Existence of periodic solutions to a non-autonomous allelopathic phytoplankton model with fear effect

Satyam Narayan Srivastava; Alexander Domoshnitsky; Seshadev Padhi; Rana D. Parshad; John R. Graef; Satyam Narayan Srivastava; Alexander Domoshnitsky; Seshadev Padhi; Rana D. Parshad; John R. Graef

doi:10.3934/era.2026033

Electronic Research Archive

2026, Volume 34, Issue 2: 723-737. doi: 10.3934/era.2026033

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Existence of periodic solutions to a non-autonomous allelopathic phytoplankton model with fear effect

1.
Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Brno 60200, Czech Republic
2.
Department of Mathematics, Ariel University, Ariel 40700, Israel
3.
Department of Mathematics, Birla Institute of Technology, Ranchi 835215, India
4.
Department of Mathematics, Iowa State University, Ames 50011, IA, USA
5.
Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga 37401, TN, USA

Received: 04 November 2025 Revised: 15 December 2025 Accepted: 04 January 2026 Published: 22 January 2026

The authors considered a non-autonomous allelopathic phytoplankton competition model that incorporates the influence of fear effects that are often observed in natural biological phenomena. Based on Mawhin's coincidence degree theory, sufficient conditions for the existence of periodic solutions were obtained. The authors validated their findings with an example and simulations showing that the constant coefficient case leads to dynamics that are neither steady state nor $ T $-periodic, while the periodic coefficient case generates sustained $ T $-periodic oscillations.
- allelopathy,
- phytoplankton,
- competition model,
- fear effects,
- existence of periodic solutions,
- coincidence degree theory
Citation: Satyam Narayan Srivastava, Alexander Domoshnitsky, Seshadev Padhi, Rana D. Parshad, John R. Graef. Existence of periodic solutions to a non-autonomous allelopathic phytoplankton model with fear effect[J]. Electronic Research Archive, 2026, 34(2): 723-737. doi: 10.3934/era.2026033

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Abstract

The authors considered a non-autonomous allelopathic phytoplankton competition model that incorporates the influence of fear effects that are often observed in natural biological phenomena. Based on Mawhin's coincidence degree theory, sufficient conditions for the existence of periodic solutions were obtained. The authors validated their findings with an example and simulations showing that the constant coefficient case leads to dynamics that are neither steady state nor $ T $-periodic, while the periodic coefficient case generates sustained $ T $-periodic oscillations.

References

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