State-feedback control for exponential asymptotic stability of a fractional-order predator–prey model
This paper investigates a fractional-order (FO) predator–prey (PP) model that incorporates a constant proportion of prey refuge, linear (constant-effort) harvesting of the prey, and a generalist predator whose survival is supported by alternative resources. The predator’s consumption is described by a ratiodependent functional response, and long-term memory effects are captured via the Caputo fractional derivative. All equilibria of the system (extinction, preyonly, predator-only, and interior) are explicitly identified, and their existence conditions are derived in terms of model parameters. To counter the destabilizing interplay of strong refuge, harvesting, and fractional memory, a novel state-feedback control strategy is designed using Lyapunov stability theory. We rigorously prove that, under a suitable symmetry condition between control weights, the proposed control functions corresponding to adaptive adjustments of prey harvesting and predator culling or supplementary feeding render the desired interior equilibrium exponentially asymptotically stable (EAS), with explicit exponential convergence rates. Numerical simulations confirm that while the uncontrolled system may drive the predator to extinction under extreme refuge (e.g., 99.995% prey protection), the proposed feedback controller successfully stabilises the prey-only boundary equilibrium, with the error variables and the Lyapunov function (LF) exhibiting strict exponential decay. Quantitative characterisation of the exponential decay rate provides a predictable timeline for ecological recovery. The findings highlight the role of memory in ecological dynamics and provide a biologically interpretable management framework for achieving robust coexistence in heavily perturbed ecosystems.
1. Lotka AJ. Analytical note on certain rhythmic relations in organic systems. Proc Natl Acad Sci USA. 1920;6(7):410-415. https://doi.org/10.1073/pnas.6.7.410
2. Sih A. Prey refuges and predator-prey stability. Theor Popul Biol. 1987;31(1):1-12. https://doi.org/10.1016/0040-5809(87)90019-0
3. McNair JN. The effects of refuges on predator-prey interactions: a reconsideration. Theor Popul Biol. 1986;29(1):38-63. https://doi.org/10.1016/0040-5809(86)90004-3
4. Clark CW. Mathematical models in the economics of renewable resources. SIAM Rev. 1979;21(1):81-99. https://doi.org/10.1137/1021006
5. Aguegboh NS, Onyiaji N, Okeke CA, Daniel NU, Walter O, Diallo B. Analysis of a fractional-order prey-predator model with prey refuge and predator harvest using the consumption number: Holling type III functional response. Comput Math Biophys. 2025;13(1):20250023. https://doi.org/10.1515/cmb-2025-0023
6. Afiyah SN, Akanni JO. Memory effects in eco-epidemiology: A dynamical systems approach to fear-disease-harvesting interactions. Stat Optim Inf Comput. 2026;15(1):683-710. https://doi.org/10.19139/soic-2310-5070-2720
7. Xu C, Balci E. Hunting cooperation and gestation delay in a prey-predator model with fractional derivative. J Appl Anal Comput. 2026;16:1035-1053. https://doi.org/10.11948/20250147
8. Rahman HH, Faraj BM. Memory effects and sustainable harvesting in a fractional-order predator-prey model with prey refuge and nonlinear harvesting. Desimal J Mat. 2025;8(2):173-184. https://doi.org/10.24042/94h4r762
9. Hammouch Z, Jamil MO, Unlu C. Dynamics investigation and numerical simulation of fractional-order predator-prey model with Holling type II functional response. Discrete Contin Dyn Syst Ser S. 2025;18(5):1230-1266. https://doi.org/10.3934/dcdss.2024181
10. Al-Betar MA, Braik MS, Shambour QY, Al-Naymat G, Porntaveetus T. Ameliorated elk herd optimizer for global optimization and engineering problems. Artif Intell Rev. 2025;58(11):360. https://doi.org/10.1007/s10462-025-11360-1
11. Braik M, Al-Betar MA, Mahdi MA, Al-Shalabi M, Ahamad S, Saad SA. Enhancement of satellite images based on CLAHE and augmented elk herd optimizer. Artif Intell Rev. 2025;58(2):38. https://doi.org/10.1007/s10462-024-11022-8
12. Hassell MP, May RM. Stability in insect host-parasite models. J Anim Ecol. 1973;42(3):693-726. https://doi.org/10.2307/3133
13. Jang SR, Yousef AM. Effects of prey refuge and predator cooperation on a predator-prey system. J Biol Dyn. 2023;17(1):2242372. https://doi.org/10.1080/17513758.2023.2242372
14. Jebril IH. Linear and nonlinear control for complete synchronization of fractional-order discrete reaction-diffusion systems. Int J Anal Appl. 2026;24:26. https://doi.org/10.28924/2291-8639-24-2026-26
15. Brauer F, Soudack AC. Stability regions and transition phenomena for harvested predator-prey systems. J Math Biol. 1979;7(3):319-337. https://doi.org/10.1007/BF00275152
16. Huxel GR, McCann K. Food web stability: the influence of the predator's dietary breadth. Am Nat. 1998;152(4):502-513. https://doi.org/10.1086/286182
17. Anber A, Dahmani Z. The Laplace decomposition method for solving nonlinear conformable fractional evolution equations. Int J Open Probl Comput Math. 2024;17(1):67-81. Accessed Jul 6, 2026. https://www.ijopcm.org/Vol/2024/2024.1.7.pdf
18. Arditi R, Ginzburg LR. Coupling in predator-prey dynamics: ratio-dependence. J Theor Biol. 1989;139(3):311-326. https://doi.org/10.1016/S0022-5193(89)80211-5
19. Arditi R, Akçakaya HR. Underestimation of mutual interference of predators. Oecologia. 1990;83(3):358-361. https://doi.org/10.1007/BF00317560
20. Sanchirico JN, Wilen JE. A bioeconomic model of marine reserve creation. J Environ Econ Manage. 2001;42(3):257-276. https://doi.org/10.1006/jeem.2000.1162
21. Lauck T, Clark CW, Mangel M, Munro GR. Implementing the precautionary principle in fisheries management through marine reserves. Ecol Appl. 1998;8(S1):S72-S78. https://doi.org/10.4324/9780429288500-12
22. El-Sayed AMA, El-Mesiry AEM, El-Saka HAA. On the fractional-order logistic equation. Appl Math Lett. 2007;20(7):817-823. https://doi.org/10.1016/j.aml.2006.08.013
23. Li HL, Zhang L, Hu C, Jiang YL. Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge. J Appl Math Comput. 2017;54(1-2):435-449. https://doi.org/10.1007/s12190-016-1017-8
24. Ahmed E, El-Sayed AMA, El-Saka HAA. Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models. J Math Anal Appl. 2007;325(1):542-553. https://doi.org/10.1016/j.jmaa.2006.01.087
25. Caputo M. Linear models of dissipation whose Q is almost frequency independent – II. Geophys J Int. 1967;13(5):529-539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x
26. Podlubny I. Geometric and physical interpretation of fractional integration and fractional differentiation. Fract Calc Appl Anal. 2002;5(4):367-386. https://doi.org/10.48550/arXiv.math/0110241
27. Jebril IH, Batiha I. A stable and convergent implicit finite difference scheme for variable-order time-fractional convection-diffusion equations. Int J Open Probl Comput Math. 2026;19(1):73-94. Accessed Jul 6, 2026. https://ijopcm.icsrs.uk/index.php/journal/article/view/54/45
28. Elsadany AA, Matouk AE. Dynamical behaviors of fractional-order Lotka-Volterra predator-prey model and its discretization. J Appl Math Comput. 2015;49(1-2):269-283. https://doi.org/10.1007/s12190-014-0838-6
29. Li CP, Zhang FR. A survey on the stability of fractional differential equations. Eur Phys J Spec Top. 2011;193(1):27-47. https://doi.org/10.1140/epjst/e2011-01379-1
30. Gómez-Aguilar JF. New bilingualism model based on fractional operators with Mittag-Leffler kernel. J Math Sociol. 2017;41(3):172-184. https://doi.org/10.1080/0022250X.2017.1356828
31. Ozair M, Hussain T, Aslam A, Anees R, Tanveer M, Gómez-Aguilar JF. Management of pine forests by assessment of insect pests and nematodes. Eur Phys J Plus. 2021;136(9):1-23. https://doi.org/10.1140/epjp/s13360-021-01934-7
32. Yang Z, Xiao M, Wang Z, Sun Y, Yang X, Gómez-Aguilar JF, Cao J. Higher-order interactions in the spatio-temporal dynamics of Leslie-Gower predator-prey systems. Comput Appl Math. 2026;45(8):344. https://doi.org/10.1007/s40314-026-03740-2
33. Chen L, Chen F. Global analysis of a harvested predator-prey model incorporating a constant prey refuge. Int J Bifurc Chaos. 2010;3(2):205-223. https://doi.org/10.1142/S1793524510000957
34. García CC. Impact of prey refuge in a discontinuous Leslie-Gower model with harvesting and alternative food for predators and linear functional response. Math Comput Simul. 2023;206:147-165. https://doi.org/10.1016/j.matcom.2022.11.013
35. Lv Y, Zhang Z, Yuan R, Pei Y. Effect of harvesting and prey refuge in a prey-predator system. J Biol Syst. 2014;22(1):133-150. https://doi.org/10.1142/S0218339014500089
36. Ilmiyah NN, Alghofari AR. Dynamical analysis of a harvested predator-prey model with ratio-dependent response function and prey refuge. Appl Math Sci. 2014;8(101-104):5027-5037. https://doi.org/10.12988/ams.2014.46424
37. Devi S. Nonconstant prey harvesting in ratio-dependent predator-prey system incorporating a constant prey refuge. Int J Biomath. 2012;5(2):1250021. https://doi.org/10.1142/S1793524511001635
38. Hakim L, Habibi AR. Dynamic behavior of predator-prey with ratio dependent, refuge in prey and harvest from predator. ZERO J Sains Mat Terap. 2019;3(1):23-32. https://doi.org/10.30829/zero.v3i1.5878
39. Yadav KS, Lata P, Kumari M, Jain S, Agarwal P, Gour MM. Qualitative stability and semi-analytical solutions of fractional-order hepatitis B model under Riemann-Liouville fractional derivative. Math Open. 2026;5:2650001. https://doi.org/10.1142/S281100722650001X
40. Ali I, Zhang H, Ali Shah SM, Alkhazzan A, Sabbar Y. An epidemiological stochastic predator-prey model with prey refuge and harvesting. Chin Phys B. 2026;35(2):020503. https://doi.org/10.1088/1674-1056/adf17f
41. Haque MM, Sarwardi S. Dynamics of a harvested prey-predator model with prey refuge dependent on both species. Int J Bifurc Chaos. 2018;28(12):1830040. https://doi.org/10.1142/S0218127418300409
42. Guin LN, Acharya S. Dynamic behaviour of a reaction-diffusion predator-prey model with both refuge and harvesting. Nonlinear Dyn. 2017;88:1501-1533. https://doi.org/10.1007/s11071-016-3326-8
43. Fan H, Wen H, Shi K, Xiao J. New fixed-time synchronization criteria for fractional-order fuzzy cellular neural networks with bounded uncertainties and transmission delays via multi-module control schemes. Fractal Fract. 2026;10(2):91. https://doi.org/10.3390/fractalfract10020091
44. Balci E. Varying capacity and harvesting in a prey-predator system with memory effect. J Adv Res Nat Appl Sci. 2025;11(1):12-26. https://doi.org/10.28979/jarnas.1611875
45. Adel AW, Günerhan H, Nisar KS, Agarwal P, El-Mesady A. Designing a novel fractional order mathematical model for COVID-19 incorporating lockdown measures. Sci Rep. 2024;14:2926. https://doi.org/10.1038/s41598-023-50889-5