Stability analysis of fractional-order predator-prey model with anti-predator behaviour and prey refuge

[1] S.J. Ali, N.M. Arifin, R.K. Naji, F. Ismail, N. Bachok, Analysis of ecological model with Holling
type IV functional response, Int. J. Pure Appl. Math. 106 (2016) 317–331.
[2] L. Chen, F. Chen, Qualitative analysis of a predator-prey model with Holling type II functional
response incorporating a constant prey refuge, Nonlinear Anal. Real World Appl. 11 (2010) 246–
252.
[3] F. Chen, L. Chen, X. Xie, On a Leslie-Gower predator-prey model incorporating a prey refuge,
Nonlinear Anal. Real World Appl. 10(5) (2009) 2905–2908.
[4] Y. Choh, M. Lgnacio, M.W. Sabelis, A. Janssen, Predator-prey role reversals, juvenile experience
and adult antipredator behaviour, Sci. Rep. 2(728) (2012) 1–6.
[5] K. Diethelm, N.J. Ford, A.D. Freed, A predictor-corrector approach for the numerical solution of
fractional differential equations, Nonlinear Dynam. 29 (2002) 3–22.
[6] E. Diz-Pita, J. Llibre, M. Victoria Otero-Espinar, Global phase portraits of a predator-prey system,
Electron. J. Qual. Theory Differ. Equ. (2022) 1–13.
[7] G. Gause, N. Smaragdova, A. Witt, Further studies of interaction between predators and prey, J.
Anim. Ecol. 5 (1936) 1–18.
[8] G. Gause, The Struggle for Existence, Williams Wilkins Co., Balitmore, MD, USA, 1934.
[9] J. Ghosh, B. Sahoo, S. Poria, Prey-predator dynamics with prey refuge providing additional food to
predator, Chaos Solitons Fractals 96 (2017) 110–119.
[10] H. Molla, S. Sarwardi, S.R. Smith, M. Haque, Dynamics of adding variable prey refuge and an
Allee effect to a predator-prey model, Alex. Eng. J. 61(6) (2022) 4175–4188.
[11] S. Jana, M. Chakraborty, K. Chakraborty, T. Kar, Global stability and bifurcation of time delayed
prey-predator system incorporating prey refuge, Math. Comput. Simul. 85 (2012) 57–77.
[12] T.K. Kar, Stability analysis of a predator-prey model incorporating a prey refuge, Commun. Non-
linear Sci. Numer. Simul. 10 (2006) 681–691.
[13] A. Kilbas, H. Srivastava, J. Trujillo, Theory and Application of Fractional Differential Equations,
Elsevier, New York, 2006.
[14] A. Kumar, B. Dubey, Modeling the effect of fear in a preypredator system with prey refuge and
gestation delay, Internat. J. Bifur. Chaos 29 (2019) 1950195.
[15] Y. Lan, J. Shi, H. Fang, Hopf bifurcation and control of a fractional-order delay stage structure
prey-predator model with two fear effects and prey refuge, Symmetry 14(7) (2022) 1408.
[16] A.J. Lotka, Elements of Physical Biology, Nature, 1925.

[17] W. Lu, Y. Xia, Y. Bai, Periodic solution of a stage-structured predator-prey model incorporating
prey refuge, Math. Biosci. Eng. 4 (2020) 3160–3174.
[18] S. Mondal, G.P. Samanta, Dynamics of a delayed predator-prey interaction incorporating nonlinear
prey refuge under the influence of fear effect and additional food, J. Phys. A 53 (2020) 1-46.
[19] I. Petras, Fractional-Order Nonlinear Systems: Modeling Analysis and Simulation, Higher Educa-
tion Press, Beijing, 2011.
[20] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[21] M. Sambath, P. Ramesh, K. Balachandran, Asymptotic behavior of the fractional order three species
prey-predator model, Int. J. Nonlinear Sci. Numer. Simul. 19 (2018) 721–733.
[22] S. Sharma, G.P. Samanta, A Leslie-Gower predator-prey model with disease in prey incorporating
a prey refuge, Chaos Solitons Fractals 70 (2015) 69–84.
[23] S. Sirisubtawee, N. Khansai, A. Charoenloedmongkhon, Investigation on dynamics of an impul-
sive predatorprey system with generalized Holling type IV functional response and anti-predator
behavior, Adv. Difference Equ. 160 (2021).
[24] S.G. Mortoja, P. Panja, S.K. Mondal, Dynamics of a predator-prey model with stage-structure on
both species and anti-predator behavior, Inform. Med. Unlocked 10 (2018) 50-57.
[25] G. Tang, W. Qin, Backward bifurcation of predator-prey model with anti-predator behaviors, Adv.
Difference Equ. 8 (2019).
[26] B. Tang, Y. Xiao, Bifurcation analysis of a predator-prey model with anti-predator behaviour,
Chaos Solitons Fractals, 70 (2015) 58-68.
[27] Z. Tian-Wei-Tian, Multiplicity of positive almost periodic solutions in a delayed HassellVarley-type
predatorprey model with harvesting on prey, Math. Methods Appl. Sci. 37 (2014) 686–697.
[28] J.P. Tripathi, S. Abbas, M. Thakur, Dynamical analysis of a prey-predator model with Beddington-
DeAngelis type function response incorporating a prey refuge, Nonlinear Dynam. 80 (2015) 177–
196.
[29] C. Vargas-De-Len, Volterra-type Lyapunov functions for fractional-order epidemic systems, Com-
mun. Nonlinear Sci. Numer. Simul. 24 (2015) 75-85.

[30] V. Volterra, Variazioni e fluttuazioni del numero dindividui in specie animali conviventi, Memoire
della R. Accademia Nazionale dei Lincei. 2 (1926) 31–113.
[31] Y. Xie, J. Lu, Z. Wang, Stability analysis of a fractional-order diffused prey-predator model with
prey refuges, Phys. A 526 (2019) 120773.
[32] Zaid M. Odibat, Nabil T. Shawagfeh, Generalized Taylors formula, Appl. Math. Comput. 186
(2007) 286-293.
[33] R. Zou, S. Guo, Dynamics of a Leslie-Gower predator-prey system with cross-diffusion, Electron.
J. Qual. Theory Differ. Equ. (2020) 1-33.