{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,14]],"date-time":"2026-10-14T18:45:06Z","timestamp":1792003506411,"version":"4.4.0"},"reference-count":62,"publisher":"Public Library of Science (PLoS)","issue":"10","license":[{"start":{"date-parts":[[2026,10,14]],"date-time":"2026-10-14T00:00:00Z","timestamp":1791936000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["www.plosone.org"],"crossmark-restriction":false},"short-container-title":["PLoS One"],"abstract":"<jats:p>Wild prey are shaped by predators long before they are caught. Fear alone suppresses reproduction, physical refuges reduce exposure, and many species strike back directly at their attackers, imposing real costs on the hunter. To investigate the combined effects of these behavioral mechanisms, we develop a Caputo fractional-order predator\u2013prey model incorporating fear-induced reproductive suppression, proportional prey refuge, active anti-predator retaliation, a Holling type II functional response, and nonlinear saturation harvesting. The fractional memory order accounts for hereditary effects in ecological interactions, with the classical integer-order system recovered as a special case. Fundamental dynamical properties of the model, including non-negativity, uniform boundedness, and existence and uniqueness of solutions, are established. All biologically feasible equilibria are characterized, and local and global asymptotic stability conditions are derived using Matignon\u2019s criterion and Lyapunov function techniques. The analysis further reveals that the coexistence equilibrium can lose stability through a Hopf bifurcation induced by the fractional memory order, and a critical threshold for the onset of this bifurcation is derived. Numerical evaluation of the first Lyapunov coefficient provides evidence that the bifurcation is supercritical, leading to the emergence of stable periodic oscillations. Applying Pontryagin\u2019s Maximum Principle, we derive a bang\u2013singular\u2013bang optimal harvesting strategy that improves discounted net revenue while maintaining ecologically sustainable population levels. Numerical simulations support the theoretical results and illustrate the combined influence of fear, refuge, anti-predator behavior, harvesting, and memory on ecosystem dynamics. The findings demonstrate that the fractional memory order acts as an effective regulator of ecological stability, providing a biologically interpretable mechanism for controlling oscillatory behavior that has no counterpart in classical integer-order predator\u2013prey models.<\/jats:p>","DOI":"10.1371\/journal.pone.0360316","type":"journal-article","created":{"date-parts":[[2026,10,14]],"date-time":"2026-10-14T17:51:31Z","timestamp":1792000291000},"page":"e0360316","update-policy":"https:\/\/doi.org\/10.1371\/journal.pone.corrections_policy","source":"Crossref","is-referenced-by-count":0,"title":["Dynamics and optimal harvesting of a fractional predator\u2013prey system with fear and anti-predator mechanisms"],"prefix":"10.1371","volume":"21","author":[{"given":"Mohan Kumar","family":"Govindan","sequence":"first","affiliation":[{"name":"Vellore Institute of Technology-Chennai Campus","place":["Chennai, Tamil Nadu, India"],"department":["Department of Mathematics, School of Advanced 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Zhang","year":"2022","journal-title":"Chaos"},{"key":"pone.0360316.ref016","volume-title":"Mathematical Bioeconomics: The Optimal Management of Renewable Resources","author":"CW Clark","year":"1990"},{"issue":"2","key":"pone.0360316.ref017","doi-asserted-by":"crossref","first-page":"2194","DOI":"10.1002\/mma.10431","article-title":"Nonlinear dynamics in a fear\u2010driven predator\u2013prey system: Bistability, bifurcations, hydra effect, and optimal harvesting","volume":"48","author":"AK Umrao","year":"2024","journal-title":"Math Methods App Sci"},{"key":"pone.0360316.ref018","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1007\/s12190-020-01491-9","article-title":"Bifurcations in a modified Leslie\u2013Gower predator\u2013prey discrete model with Michaelis\u2013Menten prey harvesting","volume":"67","author":"A Singh","year":"2021","journal-title":"J Appl Math Comput"},{"issue":"1","key":"pone.0360316.ref019","article-title":"Analysis of the stability of a predator-prey 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Wang","year":"2024","journal-title":"Qual Theory Dyn Syst"},{"key":"pone.0360316.ref027","article-title":"Dynamical analysis and neural network approximation of a fractional-order predator\u2013prey model with disease transmission","author":"M Govindan","year":"2026","journal-title":"Sci Rep"},{"issue":"1","key":"pone.0360316.ref028","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1007\/s11071-014-1439-5","article-title":"Hopf bifurcation analysis of a new commensurate fractional-order hyperchaotic system","volume":"78","author":"X Li","year":"2014","journal-title":"Nonlinear Dyn"},{"issue":"19","key":"pone.0360316.ref029","doi-asserted-by":"crossref","first-page":"26949","DOI":"10.1007\/s11071-025-11499-0","article-title":"Dynamic analysis of a fractional order three-dimensional predator-prey system with competitive and fear effects","volume":"113","author":"B Xie","year":"2025","journal-title":"Nonlinear 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principle for general Caputo fractional optimal control problems with Bolza cost and terminal constraints","volume":"26","author":"M Bergounioux","year":"2020","journal-title":"ESAIM: Control Optim Calc Var"},{"key":"pone.0360316.ref041","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1086\/284719","article-title":"Antipredator behavior and the population dynamics of simple predator\u2013prey systems","volume":"130","author":"AR Ives","year":"1987","journal-title":"Am Nat"},{"issue":"4","key":"pone.0360316.ref042","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1007\/s11071-012-0475-2","article-title":"A fractional calculus based model for the simulation of an outbreak of dengue fever","volume":"71","author":"K Diethelm","year":"2012","journal-title":"Nonlinear Dyn"},{"issue":"6","key":"pone.0360316.ref043","doi-asserted-by":"crossref","first-page":"4779","DOI":"10.1016\/j.aej.2021.10.030","article-title":"A new comparative study on the general fractional model of COVID-19 with isolation and quarantine effects","volume":"61","author":"D Baleanu","year":"2022","journal-title":"Alexandria Eng J"},{"key":"pone.0360316.ref044","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1007\/s12190-016-1017-8","article-title":"Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge","volume":"54","author":"H-L Li","year":"2016","journal-title":"J Appl Math Comput"},{"key":"pone.0360316.ref045","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1016\/j.nonrwa.2016.05.010","article-title":"Stability and bifurcation analysis in a predator\u2013prey system with Michaelis\u2013Menten type predator harvesting","volume":"33","author":"D Hu","year":"2017","journal-title":"Nonlinear Anal: Real World Appl"},{"issue":"1","key":"pone.0360316.ref046","first-page":"286","article-title":"Generalized Taylor\u2019s formula","volume":"186","author":"ZM Odibat","year":"2007","journal-title":"Appl Math Comput"},{"issue":"5","key":"pone.0360316.ref047","doi-asserted-by":"crossref","first-page":"1810","DOI":"10.1016\/j.camwa.2009.08.019","article-title":"Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag\u2013Leffler stability","volume":"59","author":"Y Li","year":"2010","journal-title":"Comput Math Appl"},{"issue":"1","key":"pone.0360316.ref048","first-page":"963","article-title":"Stability results for fractional differential equations with applications to control processing.","volume":"2","author":"D Matignon","year":"1996","journal-title":"Computational Engineering in Systems Applications"},{"issue":"1","key":"pone.0360316.ref049","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physleta.2006.04.087","article-title":"On some Routh\u2013Hurwitz conditions for fractional order differential equations and their applications in Lorenz, R\u00f6ssler, Chua and Chen systems","volume":"358","author":"E Ahmed","year":"2006","journal-title":"Phys Lett A"},{"issue":"9","key":"pone.0360316.ref050","doi-asserted-by":"crossref","first-page":"2951","DOI":"10.1016\/j.cnsns.2014.01.022","article-title":"Lyapunov functions for fractional order systems","volume":"19","author":"N Aguila-Camacho","year":"2014","journal-title":"Commun Nonlinear Sci Numer Simul"},{"key":"pone.0360316.ref051","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.cnsns.2014.12.013","article-title":"Volterra-type Lyapunov functions for fractional-order epidemic systems","volume":"24","author":"C Vargas-De-Le\u00f3n","year":"2015","journal-title":"Commun Nonlinear Sci Numer Simul"},{"key":"pone.0360316.ref052","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/j.nonrwa.2015.05.014","article-title":"The effect of vaccines on backward bifurcation in a fractional order HIV model","volume":"26","author":"J Huo","year":"2015","journal-title":"Nonlinear Anal: Real World 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