Fear and pressure can tip a plankton-fish model into collapse

A mathematical modelling study published in BMC Ecology and Evolution examines how environmental pressure and the so-called fear effect shape the dynamics of a marine plankton-fish ecosystem. The work, published on 16 October 2026 as an open access research article, builds a prey-predator model linking phytoplankton, zooplankton and fish through non-linear ecological interactions.

The model incorporates several fundamental ecological processes: logistic growth of phytoplankton, grazing of zooplankton on phytoplankton, and predation by fish on zooplankton. The analysis first establishes that the model is mathematically well behaved, demonstrating the positivity and boundedness of the system so that populations remain biologically meaningful over time.

The researchers then determine all equilibrium points of the system and analyse local stability around each of them. This classical dynamical-systems approach allows them to map out when the ecosystem settles into a steady state and when it begins to oscillate or break down as parameters change.

The central results concern bifurcations, the mathematical points at which a system’s qualitative behaviour changes abruptly. According to the study, rising environmental pressure could drive the system to extinction through a branch point bifurcation, in which the long-term persistence of populations is lost as pressure grows.

The fear effect produces its own distinct signature. The study explores the impact of fear costs on the zooplankton population and concludes that greater fear costs can destabilise the system through another Hopf bifurcation.

The theoretical calculations are confirmed by numerical simulations carried out using Mathematica and MATLAB software, providing computational verification of the analytical findings about stability, extinction and oscillatory behaviour.

As a theoretical contribution, the study does not report field measurements or experimental observations; its conclusions concern the behaviour of the proposed mathematical model rather than a specific marine ecosystem. Nevertheless, the work adds to a growing modelling literature on fear effects in plankton systems, and the journal lists related recent articles on fear factors in plankton-fish models with prey refuge and additional food, on monitoring and modelling marine zooplankton in a changing climate, and on toxin effects in diffusive zooplankton-phytoplankton models.

By identifying which combinations of environmental pressure and fear cost lead to extinction, oscillations or stability, the model offers a structured framework for thinking about how non-linear interactions and predator-induced fear can jointly govern the fate of a three-level marine food chain.

Subject of Research: A mathematical prey-predator model of phytoplankton, zooplankton and fish under environmental pressure and fear effects

Article Title: Modelling the impact of environmental pressure and fear effect on a plankton-fish ecosystem

Article References: Barman, K., Phukan, A., Choudhary, S. K., Dehingia, K., Barman, K. K., & Sarmah, H. K. (2026). Modelling the impact of environmental pressure and fear effect on a plankton-fish ecosystem. BMC Ecology and Evolution. https://doi.org/10.1186/s12862-026-02595-9

Image Credits: AI Generated

DOI: 10.1186/s12862-026-02595-9

Keywords: plankton, fish, zooplankton, phytoplankton, prey-predator model, fear effect, environmental pressure, bifurcation, Hopf bifurcation, stability analysis, mathematical ecology, numerical simulation