doi: 10.3934/dcdsb.2021292
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021292
References:
[1]

A. S. AcklehM. I. HossainA. 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.  Google Scholar

[2]

I. AliU. 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.  Google Scholar

[3]

L. J. S. Allen, An Introduction to Mathematical Biology, Pearson/Prentice Hall, Upper Saddle River, NJ, 2007. Google Scholar

[4] E. S. Allman and J. A. Rhodes, Mathematical Models in Biology: An Introduction, Cambridge University Press, Cambridge, 2004.   Google Scholar
[5] D. K. Arrowsmith and C. M. Place, An Introduction to Dynamical Systems, Cambridge University Press, 1990.   Google Scholar
[6]

Q. ChenZ. 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.  Google Scholar

[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.  Google Scholar

[8]

L. FeiX. 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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[11]

J. HuangS. 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.  Google Scholar

[12]

A. Q. KhanJ. 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.  Google Scholar

[13] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 2004.  doi: 10.1007/978-1-4757-3978-7.  Google Scholar
[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.  Google Scholar

[15]

A. Lotka, Elements of Physical Biology, Williams Winlkins Baltimore, 1925. Google Scholar

[16]

X. LuoG. ChenB. Wang and J. Fang, Hybrid control of period-doubling bifurcation and chaos in discrete nonlinear dynamical systems, Chaos Soli. Fract., 18 (2003), 775-783.   Google Scholar

[17] J. Maynard Smith, Mathematical Ideas in Biology, Cambridge University Press, Cambridge, 1968.  doi: 10.1017/CBO9780511565144.  Google Scholar
[18] J. D. Murray, Mathematical Biology: I. An Introduction, Third Edition, Springer-Verlag, New York, 2002.   Google Scholar
[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.  Google Scholar

[20] V. Volterra, Leçons Sur La Théorie Mathématique De La Lutte Pour La Vie, Gauthier-Villars, Paris, 1931.   Google Scholar
[21] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990.  doi: 10.1007/978-1-4757-4067-7.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

X. ZhangQ. 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.  Google Scholar

show all references

References:
[1]

A. S. AcklehM. I. HossainA. 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.  Google Scholar

[2]

I. AliU. 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.  Google Scholar

[3]

L. J. S. Allen, An Introduction to Mathematical Biology, Pearson/Prentice Hall, Upper Saddle River, NJ, 2007. Google Scholar

[4] E. S. Allman and J. A. Rhodes, Mathematical Models in Biology: An Introduction, Cambridge University Press, Cambridge, 2004.   Google Scholar
[5] D. K. Arrowsmith and C. M. Place, An Introduction to Dynamical Systems, Cambridge University Press, 1990.   Google Scholar
[6]

Q. ChenZ. 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.  Google Scholar

[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.  Google Scholar

[8]

L. FeiX. 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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[11]

J. HuangS. 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.  Google Scholar

[12]

A. Q. KhanJ. 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.  Google Scholar

[13] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 2004.  doi: 10.1007/978-1-4757-3978-7.  Google Scholar
[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.  Google Scholar

[15]

A. Lotka, Elements of Physical Biology, Williams Winlkins Baltimore, 1925. Google Scholar

[16]

X. LuoG. ChenB. Wang and J. Fang, Hybrid control of period-doubling bifurcation and chaos in discrete nonlinear dynamical systems, Chaos Soli. Fract., 18 (2003), 775-783.   Google Scholar

[17] J. Maynard Smith, Mathematical Ideas in Biology, Cambridge University Press, Cambridge, 1968.  doi: 10.1017/CBO9780511565144.  Google Scholar
[18] J. D. Murray, Mathematical Biology: I. An Introduction, Third Edition, Springer-Verlag, New York, 2002.   Google Scholar
[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.  Google Scholar

[20] V. Volterra, Leçons Sur La Théorie Mathématique De La Lutte Pour La Vie, Gauthier-Villars, Paris, 1931.   Google Scholar
[21] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990.  doi: 10.1007/978-1-4757-4067-7.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

X. ZhangQ. 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.  Google Scholar

Figure 1.  phase portraits of system (5.1) when parameter $ r_0 $ is set to different values
Figure 2.  transcritical bifurcation graphs of system (5.1) for $ r_0\in [0.5, 2] $
Figure 3.  phase portraits of system (5.2) when parameter $ r_0 $ is set to different values
Figure 4.  transcritical bifurcation graphs of system (5.2) for $ r_0\in [0.5, 4] $
Figure 5.  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)
Figure 6.  time-series and phase-plane graphs for system (5.3) with θ=0.99
Figure 7.  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)
Figure 8.  time-series and phase-plane graphs for system (5.5) with θ=0.99
[1]

Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065

[2]

Yunshyong Chow, Sophia Jang. Neimark-Sacker bifurcations in a host-parasitoid system with a host refuge. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1713-1728. doi: 10.3934/dcdsb.2016019

[3]

Jicai Huang, Sanhong Liu, Shigui Ruan, Xinan Zhang. Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041

[4]

Jicai Huang, Yijun Gong, Shigui Ruan. Bifurcation analysis in a predator-prey model with constant-yield predator harvesting. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2101-2121. doi: 10.3934/dcdsb.2013.18.2101

[5]

Ming Liu, Dongpo Hu, Fanwei Meng. Stability and bifurcation analysis in a delay-induced predator-prey model with Michaelis-Menten type predator harvesting. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3197-3222. doi: 10.3934/dcdss.2020259

[6]

Qing Zhu, Huaqin Peng, Xiaoxiao Zheng, Huafeng Xiao. Bifurcation analysis of a stage-structured predator-prey model with prey refuge. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2195-2209. doi: 10.3934/dcdss.2019141

[7]

Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002

[8]

Xiao He, Sining Zheng. Bifurcation analysis and dynamic behavior to a predator-prey model with Beddington-DeAngelis functional response and protection zone. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4641-4657. doi: 10.3934/dcdsb.2020117

[9]

Eric Avila-Vales, Gerardo García-Almeida, Erika Rivero-Esquivel. Bifurcation and spatiotemporal patterns in a Bazykin predator-prey model with self and cross diffusion and Beddington-DeAngelis response. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 717-740. doi: 10.3934/dcdsb.2017035

[10]

Bing Zeng, Shengfu Deng, Pei Yu. Bogdanov-Takens bifurcation in predator-prey systems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3253-3269. doi: 10.3934/dcdss.2020130

[11]

Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507

[12]

Xiaoyuan Chang, Junjie Wei. Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge. Mathematical Biosciences & Engineering, 2013, 10 (4) : 979-996. doi: 10.3934/mbe.2013.10.979

[13]

Kie Van Ivanky Saputra, Lennaert van Veen, Gilles Reinout Willem Quispel. The saddle-node-transcritical bifurcation in a population model with constant rate harvesting. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 233-250. doi: 10.3934/dcdsb.2010.14.233

[14]

Haiying Jing, Zhaoyu Yang. The impact of state feedback control on a predator-prey model with functional response. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 607-614. doi: 10.3934/dcdsb.2004.4.607

[15]

Hongyong Zhao, Daiyong Wu. Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3271-3284. doi: 10.3934/dcdss.2020129

[16]

Zuolin Shen, Junjie Wei. Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences & Engineering, 2018, 15 (3) : 693-715. doi: 10.3934/mbe.2018031

[17]

Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045

[18]

Wenjie Li, Lihong Huang, Jinchen Ji. Globally exponentially stable periodic solution in a general delayed predator-prey model under discontinuous prey control strategy. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2639-2664. doi: 10.3934/dcdsb.2020026

[19]

Yanfei Du, Ben Niu, Junjie Wei. A predator-prey model with cooperative hunting in the predator and group defense in the prey. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021298

[20]

Peng Feng. On a diffusive predator-prey model with nonlinear harvesting. Mathematical Biosciences & Engineering, 2014, 11 (4) : 807-821. doi: 10.3934/mbe.2014.11.807

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (106)
  • HTML views (64)
  • Cited by (0)

Other articles
by authors

[Back to Top]