozepur, Punjab, India-152004

$^{2}$ Research Scholar, Shaheed Bhagat Singh State University, Ferozepur,Punjab, India-152004, Department of Mathematics, GGN Khalsa College, Civil lines, Ludhiana, Punjab, India-141001

$^{3}$ Department of Mathematics,Mahadevananda Mahavidyalaya, Manirampore-Barrackpore,Kolkata, India

$^{4}$ Professor of Mathematics, School of sciences, UP Rajarshi Tandon Open University, Shantipuram, Prayagraj -211021

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Abstract

In this study, we explore an ecological-epidemiological model involving phytoplankton and zooplankton with simultaneous harvesting of all species and a consideration of time delay. We incorporate the Holling type IV functional response to represent interactions between susceptible and infected phytoplankton, while phytoplankton predation is modeled using the Holling type I functional response. A unified harvesting effort (E) is applied to all species. We establish the positivity and boundedness of the solution, conduct feasibility and stability analyses for potential steady states, and investigate the existence of a bionomic equilibrium and an optimal harvesting policy using Pontryagin's maximal principle. Our findings indicate the stability of the trivial steady state when E > BTP (BTP=Biotechnical productivity of phytoplankton), with other states becoming asymptotically stable under specific conditions. A Hopf bifurcation analysis is conducted using harvesting and time delay as bifurcation parameters. Notably, both harvesting and time delay are highly sensitive to system dynamics, capable of inducing chaos. Elevating control parameters, such as the harvesting coefficient and the recovery rate of infected phytoplankton, as well as the growth rate of zooplankton derived from the predation of susceptible phytoplankton, plays a crucial role in stabilizing the system and mitigating chaos. Numerical simulations visually illustrate our theoretical results.

Acknowledgments

The authors are thankful to the anonymous referee for his/her suggestions to improve the quality of the paper. We are also thankful to the editor for his/her helpful comments. Further, the authors acknowledge the Shaheed Bhagat Singh State University, Ferozepur, Punjab for providing research support in Numerical computational lab.

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