https://doi.org/10.1140/epjp/s13360-025-06779-y
Regular Article
Bi-stability behaviors of plankton–fish model taking phytoplankton as a basic nutritional source for fish
1
Department of Applied Sciences and Humanities, Shaheed Bhagat Singh State University, 152004, Ferozepur, Punjab, India
2
Department of Mathematics, Guru Nanak College, 152026, Sri Muktsar Sahib, Punjab, India
3
Department of Mathematics, School of Computer Science and Artificial Intelligence, SR University, 506371, Warangal, Telangana, India
Received:
17
June
2025
Accepted:
21
August
2025
Published online:
9
September
2025
In this paper, we have proposed three population model with the assumption that some phytoplankton are consumed directly by the fish population for their growth. The phytoplankton species release toxins in water, which are harmful to predators. The other adverse effect on the plankton system is pollution in water bodies. This study involves the positivity, boundedness of solutions, feasibility of various equilibrium points, local and global stability analysis, bifurcation analysis, viz., the occurrence of Hopf, transcritical, saddle-node bifurcations, and to explore the bi-stability regions with respect to the key parameters of the system. Our findings suggest that the populations coexist for low values of the toxin released by phytoplankton; however, an exchange of stability to a fish-free equilibrium state occurs for higher values. The coexistence of all populations, in relation to the conversion rate of phytoplankton to fish, occurs over a very small interval. Beyond a critical value, the zooplankton-free equilibrium point gains stability, indicating direct interactions between phytoplankton and fish species. Lastly, the environmental toxins also impact the coexistence of three species in water bodies. The outcomes of numerical simulations exhibit strong concordance with the corresponding theoretical predictions.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
