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Stability and bifurcation analysis of a population dynamic model with Allee effect via piecewise constant argument method

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Abstract

This work investigates the complex dynamics of a discrete-time predator–prey system with a nonlinear Allee effect. We obtain the discrete system using the piecewise constant argument method. The piecewise argument approach produces a more dynamically consistent discrete system than other numerical techniques for discretization. We investigate the presence and stability of fixed points. Furthermore, we have demonstrated that the system undergoes Neimark–Sacker bifurcation at the positive fixed point by utilizing the Allee effect constant as the bifurcation parameter. To reduce bifurcation and chaos, we use feedback and hybrid control strategies. Our numerical simulations demonstrate the importance of the Allee effect in determining the system’s behavior. The findings indicate that an adequate Allee effect might improve social connections and cooperation across populations. However, a significant Allee effect on prey can destabilize the positive fixed point, thus resulting in the extinction of predator and prey populations.

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References

  1. Lotka, A.J.: Elements of physical biology. Sci. Progr. Twentieth Century 1919–1933(21), 341–343 (1926)

    Google Scholar 

  2. Volterra, V.: Fluctuations in the abundance of a species considered mathematically. Nature 118, 558–560 (1926). https://doi.org/10.1038/118558a0

    Article  Google Scholar 

  3. Pal, S., Pal, N., Samanta, S., Chattopadhyay, J.: Effect of hunting cooperation and fear in a predator-prey model. Ecol. Complex. 39, 100770 (2019). https://doi.org/10.1016/j.ecocom.2019.100770

    Article  Google Scholar 

  4. Kumar, S., Kharbanda, H.: Chaotic behavior of predator-prey model with group defense and non-linear harvesting in prey. Chaos Solitons Fract. 119, 19–28 (2019). https://doi.org/10.1016/j.chaos.2018.12.011

    Article  MathSciNet  Google Scholar 

  5. Zhou, Y., Sun, W., Song, Y., Zheng, Z., Lu, J., Chen, S.: Hopf bifurcation analysis of a predator-prey model with Holling-II type functional response and a prey refuge. Nonlinear Dyn. 97, 1439–1450 (2019). https://doi.org/10.1007/s11071-019-05063-w

    Article  Google Scholar 

  6. Akhtar, S., Ahmed, R., Batool, M., Shah, N.A., Chung, J.D.: Stability, bifurcation and chaos control of a discretized Leslie prey-predator model. Chaos Solitons Fract. 152, 111345 (2021). https://doi.org/10.1016/j.chaos.2021.111345

    Article  MathSciNet  Google Scholar 

  7. Deng, H., Chen, F., Zhu, Z., Li, Z.: Dynamic behaviors of Lotka–Volterra predator-prey model incorporating predator cannibalism. Adv. Differ. Equ. 2019, 359 (2019). https://doi.org/10.1186/s13662-019-2289-8

    Article  MathSciNet  Google Scholar 

  8. Naik, P.A., Amer, M., Ahmed, R., Qureshi, S., Huang, Z.: Stability and bifurcation analysis of a discrete predator-prey system of ricker type with refuge effect. Math. Biosci. Eng. 21(3), 4554–4586 (2024). https://doi.org/10.3934/mbe.2024201

    Article  MathSciNet  Google Scholar 

  9. Owolabi, K.M., Pindza, E., Karaagac, B., Oguz, G.: Laplace transform-homotopy perturbation method for fractional time diffusive predator-prey models in ecology. Partial Differ. Equ. Appl. Math. 9, 100607 (2024). https://doi.org/10.1016/j.padiff.2023.100607

    Article  Google Scholar 

  10. Mahapatra, G.S., Santra, P.K., Bonyah, E.: Dynamics on effect of prey refuge proportional to predator in discrete-time prey-predator model. Complexity 2021, 6209908 (2021). https://doi.org/10.1155/2021/6209908

    Article  Google Scholar 

  11. Owolabi, K.M., Jain, S.: Spatial patterns through diffusion-driven instability in modified predator-prey models with chaotic behaviors. Chaos Solitons Fract. 174, 113839 (2023). https://doi.org/10.1016/j.chaos.2023.113839

    Article  MathSciNet  Google Scholar 

  12. Mukherjee, M., Pal, D., Mahato, S., Bonyah, E.: Prey-predator optimal harvesting mathematical model in the presence of toxic prey under interval uncertainty. Sci. Afr. 21, e01837 (2023). https://doi.org/10.1016/j.sciaf.2023.e01837

    Article  Google Scholar 

  13. Stephens, P.A., Sutherland, W.J., Freckleton, R.P.: What is the Allee effect? Oikos 87(1), 185–190 (1999). https://doi.org/10.2307/3547011

    Article  Google Scholar 

  14. Stephens, P.A., Sutherland, W.J.: Consequences of the Allee effect for behaviour, ecology and conservation. Trends Ecol. Evol. 14(10), 401–405 (1999). https://doi.org/10.1016/s0169-5347(99)01684-5

    Article  Google Scholar 

  15. Dennis, B.: Allee effects: population growth, critical density, and the chance of extinction. Nat. Resour. Model. 3(4), 481–538 (1989). https://doi.org/10.1111/j.1939-7445.1989.tb00119.x

    Article  MathSciNet  Google Scholar 

  16. Kramer, A.M., Dennis, B., Liebhold, A.M., Drake, J.M.: The evidence for Allee effects. Popul. Ecol. 51(3), 341–354 (2009). https://doi.org/10.1007/s10144-009-0152-6

    Article  Google Scholar 

  17. Courchamp, F., Berec, L., Gascoigne, J.: Allee Effects in Ecology and Conservation. Oxford University Press, Oxford (2008). https://doi.org/10.1093/acprof:oso/9780198570301.001.0001

    Book  Google Scholar 

  18. Allee, W.C.: Animal Aggregations, A Study in General Sociology. The University of Chicago Press, Chicago (1931). https://doi.org/10.5962/bhl.title.7313

    Book  Google Scholar 

  19. Vinoth, S., Sivasamy, R., Sathiyanathan, K., Unyong, B., Rajchakit, G., Vadivel, R., Gunasekaran, N.: The dynamics of a Leslie type predator-prey model with fear and Allee effect. Adv. Differ. Equ. 2021, 338 (2021). https://doi.org/10.1186/s13662-021-03490-x

    Article  MathSciNet  Google Scholar 

  20. Du, Y., Niu, B., Wei, J.: Dynamics in a predator-prey model with cooperative hunting and Allee effect. Mathematics 9(24), 3193 (2021). https://doi.org/10.3390/math9243193

    Article  Google Scholar 

  21. Shang, Z., Qiao, Y.: Bifurcation analysis of a Leslie-type predator-prey system with simplified Holling type IV functional response and strong Allee effect on prey. Nonlinear Anal. Real World Appl. 64, 103453 (2022). https://doi.org/10.1016/j.nonrwa.2021.103453

    Article  MathSciNet  Google Scholar 

  22. Fang, K., Zhu, Z., Chen, F., Li, Z.: Qualitative and bifurcation analysis in a Leslie–Gower model with Allee effect. Qual. Theory Dyn. Syst. 21, 86 (2022). https://doi.org/10.1007/s12346-022-00591-0

    Article  MathSciNet  Google Scholar 

  23. Ahmed, R., Akhtar, S., Farooq, U., Ali, S.: Stability, bifurcation, and chaos control of predator-prey system with additive Allee effect. Commun. Math. Biol. Neurosci. (2023). https://doi.org/10.28919/cmbn/7824

  24. Isik, S.: A study of stability and bifurcation analysis in discrete-time predator-prey system involving the Allee effect. Int. J. Biomath. 12(1), 1950011 (2019). https://doi.org/10.1142/s1793524519500116

    Article  MathSciNet  Google Scholar 

  25. Zhao, M., Du, Y.: Stability and bifurcation analysis of an amensalism system with Allee effect. Adv. Differ. Equ. 2020, 341 (2020). https://doi.org/10.1186/s13662-020-02804-9

    Article  MathSciNet  Google Scholar 

  26. Cai, J., Pinto, M., Xia, Y.: Stability and bifurcation analysis of a commensal model with Allee effect and herd behavior. Int. J. Bifurc. Chaos 32(14), 2250217 (2022). https://doi.org/10.1142/s0218127422502170

    Article  MathSciNet  Google Scholar 

  27. Hamada, M.Y.: El-Azab, T., El-Metwally, H.: Allee effect in a Ricker type predator-prey model. J. Math. Comput. Sci. 29(3), 239–251 (2022). https://doi.org/10.22436/jmcs.029.03.03

  28. Biswas, S., Pal, D., Santra, P.K., Bonyah, E., Mahapatra, G.S.: Dynamics of a three-patch prey-predator system with the impact of dispersal speed incorporating strong Allee effect on double prey. Discret. Dyn. Nat. Soc. 2022, 7919952 (2022). https://doi.org/10.1155/2022/7919952

    Article  Google Scholar 

  29. Ma, Y., Zhao, M., Du, Y.: Impact of the strong Allee effect in a predator-prey model. AIMS Math. 7(9), 16296–16314 (2022). https://doi.org/10.3934/math.2022890

    Article  MathSciNet  Google Scholar 

  30. Khabyah, A.A., Ahmed, R., Akram, M.S., Akhtar, S.: Stability, bifurcation, and chaos control in a discrete predator-prey model with strong Allee effect. AIMS Math. 8(4), 8060–8081 (2023). https://doi.org/10.3934/math.2023408

    Article  MathSciNet  Google Scholar 

  31. Liu, W., Cai, D.: Bifurcation, chaos analysis and control in a discrete-time predator-prey system. Adv. Differ. Equ. 2019, 11 (2019). https://doi.org/10.1186/s13662-019-1950-6

    Article  MathSciNet  Google Scholar 

  32. Li, Y., Zhang, F., Zhuo, X.: Flip bifurcation of a discrete predator-prey model with modified Leslie–Gower and Holling-type III schemes. Math. Biosci. Eng. 17(3), 2003–2015 (2020). https://doi.org/10.3934/mbe.2020106

    Article  MathSciNet  Google Scholar 

  33. Ghosh, B.: Multistability, chaos and mean population density in a discrete-time predator-prey system. Chaos Solitons Fract. 162, 112497 (2022). https://doi.org/10.1016/j.chaos.2022.112497

    Article  MathSciNet  Google Scholar 

  34. Yousef, A., Algelany, A.M., Elsadany, A.: Codimension one and codimension two bifurcations in a discrete Kolmogorov-type predator-prey model. J. Comput. Appl. Math. 428, 115171 (2023). https://doi.org/10.1016/j.cam.2023.115171

    Article  MathSciNet  Google Scholar 

  35. Khan, A.Q., Alsulami, I.M.: Complicate dynamical analysis of a discrete predator-prey model with a prey refuge. AIMS Math. 8(7), 15035–15057 (2023). https://doi.org/10.3934/math.2023768

    Article  MathSciNet  Google Scholar 

  36. Yavuz, M., Sene, N.: Complex dynamics of a predator-prey system with Gompertz growth and herd behavior. Fractal Fract. 4(3), 35 (2020). https://doi.org/10.3390/fractalfract4030035

    Article  Google Scholar 

  37. Lin, Q.: Allee effect increasing the final density of the species subject to the Allee effect in a Lotka–Volterra commensal symbiosis model. Adv. Differ. Equ. 2018, 196 (2018). https://doi.org/10.1186/s13662-018-1646-3

    Article  MathSciNet  Google Scholar 

  38. Khan, A.Q., Ahmad, I., Alayachi, H.S., Noorani, M.S.M., Khaliq, A.: Discrete-time predator-prey model with flip bifurcation and chaos control. Math. Biosci. Eng. 17(5), 5944–5960 (2020). https://doi.org/10.3934/mbe.2020317

    Article  MathSciNet  Google Scholar 

  39. AlSharawi, Z., Pal, S., Pal, N., Chattopadhyay, J.: A discrete-time model with non-monotonic functional response and strong Allee effect in prey. J. Differ. Equ. Appl. 26(3), 404–431 (2020). https://doi.org/10.1080/10236198.2020.1739276

    Article  MathSciNet  Google Scholar 

  40. Ahmed, R., Ahmad, A., Ali, N.: Stability analysis and Neimark–Sacker bifurcation of a nonstandard finite difference scheme for Lotka–Volterra prey-predator model. Commun. Math. Biol. Neurosci. 2022, 61 (2022). https://doi.org/10.28919/cmbn/7534

    Article  Google Scholar 

  41. Kangalgil, F.: Neimark-Sacker bifurcation and stability analysis of a discrete-time prey-predator model with Allee effect in prey. Adv. Differ. Equ. 2019, 92 (2019). https://doi.org/10.1186/s13662-019-2039-y

    Article  MathSciNet  Google Scholar 

  42. Murakami, K.: Stability and bifurcation in a discrete-time predator–prey model. J. Differ. Equ. Appl. 13(10), 911–925 (2007). https://doi.org/10.1080/10236190701365888

    Article  MathSciNet  Google Scholar 

  43. Suleman, A., Khan, A.Q., Ahmed, R.: Bifurcation analysis of a discrete Leslie–Gower predator-prey model with slow-fast effect on predator. Math. Methods Appl. Sci. (2024). https://doi.org/10.1002/mma.10032

    Article  Google Scholar 

  44. Ahmed, R., Tahir, N., Shah, N.A.: An analysis of the stability and bifurcation of a discrete-time predator-prey model with the slow-fast effect on the predator. Chaos Interdiscipl. J. Nonlinear Sci. 34, 033127 (2024). https://doi.org/10.1063/5.0185809

    Article  MathSciNet  Google Scholar 

  45. Naik, P.A., Eskandari, Z., Yavuz, M., Zu, J.: Complex dynamics of a discrete-time Bazykin–Berezovskaya prey-predator model with a strong Allee effect. J. Comput. Appl. Math. 413, 114401 (2022). https://doi.org/10.1016/j.cam.2022.114401

    Article  MathSciNet  Google Scholar 

  46. Rana, S.M.S.: Dynamics and chaos control in a discrete-time ratio-dependent Holling–Tanner model. J. Egyptian Math. Soc. 27, 48 (2019). https://doi.org/10.1186/s42787-019-0055-4

    Article  MathSciNet  Google Scholar 

  47. Baydemir, P., Merdan, H., Karaoglu, E., Sucu, G.: Complex dynamics of a discrete-time prey-predator system with Leslie type: stability, bifurcation analyses and chaos. Int. J. Bifurc. Chaos 30(10), 2050149 (2020). https://doi.org/10.1142/s0218127420501497

    Article  MathSciNet  Google Scholar 

  48. Zhao, M., Li, C., Wang, J.: Complex dynamic behaviors of a discrete-time predator-prey system. J. Appl. Anal. Comput. 7(2), 478–500 (2017). https://doi.org/10.11948/2017030

    Article  MathSciNet  Google Scholar 

  49. Naik, P.A., Eskandari, Z., Avazzadeh, Z., Zu, J.: Multiple bifurcations of a discrete-time prey-predator model with mixed functional response. Int. J. Bifurc. Chaos 32(4), 2250050 (2022). https://doi.org/10.1142/s021812742250050x

    Article  MathSciNet  Google Scholar 

  50. Eskandari, Z., Avazzadeh, Z., Ghaziani, R.K., Li, B.: Dynamics and bifurcations of a discrete-time Lotka–Volterra model using nonstandard finite difference discretization method. Math. Methods Appl. Sci. (2022). https://doi.org/10.1002/mma.8859

    Article  Google Scholar 

  51. Naik, P.A., Eskandari, Z., Madzvamuse, A., Avazzadeh, Z., Zu, J.: Complex dynamics of a discrete-time seasonally forced SIR epidemic model. Math. Methods Appl. Sci. 46(6), 7045–7059 (2023). https://doi.org/10.1002/mma.8955

    Article  MathSciNet  Google Scholar 

  52. Jing, Z., Yang, J.: Bifurcation and chaos in discrete-time predator–prey system. Chaos Solitons Fract. 27(1), 259–277 (2006). https://doi.org/10.1016/j.chaos.2005.03.040

    Article  MathSciNet  Google Scholar 

  53. Din, Q., Haider, K.: Discretization, bifurcation analysis and chaos control for Schnakenberg model. J. Math. Chem. 58, 1615–1649 (2020). https://doi.org/10.1007/s10910-020-01154-x

    Article  MathSciNet  Google Scholar 

  54. Gao, M., Chen, L., Chen, F.: Dynamical analysis of a discrete two-patch model with the Allee effect and nonlinear dispersal. Math. Biosci. Eng. 21(4), 5499–5520 (2024). https://doi.org/10.3934/mbe.2024242

    Article  MathSciNet  Google Scholar 

  55. Sharma, V.S., Singh, A., Elsonbaty, A., Elsadany, A.A.: Codimension-one and -two bifurcation analysis of a discrete-time prey-predator model. Int. J. Dyn. Control. 11, 2691–2705 (2023). https://doi.org/10.1007/s40435-023-01177-7

    Article  MathSciNet  Google Scholar 

  56. Shabbir, M.S., Din, Q., Safeer, M., Khan, M.A., Ahmad, K.: A dynamically consistent nonstandard finite difference scheme for a predator-prey model. Adv. Differ. Equ. 2019, 381 (2019). https://doi.org/10.1186/s13662-019-2319-6

    Article  MathSciNet  Google Scholar 

  57. Moghadas, S., Alexander, M., Corbett, B.: A non-standard numerical scheme for a generalized Gause-type predator-prey model. Physica D 188, 134–151 (2004). https://doi.org/10.1016/s0167-2789(03)00285-9

    Article  MathSciNet  Google Scholar 

  58. Tassaddiq, A., Shabbir, M.S., Din, Q., Naaz, H.: Discretization, bifurcation, and control for a class of predator-prey interactions. Fract. Fract. 6(1), 31 (2022). https://doi.org/10.3390/fractalfract6010031

    Article  Google Scholar 

  59. Zhou, Q., Chen, F., Lin, S.: Complex dynamics analysis of a discrete amensalism system with a cover for the first species. Axioms 11(8), 365 (2022). https://doi.org/10.3390/axioms11080365

    Article  Google Scholar 

  60. Mukherjee, D.: Global stability and bifurcation analysis in a discrete-time two prey one predator model with help. Int. J. Model. Simul. 43(5), 752–763 (2023). https://doi.org/10.1080/02286203.2022.2121676

    Article  Google Scholar 

  61. Lin, S., Chen, F., Li, Z., Chen, L.: Complex dynamic behaviors of a modified discrete Leslie–Gower predator-prey system with fear effect on prey species. Axioms 11(10), 520 (2022). https://doi.org/10.3390/axioms11100520

    Article  Google Scholar 

  62. Ahmed, R., Rafaqat, M., Siddique, I., Arefin, M.A.: Complex dynamics and chaos control of a discrete-time predator–prey model. Discret. Dyn. Nat. Soc. 2023, 8873611 (2023). https://doi.org/10.1155/2023/8873611

    Article  Google Scholar 

  63. Luo, A.C.J.: Regularity and Complexity in Dynamical Systems. Springer, Cham (2012). https://doi.org/10.1007/978-1-4614-1524-4

    Book  Google Scholar 

  64. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 42. Springer, New York (1983). https://doi.org/10.1007/978-1-4612-1140-2

    Book  Google Scholar 

  65. Wiggins, S., Golubitsky, M.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, vol. 2. Springer-Verlag, Cham (2003). https://doi.org/10.1007/b97481

    Book  Google Scholar 

  66. Chen, G., Dong, X.: From Chaos to Order: Methodologies, Perspectives and Applications. World Scientific, Singapore (1998). https://doi.org/10.1142/3033

    Book  Google Scholar 

  67. Lei, C., Han, X., Wang, W.: Bifurcation analysis and chaos control of a discrete-time prey-predator model with fear factor. Math. Biosci. Eng. 19(7), 6659–6679 (2022). https://doi.org/10.3934/mbe.2022313

    Article  MathSciNet  Google Scholar 

  68. Luo, X.S., Chen, G., Wang, B.H., Fang, J.Q.: Hybrid control of period-doubling bifurcation and chaos in discrete nonlinear dynamical systems. Chaos Solitons Fract. 18(4), 775–783 (2003). https://doi.org/10.1016/s0960-0779(03)00028-6

    Article  Google Scholar 

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Acknowledgements

This study was supported by the Scientific Research Project for High-Level Talents of Youjiang Medical University for Nationalities, Baise, Guangxi, China under Grant number yy2023rcky002.

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Correspondence to Parvaiz Ahmad Naik or Zohreh Eskandari.

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Appendices

Appendix A

$$\begin{aligned} a_{11}&=\biggl (\epsilon ^2 \eta ^6 \sigma ^8+\epsilon \eta ^5 \sigma ^6 ((-1+2 \epsilon ) \sigma ^2-2 \epsilon \omega -6 \epsilon \sigma \omega )+\eta ^4 \sigma ^4 ((-1-4 \epsilon +\epsilon ^2) \sigma ^4\\&\quad +5 (1-2 \epsilon ) \epsilon \sigma ^3 \omega +\epsilon ^2 \omega ^2 +10 \epsilon ^2 \sigma \omega ^2+\epsilon \sigma ^2 \omega (4-6 \epsilon +15 \epsilon \omega ))\\&\quad -\eta ^3 \sigma ^3 ((3+5 \epsilon ) \sigma ^5+4 (-1-4 \epsilon +\epsilon ^2) \sigma ^4 \omega +4 \epsilon ^2 \omega ^3+4 \epsilon \sigma ^2 \omega ^2 (4 +\epsilon (-6+5 \omega ))\\&\quad +\epsilon \sigma \omega ^2 (3+4 \epsilon (-1+5 \omega ))-\sigma ^3 \omega (1-2 \epsilon (-7+5 \omega )+4 \epsilon ^2 (-1+5 \omega )))\\&\quad +\omega (\sigma ^7+\epsilon \sigma (3 +2 \epsilon (-2+\omega )) \omega ^4+\epsilon ^2 \omega ^5+\sigma ^6 (2-(3+2 \epsilon ) \omega )\\&\quad +\sigma ^5 \omega (-5+3 \omega +\epsilon (-8+5 \omega ))-\sigma ^3 \omega ^2 (2+\omega +2 \epsilon ^2 (-2+\omega ) \omega \\&\quad -\epsilon (16-14 \omega +\omega ^2))+\sigma ^2 \omega ^3 (1+4 \epsilon (-3+\omega )+\epsilon ^2 (4-6 \omega +\omega ^2))\\&\quad +\sigma ^4 \omega (1+4\omega -\omega ^2+\epsilon ^2 \omega ^2-2 \epsilon (4-9 \omega +2 \omega ^2)))-\eta \sigma (\sigma ^7-2 (3+2 \epsilon ) \sigma ^6 \omega +4 \epsilon ^2 \omega ^5 \\&\quad +4 \sigma ^4 \omega ^2(2+\epsilon (9-4 \omega )-\omega +\epsilon ^2 \omega )+\epsilon \sigma \omega ^4 (9+2 \epsilon (-6+5 \omega ))+\sigma ^5 \omega \\&\quad (-5+9 \omega +\epsilon (-8+15 \omega )) +2 \sigma ^2 \omega ^3 (1+4 \epsilon (-3+2 \omega ) +\epsilon ^2 (4-12 \omega +3 \omega ^2))+\sigma ^3 \omega ^2\\&\quad (-2-3 \omega -2 \epsilon ^2 \omega (-6+5 \omega )+\epsilon (16-42 \omega +5 \omega ^2)))+\eta ^2 \sigma ^2 (-(3+2 \epsilon ) \sigma ^6 \\&\quad +3 (3+5 \epsilon ) \sigma ^5 \omega +6 \epsilon ^2 \omega ^4+2 \sigma ^4 \omega (2+\epsilon (9-12 \omega )-3 \omega +3 \epsilon ^2 \omega )+\epsilon \sigma \omega ^3\\&\quad (9+4 \epsilon (-3+5 \omega ))+\sigma ^3 \omega ^2 (-3 +2 \epsilon (-21+5 \omega )-4 \epsilon ^2 (-3+5 \omega ))\\&\quad +\sigma ^2 \omega ^2 (1+12 \epsilon (-1+2 \omega )+\epsilon ^2 (4-36 \omega +15 \omega ^2)))\biggr )\bigg /\\&\quad \biggl ((\sigma (\sigma +\eta \sigma -\omega ) \omega +\epsilon (\eta ^3 \sigma ^4+\eta ^2 \sigma ^2 (\sigma ^2-\omega -3 \sigma \omega )+\omega ^2 (2 \sigma +\sigma ^2-\omega -\sigma \omega )\\&\quad +\eta \sigma \omega (-2 \sigma -2 \sigma ^2+2 \omega +3 \sigma \omega )))^2\biggr ), \\ a_{1}&=-\frac{ \sigma }{2 k^2 (\eta \sigma -\omega )}, \ a_{2}=\frac{ \sigma }{6 k^3 \eta \sigma -6 k^3 \omega }, \\ a_{3}&=\bigg ((1+(1-2 \beta -2 \epsilon ) \eta -3 (\beta +\epsilon ) \eta ^2+(\beta +\epsilon )^2 \eta ^3) \sigma ^3-(1-2 \epsilon -6 \epsilon \eta +3 \beta ^2 \eta ^2+3 \epsilon ^2 \eta ^2\\&+\beta (-2-6 \eta +6 \epsilon \eta ^2)) \sigma ^2 \omega +3 (\beta +\epsilon ) (-1+\beta \eta +\epsilon \eta ) \sigma \omega ^2-(\beta +\epsilon )^2 \omega ^3)\bigg )\bigg /\\&\quad \bigg (2 k^2 (\eta \sigma -\omega ) ((-1+\beta \eta +\epsilon \eta ) \sigma -(\beta +\epsilon ) \omega )^2\bigg ),\\ a_{4}&= \bigg (-(1+(1-2 \beta -2 \epsilon ) \eta -3 (\beta +\epsilon ) \eta ^2+(\beta +\epsilon )^2 \eta ^3) \sigma ^3+(1-2 \epsilon -6 \epsilon \eta \\&\quad +3 \beta ^2 \eta ^2+3 \epsilon ^2 \eta ^2+\beta (-2-6 \eta +6 \epsilon \eta ^2)) \sigma ^2 \omega \\&\quad -3 (\beta +\epsilon ) (-1+\beta \eta +\epsilon \eta ) \sigma \omega ^2+(\beta +\epsilon )^2 \omega ^3)\bigg )\bigg /\\&\quad \bigg (k (\eta \sigma -\omega ) ((-1+\beta \eta +\epsilon \eta ) \sigma -(\beta +\epsilon ) \omega )^2\bigg ), \\ a_{5}&=\bigg ((-1+(-2+4 \beta +4 \epsilon ) \eta -2 (2 \beta ^2+\epsilon (-5+2 \epsilon )+\beta (-5+4 \epsilon )) \eta ^2\\&\quad -16 (\beta +\epsilon )^2 \eta ^3+3 (\beta +\epsilon )^2 (-1+2 \beta +2 \epsilon ) \eta ^4+2 (\beta +\epsilon )^3 \eta ^5) \sigma ^5\\&\quad -2 (-1+2 \epsilon (1+5 \eta )+\beta ^3 \eta ^3 (12+5 \eta )+\epsilon ^3 \eta ^3 (12+5 \eta )-2 \epsilon ^2 \eta (2+12 \eta +3 \eta ^2)\\&\quad +\beta ^2 \eta (-4-24 \eta +6 (-1+6 \epsilon ) \eta ^2+15 \epsilon \eta ^3)+\beta (2+(10-8 \epsilon ) \eta -48 \epsilon \eta ^2+12 \epsilon (-1+3 \epsilon ) \eta ^3\\&\quad +15 \epsilon ^2 \eta ^4)) \sigma ^4 \omega +2 (2 \beta ^3 \eta ^2 (9+5 \eta )+\beta ^2 (-2-24 \eta +9 (-1+6 \epsilon ) \eta ^2+30 \epsilon \eta ^3)\\&\quad +\beta (5+6 \epsilon ^2 \eta ^2 (9+5 \eta ) -2 \epsilon (2+24 \eta +9 \eta ^2))+\epsilon (5+2 \epsilon ^2 \eta ^2 (9+5 \eta )\\&\quad -\epsilon (2+24 \eta +9 \eta ^2))) \sigma ^3 \omega ^2-4 (\beta +\epsilon )^2 (-4+(-3+6 \beta +6 \epsilon ) \eta +5 (\beta +\epsilon ) \eta ^2) \sigma ^2 \omega ^3\\&\quad +(\beta +\epsilon )^2 (-3+2 \beta (3+5 \eta )+2 \epsilon (3+5 \eta )) \sigma \omega ^4-2 (\beta +\epsilon )^3 \omega ^5)\bigg )\bigg /\\&\bigg (2 (\eta \sigma -\omega ) ((-1+\beta \eta +\epsilon \eta ) \sigma -(\beta +\epsilon ) \omega )^4\bigg ),\\ a_{6}&=\bigg ((1+(2-4 \beta -4 \epsilon ) \eta +2 (2 \beta ^2+\epsilon (-5+2 \epsilon )+\beta (-5+4 \epsilon )) \eta ^2+16 (\beta +\epsilon )^2 \eta ^3\\&\quad -3 (\beta +\epsilon )^2 (-1+2 \beta +2 \epsilon ) \eta ^4-2 (\beta +\epsilon )^3\eta ^5) \sigma ^5+2 (-1+2 \epsilon (1+5 \eta )+\beta ^3 \eta ^3 (12+5 \eta )\\&\quad +\epsilon ^3 \eta ^3 (12+5 \eta )-2 \epsilon ^2 \eta (2+12 \eta +3 \eta ^2)+\beta ^2 \eta (-4-24 \eta +6 (-1+6 \epsilon ) \eta ^2+15 \epsilon \eta ^3)\\&\quad +\beta (2+(10-8 \epsilon ) \eta -48 \epsilon \eta ^2+12 \epsilon (-1+3 \epsilon ) \eta ^3+15 \epsilon ^2 \eta ^4)) \sigma ^4 \omega -2 (2 \beta ^3 \eta ^2 (9+5 \eta )\\&\quad +\beta ^2 (-2-24 \eta +9 (-1+6 \epsilon ) \eta ^2+30 \epsilon \eta ^3)+\beta (5+6 \epsilon ^2 \eta ^2 (9+5 \eta )-2 \epsilon (2+24 \eta +9 \eta ^2))\\&\quad +\epsilon (5+2 \epsilon ^2 \eta ^2 (9+5 \eta ) -\epsilon (2+24 \eta +9 \eta ^2))) \sigma ^3 \omega ^2+4 (\beta +\epsilon )^2 (-4+(-3+6 \beta +6 \epsilon ) \eta \\&\quad +5 (\beta +\epsilon ) \eta ^2) \sigma ^2 \omega ^3-(\beta +\epsilon )^2 (-3 +2 \beta (3+5 \eta )+2 \epsilon (3+5 \eta )) \sigma \omega ^4\\&+2 (\beta +\epsilon )^3 \omega ^5)\bigg )\\&\bigg /\bigg (2 k (\eta \sigma -\omega ) ((-1+\beta \eta +\epsilon \eta ) \sigma -(\beta +\epsilon ) \omega )^4\bigg ), \\ a_{7}&=-\frac{1}{6 (\beta +\epsilon -\frac{\sigma }{\eta \sigma -\omega })^6}(3 (\beta +\epsilon -\frac{\sigma }{\eta \sigma -\omega })^4+6 (\beta +\epsilon -\frac{\sigma }{\eta \sigma -\omega })^5+\frac{\sigma ^7}{(\eta \sigma -\omega )^7} \\&+\frac{3 \sigma ^5}{(\eta \sigma -\omega )^5}+\frac{9 \sigma ^5 (\beta +\epsilon -\frac{\sigma }{\eta \sigma -\omega })}{(\eta \sigma -\omega )^5} +\frac{12 \sigma ^5 (\beta +\epsilon -\frac{\sigma }{\eta \sigma -\omega })^2}{(\eta \sigma -\omega )^5}-\frac{3 \sigma ^3 (\beta +\epsilon -\frac{\sigma }{\eta \sigma -\omega })}{(\eta \sigma -\omega )^3} \\&-\frac{21 \sigma ^3 (\beta +\epsilon -\frac{\sigma }{\eta \sigma -\omega })^2}{(\eta \sigma -\omega )^3}-\frac{30 \sigma ^3 (\beta +\epsilon -\frac{\sigma }{\eta \sigma -\omega })^3}{(\eta \sigma -\omega )^3}+\frac{3 \sigma ^6}{(-\eta \sigma +\omega )^6}+\frac{6 \sigma ^6 (\beta +\epsilon -\frac{\sigma }{\eta \sigma -\omega })}{(-\eta \sigma +\omega )^6} \\&+\frac{\sigma ^4}{(-\eta \sigma +\omega )^4}-\frac{3 \sigma ^4 (\beta +\epsilon -\frac{\sigma }{\eta \sigma -\omega })^2}{(-\eta \sigma +\omega )^4}+\frac{8 \sigma ^4 (\beta +\epsilon -\frac{\sigma }{\eta \sigma -\omega })^3}{(-\eta \sigma +\omega )^4}-\frac{6 \sigma ^2 (\beta +\epsilon -\frac{\sigma }{\eta \sigma -\omega })^2}{(-\eta \sigma +\omega )^2}\\&-\frac{24 \sigma ^2 (\beta +\epsilon -\frac{\sigma }{\eta \sigma -\omega })^3}{(-\eta \sigma +\omega )^2}-\frac{24 \sigma ^2 (\beta +\epsilon -\frac{\sigma }{\eta \sigma -\omega })^4}{(-\eta \sigma +\omega )^2}+\frac{\sigma ((-1+\beta \eta +\epsilon \eta ) \sigma -(\beta +\epsilon ) \omega )^3}{(-\eta \sigma +\omega )^4}, \\ b_{1}&=\frac{(-\eta \sigma +\omega )^2}{\omega },\ b_{2}=\frac{k \sigma (\sigma +\eta \sigma -\omega ) (\eta (2+\sigma )-\omega ) (-\eta \sigma +\omega )^2}{2 \omega ^2 (-(-1+\beta \eta +\epsilon \eta ) \sigma +(\beta +\epsilon ) \omega )},\\ b_{3}&=\frac{(\eta \sigma -\omega )^3 (\eta (2+\sigma )-\omega )}{2 \omega ^2},\\ b_{4}&=\frac{k \sigma (\eta \sigma -\omega )^3 (\sigma +\eta \sigma -\omega ) (\eta ^2 (6+6 \sigma +\sigma ^2)-2 \eta (3+\sigma ) \omega +\omega ^2)}{6 \omega ^3 (-(-1+\beta \eta +\epsilon \eta ) \sigma +(\beta +\epsilon ) \omega )}. \end{aligned}$$

Appendix B

$$\begin{aligned} p(\epsilon )&=-\bigg ((-2 \epsilon ^2 \eta ^6 \sigma ^8+\epsilon \eta ^5 \sigma ^6 ((1-4 \epsilon ) \sigma ^2+4 \epsilon \omega +12 \epsilon \sigma \omega )+\eta ^4 \sigma ^4 ((1+4 \epsilon -2 \epsilon ^2) \sigma ^4\\&+5 \epsilon (-1+4 \epsilon ) \sigma ^3 \omega -2 \epsilon ^2 \omega ^2-20 \epsilon ^2 \sigma \omega ^2-6 \epsilon \sigma ^2 \omega (1-2 \epsilon +5 \epsilon \omega ))+\eta ^3 \sigma ^3 ((3+5\epsilon )\\&\sigma ^5+4 (-1-4 \epsilon +2 \epsilon ^2) \sigma ^4 \omega +8 \epsilon ^2 \omega ^3+8 \epsilon \sigma ^2 \omega ^2 (3+\epsilon (-6+5 \omega ))+\epsilon \sigma \omega ^2 (5+8 \epsilon (-1+5 \omega )) \\&-\sigma ^3 \omega (1-2 \epsilon (-9+5 \omega )+8 \epsilon ^2 (-1+5\omega )))-\omega (\sigma ^7+\epsilon \sigma (5+4 \epsilon (-2+\omega )) \omega ^4\\&+2 \epsilon ^2 \omega ^5+\sigma ^6 (2-(3+2\epsilon ) \omega )+\sigma ^5 \omega (-5+3 \omega +\epsilon (-8+5 \omega ))-\sigma ^3 \omega ^2 (4+\omega +4 \epsilon ^2(-2+\omega )\\&\omega -\epsilon (20-18 \omega +\omega ^2))+2 \sigma ^2 \omega ^3 (1+3 \epsilon (-3+\omega )+\epsilon ^2 (4-6 \omega +\omega ^2))\\&+\sigma ^4 \omega (2+4 \omega -\omega ^2+2 \epsilon ^2 \omega ^2-4 \epsilon (2-5 \omega +\omega ^2)))+\eta \sigma (\sigma ^7-2 (3+2 \epsilon ) \sigma ^6 \omega +8 \epsilon ^2 \omega ^5\\&+4 \sigma ^4 \omega ^2 (2+\epsilon (10-4 \omega )-\omega +2 \epsilon ^2 \omega )+\epsilon \sigma \omega ^4 (15+4 \epsilon (-6+5 \omega ))+\sigma ^5 \omega (-5+9 \omega \\&+\epsilon (-8+15 \omega )) +4 \sigma ^2 \omega ^3 (1+\epsilon (-9+6 \omega )+\epsilon ^2 (4-12 \omega +3 \omega ^2))+\sigma ^3 \omega ^2\\&(-4-3 \omega -4 \epsilon ^2 \omega (-6+5 \omega ) +\epsilon (20-54 \omega +5 \omega ^2)))+\eta ^2 \sigma ^2 ((3+2 \epsilon ) \sigma ^6\\&-3 (3+5 \epsilon ) \sigma ^5 \omega -12 \epsilon ^2 \omega ^4+\epsilon \sigma \omega ^3 (-15-8 \epsilon (-3+5 \omega )) \\&+\sigma ^3 \omega ^2(3+\epsilon (54-10 \omega )+8 \epsilon ^2 (-3+5 \omega ))-2 \sigma ^4 \omega (2-3 \omega +6 \epsilon ^2\\&\omega -2 \epsilon (-5+6 \omega ))-2 \sigma ^2 \omega ^2 (1+9 \epsilon (-1+2 \omega )+\epsilon ^2 (4-36 \omega +15 \omega ^2)))\bigg )\bigg /\\&\quad \bigg (\sigma (\sigma +\eta \sigma -\omega ) \omega +\epsilon (\eta ^3 \sigma ^4+\eta ^2 \sigma ^2 (\sigma ^2-\omega -3 \sigma \omega )\\&+\omega ^2(2 \sigma +\sigma ^2-\omega -\sigma \omega )+\eta \sigma \omega (-2 \sigma -2 \sigma ^2+2 \omega +3 \sigma \omega )))^2\bigg ),\\ q(\epsilon )&=\bigg (\sigma ^2 (\sigma +\eta \sigma -\omega )^2 \omega ^3+\epsilon ^2 \omega (\eta ^3 \sigma ^4+(\sigma ^2-\sigma (-2+\omega )\\&-\omega ) \omega ^2+\eta ^2 \sigma ^2 (\sigma ^2-\omega -3 \sigma \omega )+\eta \sigma \omega (-2 \sigma ^2+2 \omega +\sigma (-2+3\omega )))^2\\&+\epsilon \sigma (\eta ^6 \sigma ^9+3 \eta ^5 \sigma ^7 (\sigma ^2-\omega -2 \sigma \omega )+\eta ^3 \sigma ^3 (\sigma ^6-12 \sigma ^5 \omega +\sigma ^2 \\&(19-30 \omega ) \omega ^2+4 \sigma ^3 (12-5 \omega ) \omega ^2-3 \omega ^3-20 \sigma \omega ^3+15 \sigma ^4 \omega (-1+2 \omega ))+\eta ^4 \sigma ^5 (3 \sigma ^4-15 \sigma ^3 \omega \\&+5 \omega ^2+15 \sigma \omega ^2+3 \sigma ^2 \omega (-4+5 \omega ))+\eta \sigma \omega ^2 (3 \sigma ^6-12 \sigma ^5 (-1+\omega )-9 \omega ^3\\&-4 \sigma \omega ^2 (-6+5 \omega )+\sigma ^2 \omega (-16+57 \omega -15 \omega ^2)-2 \sigma ^3 \omega (26-24 \omega +3 \omega ^2)\\&+3 \sigma ^4 (4-15 \omega +5 \omega ^2))+\eta ^2 \sigma ^2 \omega (-3 \sigma ^6\\&+15 \sigma ^4 (3-2 \omega ) \omega +9 \omega ^3+6 \sigma ^5 (-1+3 \omega )+6 \sigma \omega ^2 (-2+5 \omega )+3 \sigma ^2 \omega ^2 (-19+10 \omega )\\&+\sigma ^3 \omega (26-72 \omega +15 \omega ^2))+\omega ^3 (-\sigma ^6+3 \sigma ^5 (-2+\omega )+3 \omega ^3+\sigma \omega ^2 (-12+5 \omega )\\&-3 \sigma ^4 (4-5 \omega +\omega ^2)+\sigma ^2 \omega (16-19 \omega +3 \omega ^2)\\&+\sigma ^3 (-8+26 \omega -12 \omega ^2+\omega ^3)))\bigg )\bigg /\bigg (\omega (\sigma (\sigma +\eta \sigma -\omega ) \omega +\epsilon (\eta ^3 \sigma ^4\\&+\eta ^2 \sigma ^2 (\sigma ^2-\omega -3 \sigma \omega )+\omega ^2 (2 \sigma +\sigma ^2-\omega -\sigma \omega )+\eta \sigma \omega (-2 \sigma -2 \sigma ^2+2 \omega +3 \sigma \omega )))^2\bigg ). \end{aligned}$$

Appendix C

$$\begin{aligned} C_1&=\frac{1}{8 k \omega ^4} \sigma \bigg (\eta ^4 \sigma ^7+2 \eta ^3 \sigma ^5 (\sigma ^2-\omega -2 \sigma \omega )+\eta ^2 \sigma ^3 (\sigma ^4-6 \sigma ^3 \omega +6 \sigma ^2 (-1+\omega ) \omega -3 \omega ^2+6 \sigma \omega ^2)\\&-2 \eta \sigma (2+\sigma ) \omega (\sigma ^4-3 \sigma ^3 \omega -\omega ^2-\sigma \omega ^2+2 \sigma ^2 \omega ^2)+\omega ^2 (\sigma ^5-2 \sigma ^4 (-2+\omega )+\sigma (8-3 \omega ) \omega \\&-4 \omega ^2+2 \sigma ^2 \omega ^2+\sigma ^3 (4-6 \omega +\omega ^2))\bigg ), \\ C_{2}&=-\frac{1}{4 k \omega ^2}\bigg (\eta ^2 \sigma ^4+\eta \sigma ^2 (\sigma ^2-\omega -2 \sigma \omega )-(2+\sigma ) \omega (\sigma ^2+\omega -\sigma \omega )\bigg ) \bigg (-\frac{1}{\omega ^4}\sigma (\eta ^4 \sigma ^7+2 \eta ^3 \sigma ^5 (\sigma ^2-\omega -2 \sigma \omega )\\&+\eta ^2 \sigma ^3 (\sigma ^4-6 \sigma ^3 \omega +6 \sigma ^2 (-1+\omega ) \omega -3 \omega ^2+6 \sigma \omega ^2)-2 \eta \sigma (2+\sigma ) \omega (\sigma ^4-3 \sigma ^3 \omega -\omega ^2-\sigma \omega ^2+2 \sigma ^2 \omega ^2)+\\&\omega ^2 (\sigma ^5-2 \sigma ^4 (-2+\omega )+\sigma (8-3 \omega ) \omega -4 \omega ^2+2 \sigma ^2 \omega ^2+\sigma ^3 (4-6 \omega +\omega ^2)))\bigg )^{1/2},\\ C_3&=-\bigg (( \sigma (\eta ^2 \sigma ^3+\omega (-\sigma ^2+\sigma (-2+\omega )+\omega )+\eta \sigma (\sigma ^2-\omega -2 \sigma \omega )) (\eta ^4 \sigma ^6+\eta ^3 \sigma ^4 (2 \sigma ^2-9 \omega -4 \sigma \omega )\\&+\omega ^2 (\sigma ^4-2 \sigma ^3 (-1+\omega )+\sigma ^2 (-11+\omega ) \omega +4 (-2+\omega ) \omega +3 \sigma \omega (-4+3 \omega ))+\eta ^2 \sigma ^2 (\sigma ^4-6 \sigma ^3 \omega +4 \omega ^2\\&+27 \sigma \omega ^2+\sigma ^2 \omega (-11+6 \omega ))-\eta \sigma \omega (2 \sigma ^4+\sigma ^3 (2-6 \omega )+8 \omega ^2+2 \sigma ^2 \omega (-11+2 \omega )\\&+3 \sigma \omega (-4+9 \omega )))\bigg )\bigg /\bigg (8 k(\eta \sigma -\omega ) (\sigma +\eta \sigma -\omega ) \omega ^4)\bigg ), \\ C_4&=-\bigg (\sigma (\eta ^2 \sigma ^3+\omega (-\sigma ^2+\sigma (-2+\omega )+\omega )+\eta \sigma (\sigma ^2-\omega -2 \sigma \omega )\bigg ) \bigg (\eta ^4 \sigma ^6+\eta ^3 \sigma ^4 (2 \sigma ^2-9 \omega -4 \sigma \omega )\\&+\omega ^2 (\sigma ^4-2 \sigma ^3 (-1+\omega )+\sigma ^2 (-11+\omega ) \omega +4 (-2+\omega ) \omega +3 \sigma \omega (-4+3 \omega ))+\eta ^2 \sigma ^2 (\sigma ^4-6 \sigma ^3 \omega +4 \omega ^2 \\&+27 \sigma \omega ^2+\sigma ^2 \omega (-11+6 \omega ))-\eta \sigma \omega (2 \sigma ^4+\sigma ^3 (2-6 \omega )+8 \omega ^2+2 \sigma ^2 \omega (-11+2 \omega )+3 \sigma \omega (-4+9 \omega ))\bigg ) \\&\bigg (-\frac{1}{\omega ^4}\sigma \bigg (\eta ^4\sigma ^7+2 \eta ^3 \sigma ^5 (\sigma ^2-\omega -2 \sigma \omega )+\eta ^2 \sigma ^3 (\sigma ^4-6 \sigma ^3 \omega +6 \sigma ^2 (-1+\omega ) \omega -3 \omega ^2+6 \sigma \omega ^2)-2 \eta \sigma (2 \\&+\sigma ) \omega (\sigma ^4-3 \sigma ^3 \omega -\omega ^2-\sigma \omega ^2+2 \sigma ^2 \omega ^2)+\omega ^2 (\sigma ^5-2 \sigma ^4 (-2+\omega )+\sigma (8-3 \omega ) \omega -4 \omega ^2+2 \sigma ^2 \omega ^2\\&+\sigma ^3 (4-6 \omega +\omega ^2))\bigg )\bigg )^{1/2}\bigg /\bigg (16 k^2 (\eta \sigma -\omega ) (\sigma +\eta \sigma -\omega ) \omega ^4\bigg ), \\ C_{5}&=-\frac{1}{48 k^2} \bigg (-\frac{1}{\omega ^4}\sigma (\eta ^4 \sigma ^7+2 \eta ^3 \sigma ^5 (\sigma ^2-\omega -2 \sigma \omega )+\eta ^2 \sigma ^3 (\sigma ^4-6 \sigma ^3 \omega +6 \sigma ^2 (-1+\omega ) \omega -3 \omega ^2\\&+6 \sigma \omega ^2)-2 \eta \sigma (2+\sigma ) \omega (\sigma ^4-3 \sigma ^3 \omega -\omega ^2-\sigma \omega ^2+2 \sigma ^2 \omega ^2)+\omega ^2 (\sigma ^5-2 \sigma ^4 (-2+\omega )\\&+\sigma (8-3 \omega ) \omega -4 \omega ^2+2 \sigma ^2 \omega ^2+\sigma ^3 (4-6 \omega +\omega ^2)))\bigg )^{3/2}, \\ C_{6}&=\frac{1}{16 k^2 \omega ^6} \sigma \bigg (\eta ^6 \sigma ^{11}+3 \eta ^5 \sigma ^9 (\sigma ^2-\omega -2 \sigma \omega )+\eta ^3 \sigma ^5 (\sigma ^6-12 \sigma ^5 \omega +3 \sigma ^2 (1-10 \omega ) \omega ^2+4 \sigma ^3 (12-5 \omega )\\&\omega ^2+11 \omega ^3+12 \sigma \omega ^3+15 \sigma ^4 \omega (-1+2 \omega ))+3 \eta ^4 \sigma ^7 (\sigma ^4-5 \sigma ^3 \omega -\omega ^2+5\\&\sigma \omega ^2+\sigma ^2 \omega (-4+5 \omega ))-(2+\sigma ) \omega ^3 (\sigma ^7+\sigma ^6 (4-3 \omega )-\sigma ^4 (-6+\omega ) \omega ^2-4 \omega ^3+\sigma \omega ^2 (8+\omega )+\sigma ^2 \omega ^2 (-12\\&+5 \omega )-\sigma ^3 \omega (-12+9 \omega +\omega ^2)+\sigma ^5 (4-9 \omega +3 \omega ^2))+\eta ^2 \sigma ^3 \omega (-3 \sigma ^7+15 \sigma ^5 (3-2 \omega ) \omega +2 \omega ^3-33\\&\sigma \omega ^3-6 \sigma ^2 \omega ^2 (-5+3 \omega )+6 \sigma ^6 (-1+3 \omega )+3 \sigma ^3 \omega ^2 (-3+10 \omega )+3 \sigma ^4 \omega (6-24 \omega +5 \omega ^2))+\eta \sigma \omega ^2 \\&(3 \sigma ^8-12 \sigma ^7 (-1+\omega )+12 \sigma ^3 (-5+\omega ) \omega ^2-8 \omega ^3-4 \sigma \omega ^3+\sigma ^2 \omega ^2(-16+33 \omega )+3 \sigma ^4 \omega (4+3 \omega -5 \omega ^2)\\&-6 \sigma ^5 \omega (6-8 \omega +\omega ^2)+3 \sigma ^6 (4-15 \omega +5 \omega ^2))\bigg ), \\ C_{7}&=-( \sigma ^2 \bigg (\eta ^2 \sigma ^3+\omega (-\sigma ^2+\sigma (-2+\omega )+\omega )+\eta \sigma (\sigma ^2-\omega -2 \sigma \omega )\bigg ) \bigg (\eta ^7 \sigma ^{10}+\eta ^6 \sigma ^8 (4 \sigma ^2-26 \omega -7 \sigma \omega )\\&+\eta ^5 \sigma ^6 (6 \sigma ^4-24 \sigma ^3 \omega +91 \omega ^2+156 \sigma \omega ^2+\sigma ^2 \omega (-82+21 \omega ))+\eta ^4 \sigma ^4 (4 \sigma ^6-30 \sigma ^5 \omega -66 \omega ^3-455 \sigma \omega ^3\\&+30 \sigma ^4 \omega (-3+2 \omega )-5 \sigma ^3 \omega ^2 (-82+7 \omega )-6 \sigma ^2 \omega ^2 (-46+65 \omega ))+\eta ^3 \sigma ^3 (\sigma ^7-16 \sigma ^6 \omega +264 \omega ^4-40 \sigma ^4 \omega ^2\\&(-9+2 \omega )+2 \sigma ^5 \omega (-19+30 \omega )+26 \sigma \omega ^3 (-12+35 \omega )+8 \sigma ^2 \omega ^3 (-138+65 \omega )+\sigma ^3 \omega ^2 (283-820 \omega +35\\&\omega ^2))-\eta ^2 \sigma ^2 \omega (3 \sigma ^7+\sigma ^6 (4-24 \omega )+6 \sigma ^5 \omega (-19+10 \omega )+12 \omega ^3 (-10+33 \omega )+26 \sigma \omega ^3 (-36+35 \omega )-\\&6 \sigma ^4 \omega (17-90 \omega +10 \omega ^2)+\sigma ^3 \omega ^2 (849-820 \omega +21 \omega ^2)+6 \sigma ^2 \omega ^2 (59-276 \omega +65 \omega ^2))+\eta \sigma \omega ^2 (3 \sigma ^7\\&+\sigma ^6 (8-16 \omega )+24 \omega ^3 (-10+11 \omega )-12 \sigma ^4 \omega (17-30 \omega +2 \omega ^2)+12 \sigma ^2 \omega ^2 (59-92 \omega +13 \omega ^2)+2 \sigma ^5 (2 \\&-57\omega +15 \omega ^2)+\sigma \omega ^2 (216-936 \omega +455 \omega ^2)+\sigma ^3 \omega (-108+849 \omega -410 \omega ^2+7 \omega ^3))-\omega ^3 (\sigma ^7-4 \sigma ^6 (-1 \\&+\omega )-2 \sigma ^4 \omega (51-45 \omega +2 \omega ^2)+\sigma ^5 (4-38 \omega +6 \omega ^2)+6 \omega ^2 (8-20 \omega +11 \omega ^2)+\sigma \omega ^2 (216-312 \omega +91 \omega ^2)\\&+\sigma ^3 \omega (-108+283 \omega -82 \omega ^2+\omega ^3)+2 \sigma ^2 \omega (-24+177 \omega -138 \omega ^2+13 \omega ^3)))\bigg ) \\&\bigg /\bigg (48 k^2 (\eta \sigma -\omega ) (\sigma +\eta \sigma -\omega )^2 \omega ^6\bigg ),\\ D_{1}&=\frac{1}{48 k^2 \omega ^6} \sigma ^2 \bigg (\eta ^2 \sigma ^3+\omega (-\sigma ^2+\sigma (-2+\omega )+\omega )+\eta \sigma (\sigma ^2-\omega -2 \sigma \omega )\bigg )^2 \bigg (\eta ^2 \sigma ^4+\eta \sigma ^2 (\sigma ^2-\omega -2 \sigma \omega )\\&+\omega (-\sigma ^3+\sigma ^2 (-2+\omega )-4 \omega +\sigma \omega )\bigg ), \\ D_{2}&=-\frac{1}{4 k \omega ^4}\sigma \bigg (\eta ^4 \sigma ^7+2 \eta ^3 \sigma ^5 (\sigma ^2-\omega -2 \sigma \omega )+\eta ^2 \sigma ^3 (\sigma ^4-6 \sigma ^3 \omega +6 \sigma ^2 (-1+\omega ) \omega -\omega ^2+6 \sigma \omega ^2)\\&-2 \eta \sigma \omega (\sigma ^5+\sigma ^4 (2-3 \omega )+2 \sigma ^3 (-3+\omega ) \omega +\omega ^2-\sigma \omega ^2+\sigma ^2 \omega (-1+3 \omega ))+\omega ^2 (\sigma ^5-2 \sigma ^4 (-2+\omega )-\sigma \\&(-4+\omega ) \omega +2 \sigma ^2 (-1+\omega ) \omega +2 \omega ^2+\sigma ^3 (4-6 \omega +\omega ^2))\bigg ), \\ D_{3}&=\bigg ( \sigma ^2 \bigg (\eta ^2 \sigma ^3+\omega (-\sigma ^2+\sigma (-2+\omega )+\omega )+\eta \sigma (\sigma ^2-\omega -2 \sigma \omega )\bigg )^2 \bigg (\eta ^2 \sigma ^4+\eta \sigma ^2 (\sigma ^2-\omega -2 \sigma \omega )+\omega (-\sigma ^3\\&+\sigma ^2 (-2+\omega )-4 \omega +\sigma \omega )\bigg )\bigg )\bigg /\bigg (8 k \omega ^6 \bigg (-\frac{1}{\omega ^4}\sigma (\eta ^4 \sigma ^7+2 \eta ^3 \sigma ^5 (\sigma ^2-\omega -2 \sigma \omega )+\eta ^2 \sigma ^3 (\sigma ^4-6 \sigma ^3 \omega \\&+6 \sigma ^2 (-1+\omega ) \omega -3 \omega ^2+6 \sigma \omega ^2)-2 \eta \sigma (2+\sigma ) \omega (\sigma ^4-3 \sigma ^3 \omega -\omega ^2-\sigma \omega ^2+2 \sigma ^2 \omega ^2)+\omega ^2 (\sigma ^5-2 \sigma ^4 (-2 \\&+\omega )+\sigma (8-3 \omega ) \omega -4 \omega ^2+2 \sigma ^2 \omega ^2+\sigma ^3 (4-6 \omega +\omega ^2)))\bigg )^{1/2}\bigg ), \\ D_{4}&=-\frac{1}{16 k^2 \omega ^4} \sigma \bigg (\eta ^2 \sigma ^3+\omega (-\sigma ^2+\sigma (-2+\omega )+\omega )+\eta \sigma (\sigma ^2-\omega -2 \sigma \omega )\bigg ) \bigg (\eta ^2 \sigma ^4+\eta \\&\sigma ^2 (\sigma ^2-\omega -2 \sigma \omega )-(2+\sigma ) \omega (\sigma ^2+\omega -\sigma \omega )\bigg ) \bigg (-\frac{1}{\omega ^4}\sigma (\eta ^4 \sigma ^7+2 \eta ^3 \sigma ^5 (\sigma ^2-\omega -2 \sigma \omega )+\eta ^2 \sigma ^3 (\sigma ^4\\&-6 \sigma ^3 \omega +6 \sigma ^2 (-1+\omega ) \omega -3 \omega ^2+6 \sigma \omega ^2)-2 \eta \sigma (2+\sigma ) \omega (\sigma ^4-3 \sigma ^3 \omega -\omega ^2-\sigma \omega ^2+2 \sigma ^2 \omega ^2)+\omega ^2 (\sigma ^5\\&-2 \sigma ^4 (-2+\omega )+\sigma (8-3 \omega ) \omega -4 \omega ^2+2 \sigma ^2 \omega ^2+\sigma ^3 (4-6 \omega +\omega ^2)))\bigg )^{1/2}, \\ D_5&=-\bigg (\sigma ^2 \bigg (\eta ^2 \sigma ^3+\omega (-\sigma ^2+\sigma (-2+\omega )+\omega )+\eta \sigma (\sigma ^2-\omega -2 \sigma \omega )\bigg ) \bigg (\eta ^6 \sigma ^9+\eta ^5 \sigma ^7 (3 \sigma ^2-10 \omega -6 \sigma \omega ) \\&+\eta ^4 \sigma ^5 (3 \sigma ^4-15 \sigma ^3 \omega +13 \omega ^2+50 \sigma \omega ^2+3 \sigma ^2 \omega (-8+5 \omega ))+\eta ^3 \sigma ^2 (\sigma ^7-12 \sigma ^6 \omega +4 \sigma ^4 (24-5 \omega ) \omega ^2 \\&-16 \omega ^3-4 \sigma \omega ^3-52 \sigma ^2 \omega ^3+6 \sigma ^5 \omega (-3+5 \omega )-5 \sigma ^3 \omega ^2 (-9+20 \omega ))+\eta ^2 \sigma \omega (-3 \sigma ^7+32 \omega ^3+4 \sigma \omega ^2 (-4 \\&+3 \omega )-6 \sigma ^5 \omega (-9+5 \omega )+2 \sigma ^6 (-2+9 \omega )+5 \sigma ^3\omega ^2 (-27+20 \omega )+2 \sigma ^2 \omega ^2 (-14+39 \omega )+3 \sigma ^4 \omega (12-48 \omega \\&+5 \omega ^2))+\omega ^3 (-\sigma ^6+4 (-2+\omega ) \omega ^2+\sigma ^5(-4+3 \omega )+\sigma ^4 (-4+18 \omega -3 \omega ^2)+\sigma ^3 \omega (36-24 \omega +\omega ^2) \\&+\sigma ^2 \omega (32-45 \omega +10 \omega ^2)+\sigma \omega (16-28 \omega +13\omega ^2))+\eta \omega ^2 (3 \sigma ^7+\sigma ^6 (8-12 \omega )+4 \sigma ^2 (14-13 \omega ) \omega ^2 \\&-12 \sigma (-2+\omega ) \omega ^2-16 \omega ^3+\sigma ^3 \omega (-32+135 \omega -50 \omega ^2)-6 \sigma ^4\omega (12-16 \omega +\omega ^2)+\sigma ^5 (4-54 \omega +15 \omega ^2))\bigg )\bigg ) \\&\bigg /\bigg (8 k (\eta \sigma -\omega ) (\sigma +\eta \sigma -\omega ) \omega ^6 \bigg (-\frac{1}{\omega ^4}\sigma (\eta ^4 \sigma ^7+2 \eta ^3 \sigma ^5 (\sigma ^2-\omega -2 \sigma \omega )+\eta ^2 \sigma ^3 (\sigma ^4-6 \sigma ^3 \omega +6 \sigma ^2 (-1\\&+\omega ) \omega -3 \omega ^2+6 \sigma \omega ^2)-2 \eta \sigma (2+\sigma ) \omega (\sigma ^4-3 \sigma ^3 \omega -\omega ^2-\sigma \omega ^2+2 \sigma ^2 \omega ^2)+\omega ^2 (\sigma ^5-2 \sigma ^4 (-2+\omega ) \\&+\sigma (8-3 \omega ) \omega -4 \omega ^2+2 \sigma ^2 \omega ^2+\sigma ^3 (4-6 \omega +\omega ^2)))\bigg )^{1/2}\bigg ),\\ D_6&=-\bigg ( \sigma ^2 \bigg (\eta ^8 \sigma ^{12}+\eta ^7 \sigma ^{10} (4 \sigma ^2-11 \omega -8 \sigma \omega )+\eta ^6 \sigma ^8 (6 \sigma ^4-28 \sigma ^3 \omega +23 \omega ^2+77 \sigma \omega ^2+\sigma ^2 \omega (-39+28 \omega ))\\&+\eta ^5 \sigma ^6 (4 \sigma ^6-36 \sigma ^5 \omega +2 \sigma ^3(117-28 \omega ) \omega ^2-17 \omega ^3-138 \sigma \omega ^3+3 \sigma ^4 \omega (-17+28 \omega )-3 \sigma ^2 \omega ^2 (-34+77 \omega ))\\&+\eta ^3 \sigma ^2 \omega (-4 \sigma ^9+4 \sigma ^3 (103-115 \omega ) \omega ^3+2 \sigma ^2 (18-85 \omega ) \omega ^3+16 \sigma (-1+\omega ) \omega ^3+48 \omega ^4-4 \sigma ^7 \omega (-29+30 \omega )\\&+\sigma ^8 (-6+40 \omega )+\sigma ^4 \omega ^2 (-186+1020 \omega -385 \omega ^2)-4\sigma ^5 \omega ^2 (147-195 \omega +14 \omega ^2)+10 \sigma ^6 \omega (8-51 \omega \\&+14 \omega ^2))+\eta ^4 \sigma ^3(\sigma ^9-20 \sigma ^8 \omega -16 \omega ^4-4 \sigma \omega ^4+85 \sigma ^2 \omega ^4-5 \sigma ^6 \omega ^2 (-51+28 \omega )+5 \sigma ^4 \omega ^3 (-102+77 \omega )\\&+\sigma ^7 \omega (-29+90 \omega )+\sigma ^3 \omega ^3 (-103+345 \omega )+\sigma ^5 \omega ^2 (147-585 \omega +70 \omega ^2))+\omega ^4 (\sigma ^8+\sigma ^7 (6-4 \omega )-4 \omega ^3 \\&(2+\omega )+\sigma ^6 (12-29 \omega +6 \omega ^2)+\sigma \omega ^2 (32-36 \omega +17 \omega ^2)+\sigma ^5 (8-80 \omega +51 \omega ^2-4 \omega ^3)+\sigma ^4 \omega (-108+147\\&\omega -39 \omega ^2+\omega ^3)+\sigma ^3 \omega (-80+186 \omega -102 \omega ^2+11 \omega ^3)+\sigma ^2 \omega (-32+112 \omega -103 \omega ^2+23 \omega ^3))+\eta ^2 \sigma \omega ^2 (6 \sigma ^9\\&+\sigma ^8 (18-40 \omega )-24 \sigma (-1+\omega ) \omega ^3-48\omega ^4+2 \sigma ^2 \omega ^3 (-54+85 \omega )-6 \sigma ^6 \omega (40-85 \omega +14 \omega ^2)+6 \sigma ^7\\&(2-29 \omega +15 \omega ^2)+3 \sigma ^4 \omega ^2 (186-340 \omega +77 \omega ^2)+\sigma ^3 \omega ^2 (112-618 \omega +345 \omega ^2)+\sigma ^5 \omega (-108\\&+882 \omega -585 \omega ^2+28 \omega ^3))-\eta \omega ^3 (4 \sigma ^9+\sigma ^8 (18-20 \omega )-16 \omega ^4-16 \sigma \omega ^4+4 \sigma ^7 (6-29 \omega +9 \omega ^2) \\&+2 \sigma ^3 \omega ^2 (112-206 \omega +69 \omega ^2)+\sigma ^2 \omega ^2 (32-108 \omega +85 \omega ^2)+\sigma ^6 (8-240 \omega +255 \omega ^2-28 \omega ^3) \\&+2 \sigma ^5 \omega (-108+294 \omega -117 \omega ^2+4 \omega ^3)+\sigma ^4 \omega (-80+558 \omega -510 \omega ^2+77 \omega ^3))\bigg )\bigg ) \\&\bigg /\bigg (16 k^2 (\eta \sigma -\omega ) (\sigma +\eta \sigma -\omega )\omega ^6\bigg ),\\ D_7&=-\bigg ( \sigma ^3 \bigg (\eta ^2 \sigma ^3+\omega (-\sigma ^2+\sigma (-2+\omega )+\omega )+\eta \sigma (\sigma ^2-\omega -2 \sigma \omega )\bigg )\bigg (\eta ^9 \sigma ^{13}+\eta ^8 \sigma ^{11} (5 \sigma ^2-27 \omega \\&-9 \sigma \omega )+\eta ^7 \sigma ^9 (10 \sigma ^4-40 \sigma ^3 \omega +117 \omega ^2+216 \sigma \omega ^2+6 \sigma ^2 \omega (-19+6 \omega ))+\eta ^6 \sigma ^7 (10 \sigma ^6-70 \sigma ^5 \omega +3 \sigma ^2\\&(167-252 \omega ) \omega ^2+42 \sigma ^3 (19-2 \omega ) \omega ^2-157 \omega ^3-819 \sigma \omega ^3+2 \sigma ^4 \omega (-93+70 \omega ))+\eta ^5 \sigma ^3 (5 \sigma ^{10}-60 \sigma ^9 \omega \\&-96 \omega ^4-48 \sigma \omega ^4+74 \sigma ^2 \omega ^4+942 \sigma ^3 \omega ^4+6 \sigma ^8 \omega (-24+35 \omega )-4 \sigma ^7 \omega ^2 (-279+70 \omega )+18 \sigma ^5 \omega ^3 (-167 \\&+84 \omega )+\sigma ^4 \omega ^3 (-836+2457 \omega )+3 \sigma ^6 \omega ^2 (271-798 \omega +42 \omega ^2))+\eta ^4 \sigma ^2 (\sigma ^{11}-25 \sigma ^{10} \omega +10 \sigma ^8 (72 \\&-35 \omega ) \omega ^2+5 \sigma ^3 (116-471 \omega ) \omega ^4+192 \sigma (-1+\omega ) \omega ^4+288 \omega ^5+3 \sigma ^9 \omega (-17+50 \omega )-2 \sigma ^2 \omega ^4 (48 \\&+185 \omega )-5 \sigma ^4 \omega ^4 (-836+819 \omega )+\sigma ^5 \omega ^3 (-1501+7515 \omega -1890 \omega ^2)-3 \sigma ^6 \omega ^3 (1355-1330 \omega +42 \omega ^2) \\&+\sigma ^7 \omega ^2 (603-2790 \omega +350 \omega ^2))+\eta ^3 \sigma \omega (-4 \sigma ^{11}-288 \omega ^5-96 \sigma \omega ^4(-4+3 \omega )-4 \sigma ^9 \omega (-51+50 \omega ) \\&+\sigma ^{10} (-6+50 \omega )-4 \sigma ^7 \omega ^2 (603-930 \omega +70 \omega ^2)+2 \sigma ^8 \omega (93-720 \omega +175\omega ^2)+4 \sigma ^2 \omega ^3 (-24+42 \omega \\&+185 \omega ^2)+4 \sigma ^5 \omega ^3 (1501-2505 \omega +378 \omega ^2)+4 \sigma ^3 \omega ^3 (-12-580 \omega +785 \omega ^2)+\sigma ^4 \omega ^3(1322-8360 \omega \\&+4095 \omega ^2)+2 \sigma ^6 \omega ^2 (-565+4065 \omega -1995 \omega ^2+42 \omega ^3))+\eta ^2 \omega ^2 (6 \sigma ^{11}+\sigma ^{10} (18-50 \omega )+192 \sigma (-1\\&+\omega ) \omega ^4+96 \omega ^5+\sigma ^4 \omega ^3 (-3966+8360 \omega -2457 \omega ^2)+4 \sigma ^2 \omega ^3 (24+18 \omega -185 \omega ^2)+6 \sigma ^9 (2-51 \omega \\&+25 \omega ^2)-6 \sigma ^8 \omega (93-240 \omega +35 \omega ^2)-3 \sigma ^3 \omega ^3 (136-1160 \omega +785 \omega ^2)+3 \sigma ^5 \omega ^2 (360-3002 \omega +2505 \omega ^2 \\&-252\omega ^3)-6 \sigma ^6 \omega ^2 (-565+1355 \omega -399 \omega ^2+6 \omega ^3)+2 \sigma ^7 \omega (-158+1809 \omega -1395 \omega ^2+70 \omega ^3))+\omega ^4 (\sigma ^9 \\&+\sigma ^8 (6-5 \omega )+\sigma ^7 (12-51 \omega +10 \omega ^2)-2 \omega ^3 (24-60 \omega +37 \omega ^2)+\sigma \omega ^2 (96-504 \omega +580 \omega ^2-157 \omega ^3)+ \\&\sigma ^2 \omega ^2 (528-1322 \omega +836 \omega ^2-117 \omega ^3)+\sigma ^5 \omega (-316+603 \omega -186 \omega ^2+5 \omega ^3)-2 \sigma ^6 (-4+93 \omega -72 \omega ^2 \\&+5 \omega ^3)+\sigma ^3 \omega (-96+1080 \omega -1501 \omega ^2+501 \omega ^3-27 \omega ^4)-\sigma ^4 \omega (264-1130 \omega +813 \omega ^2-114 \omega ^3+\omega ^4)) \\&+\eta \omega ^3 (-4 \sigma ^{10}-48 \omega ^5+\sigma ^9 (-18+25 \omega )-12 \sigma ^8 (2-17 \omega +5 \omega ^2)+2 \sigma \omega ^3 (24-132 \omega +185 \omega ^2) \\&+2 \sigma ^2 \omega ^3 (480-1160 \omega +471 \omega ^2)-4 \sigma ^6 \omega (-158+603\omega -279 \omega ^2+10 \omega ^3)+\sigma ^7 (-8+558 \omega -720 \omega ^2 \\&+70 \omega ^3)+2 \sigma ^4 \omega ^2 (-1080+3002 \omega -1503 \omega ^2+108 \omega ^3)+\sigma ^3 \omega ^2(-528+3966 \omega -4180 \omega ^2+819 \omega ^3) \\&+3 \sigma ^5 \omega (88-1130 \omega +1355 \omega ^2-266 \omega ^3+3 \omega ^4))\bigg )\bigg )\bigg /\bigg (48 k^2 (\eta \sigma -\omega ) (\sigma +\eta \sigma -\omega )^2 \omega ^8 \bigg (-\frac{1}{\omega ^4}\sigma (\eta ^4 \sigma ^7 \\&+2 \eta ^3 \sigma ^5 (\sigma ^2-\omega -2 \sigma \omega )+\eta ^2 \sigma ^3 (\sigma ^4-6 \sigma ^3 \omega +6 \sigma ^2 (-1+\omega ) \omega -3 \omega ^2+6 \sigma \omega ^2)-2 \eta \sigma (2+\sigma ) \omega (\sigma ^4-3 \sigma ^3 \omega \\&-\omega ^2-\sigma \omega ^2+2 \sigma ^2 \omega ^2)+\omega ^2 (\sigma ^5-2 \sigma ^4 (-2+\omega )+\sigma (8-3 \omega ) \omega -4 \omega ^2+2 \sigma ^2 \omega ^2+\sigma ^3 (4-6 \omega +\omega ^2)))\bigg )^{1/2}\bigg ). \end{aligned}$$

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Naik, P.A., Javaid, Y., Ahmed, R. et al. Stability and bifurcation analysis of a population dynamic model with Allee effect via piecewise constant argument method. J. Appl. Math. Comput. (2024). https://doi.org/10.1007/s12190-024-02119-y

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