# American Institute of Mathematical Sciences

October  2022, 27(10): 5701-5721. doi: 10.3934/dcdsb.2021292

## Bifurcation and control of a predator-prey system with unfixed functional responses

 1 College of Science, Nanchang Institute of Technology, Nanchang, Jiangxi 330000, China 2 School of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

* Corresponding author: Xingwu Chen

Received  May 2021 Revised  September 2021 Published  October 2022 Early access  December 2021

Fund Project: The second author is supported by NSFC grant 11871355

In this paper we investigate a discrete-time predator-prey system with not only some constant parameters but also unfixed functional responses including growth rate function of prey, conversion factor function and predation probability function. We prove that the maximal number of fixed points is $3$ and give necessary and sufficient conditions of exactly $j$($j = 1,2,3$) fixed points, respectively. For transcritical bifurcation and Neimark-Sacker bifurcation, we provide bifurcation conditions depending on these unfixed functional responses. In order to regulate the stability of this biological system, a hybrid control strategy is used to control the Neimark-Sacker bifurcation. Finally, we apply our main results to some examples and carry out numerical simulations for each example to verify the correctness of our theoretical analysis.

Citation: Lizhi Fei, Xingwu Chen. Bifurcation and control of a predator-prey system with unfixed functional responses. Discrete and Continuous Dynamical Systems - B, 2022, 27 (10) : 5701-5721. doi: 10.3934/dcdsb.2021292
##### References:
 [1] A. S. Ackleh, M. I. Hossain, A. Veprauskas and A. Zhang, Persistence and stability analysis of discrete-time predator-prey models: A study of population and evolutionary dynamics, J. Diff. Equa. Appl., 25 (2019), 1568-1603.  doi: 10.1080/10236198.2019.1669579. [2] I. Ali, U. Saeed and Q. Din, Bifurcation analysis and chaos control in a discrete-time plant quality and larch budmoth interaction model with Ricker equation, Math. Methods Appl. Sci., 42 (2019), 7395-7410.  doi: 10.1002/mma.5857. [3] L. J. S. Allen, An Introduction to Mathematical Biology, Pearson/Prentice Hall, Upper Saddle River, NJ, 2007. [4] E. S. Allman and J. A. Rhodes, Mathematical Models in Biology: An Introduction, Cambridge University Press, Cambridge, 2004. [5] D. K. Arrowsmith and C. M. Place, An Introduction to Dynamical Systems, Cambridge University Press, 1990. [6] Q. Chen, Z. Teng and Z. Hu, Bifurcation and control for a discrete-time prey-predator model with Holling-Ⅳ functional response, Int. J. Appl. Math. Comp. Sci., 23 (2013), 247-261.  doi: 10.2478/amcs-2013-0019. [7] Q. Din, Neimark-Sacker bifurcation and chaos control in Hassell-Varley model, J. Diff. Equa. Appl., 23 (2017), 741-762.  doi: 10.1080/10236198.2016.1277213. [8] L. Fei, X. Chen and B. Han, Bifurcation analysis and hybrid control of a discrete-time predator-prey model, J. Diff. Equa. Appl., 27 (2021), 102-117.  doi: 10.1080/10236198.2021.1876038. [9] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.  doi: 10.1007/978-1-4612-1140-2. [10] C. S. Holling, The functional response of predator to prey density and its role in mimicry and population regulation, Memo. Ento. Soci. Cana., 97 (1965), 5-60.  doi: 10.4039/entm9745fv. [11] J. Huang, S. Ruan and J. Song, Bifurcations in a predator-prey system of Leslie type with generalized Holling type Ⅲ functional response, J. Differential Equations, 257 (2014), 1721-1752.  doi: 10.1016/j.jde.2014.04.024. [12] A. Q. Khan, J. Ma and D. Xiao, Global dynamics and bifurcation analysis of a host-parasitoid model with strong Allee effect, J. Bio. Dyn., 11 (2017), 121-146.  doi: 10.1080/17513758.2016.1254287. [13] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 2004.  doi: 10.1007/978-1-4757-3978-7. [14] X. Liu and D. Xiao, Bifurcations in a discrete time Lotka-Volterra predator-prey system, Disc. Cont. Dyna. Syst. Seri. B, 6 (2006), 559-572.  doi: 10.3934/dcdsb.2006.6.559. [15] A. Lotka, Elements of Physical Biology, Williams Winlkins Baltimore, 1925. [16] X. Luo, G. Chen, B. Wang and J. Fang, Hybrid control of period-doubling bifurcation and chaos in discrete nonlinear dynamical systems, Chaos Soli. Fract., 18 (2003), 775-783. [17] J. Maynard Smith, Mathematical Ideas in Biology, Cambridge University Press, Cambridge, 1968.  doi: 10.1017/CBO9780511565144. [18] J. D. Murray, Mathematical Biology: I. An Introduction, Third Edition, Springer-Verlag, New York, 2002. [19] M. G. Neubert and M. Kot, The subcritical collapse of predator populations in discrete-time predator-prey models, Math. Bios., 110 (1992), 45-66.  doi: 10.1016/0025-5564(92)90014-N. [20] V. Volterra, Leçons Sur La Théorie Mathématique De La Lutte Pour La Vie, Gauthier-Villars, Paris, 1931. [21] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990.  doi: 10.1007/978-1-4757-4067-7. [22] Y. Yao, Dynamics of a prey-predator system with foraging facilitation in predators, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2050009, 24 pp. doi: 10.1142/S0218127420500091. [23] L.-G. Yuan and Q.-G. Yang, Bifurcation, invariant curve and hybrid control in a discrete-time predator-prey system, Appl. Math. Model., 39 (2015), 2345-2362.  doi: 10.1016/j.apm.2014.10.040. [24] L. Zhang and L. Zou, Bifurcations and control in a discrete predator-prey model with strong allee effect, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850062, 29 pp. doi: 10.1142/S0218127418500621. [25] X. Zhang, Q. Zhang and V. Sreeram, Bifurcation analysis and control of a discrete harvested prey-predator system with Beddington-DeAngelis functional response, J. Franklin Instit., 347 (2010), 1076-1096.  doi: 10.1016/j.jfranklin.2010.03.016.

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##### References:
 [1] A. S. Ackleh, M. I. Hossain, A. Veprauskas and A. Zhang, Persistence and stability analysis of discrete-time predator-prey models: A study of population and evolutionary dynamics, J. Diff. Equa. Appl., 25 (2019), 1568-1603.  doi: 10.1080/10236198.2019.1669579. [2] I. Ali, U. Saeed and Q. Din, Bifurcation analysis and chaos control in a discrete-time plant quality and larch budmoth interaction model with Ricker equation, Math. Methods Appl. Sci., 42 (2019), 7395-7410.  doi: 10.1002/mma.5857. [3] L. J. S. Allen, An Introduction to Mathematical Biology, Pearson/Prentice Hall, Upper Saddle River, NJ, 2007. [4] E. S. Allman and J. A. Rhodes, Mathematical Models in Biology: An Introduction, Cambridge University Press, Cambridge, 2004. [5] D. K. Arrowsmith and C. M. Place, An Introduction to Dynamical Systems, Cambridge University Press, 1990. [6] Q. Chen, Z. Teng and Z. Hu, Bifurcation and control for a discrete-time prey-predator model with Holling-Ⅳ functional response, Int. J. Appl. Math. Comp. Sci., 23 (2013), 247-261.  doi: 10.2478/amcs-2013-0019. [7] Q. Din, Neimark-Sacker bifurcation and chaos control in Hassell-Varley model, J. Diff. Equa. Appl., 23 (2017), 741-762.  doi: 10.1080/10236198.2016.1277213. [8] L. Fei, X. Chen and B. Han, Bifurcation analysis and hybrid control of a discrete-time predator-prey model, J. Diff. Equa. Appl., 27 (2021), 102-117.  doi: 10.1080/10236198.2021.1876038. [9] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.  doi: 10.1007/978-1-4612-1140-2. [10] C. S. Holling, The functional response of predator to prey density and its role in mimicry and population regulation, Memo. Ento. Soci. Cana., 97 (1965), 5-60.  doi: 10.4039/entm9745fv. [11] J. Huang, S. Ruan and J. Song, Bifurcations in a predator-prey system of Leslie type with generalized Holling type Ⅲ functional response, J. Differential Equations, 257 (2014), 1721-1752.  doi: 10.1016/j.jde.2014.04.024. [12] A. Q. Khan, J. Ma and D. Xiao, Global dynamics and bifurcation analysis of a host-parasitoid model with strong Allee effect, J. Bio. Dyn., 11 (2017), 121-146.  doi: 10.1080/17513758.2016.1254287. [13] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 2004.  doi: 10.1007/978-1-4757-3978-7. [14] X. Liu and D. Xiao, Bifurcations in a discrete time Lotka-Volterra predator-prey system, Disc. Cont. Dyna. Syst. Seri. B, 6 (2006), 559-572.  doi: 10.3934/dcdsb.2006.6.559. [15] A. Lotka, Elements of Physical Biology, Williams Winlkins Baltimore, 1925. [16] X. Luo, G. Chen, B. Wang and J. Fang, Hybrid control of period-doubling bifurcation and chaos in discrete nonlinear dynamical systems, Chaos Soli. Fract., 18 (2003), 775-783. [17] J. Maynard Smith, Mathematical Ideas in Biology, Cambridge University Press, Cambridge, 1968.  doi: 10.1017/CBO9780511565144. [18] J. D. Murray, Mathematical Biology: I. An Introduction, Third Edition, Springer-Verlag, New York, 2002. [19] M. G. Neubert and M. Kot, The subcritical collapse of predator populations in discrete-time predator-prey models, Math. Bios., 110 (1992), 45-66.  doi: 10.1016/0025-5564(92)90014-N. [20] V. Volterra, Leçons Sur La Théorie Mathématique De La Lutte Pour La Vie, Gauthier-Villars, Paris, 1931. [21] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990.  doi: 10.1007/978-1-4757-4067-7. [22] Y. Yao, Dynamics of a prey-predator system with foraging facilitation in predators, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2050009, 24 pp. doi: 10.1142/S0218127420500091. [23] L.-G. Yuan and Q.-G. Yang, Bifurcation, invariant curve and hybrid control in a discrete-time predator-prey system, Appl. Math. Model., 39 (2015), 2345-2362.  doi: 10.1016/j.apm.2014.10.040. [24] L. Zhang and L. Zou, Bifurcations and control in a discrete predator-prey model with strong allee effect, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850062, 29 pp. doi: 10.1142/S0218127418500621. [25] X. Zhang, Q. Zhang and V. Sreeram, Bifurcation analysis and control of a discrete harvested prey-predator system with Beddington-DeAngelis functional response, J. Franklin Instit., 347 (2010), 1076-1096.  doi: 10.1016/j.jfranklin.2010.03.016.
phase portraits of system (5.1) when parameter $r_0$ is set to different values
transcritical bifurcation graphs of system (5.1) for $r_0\in [0.5, 2]$
phase portraits of system (5.2) when parameter $r_0$ is set to different values
transcritical bifurcation graphs of system (5.2) for $r_0\in [0.5, 4]$
phase portraits and bifurcation graphs of system (5.2) for $\gamma$ vary in the small neighborhood of $\gamma = 7.961845698$, the initial value is (0.25, 0.75)
time-series and phase-plane graphs for system (5.3) with θ=0.99
phase portraits and bifurcation graphs of system (5.4) when $m$ vary in the small neighborhood of $m = 6$, the initial value is (0.999, 0.999)
time-series and phase-plane graphs for system (5.5) with θ=0.99
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