In this paper, we analyze a general predator-prey modelwith state feedback impulsive harvesting strategies in which the prey species displays a strongAllee effect. We firstly show the existence of order-$1$ heteroclinic cycle and order-$1$ positive periodic solutions by using the geometric theory of differential equations for the unperturbed system. Based on the theory of rotated vector fields, the order-$1$ positive periodic solutions and heteroclinic bifurcation are studied for the perturbed system. Finally, some numerical simulations are provided to illustrate our main results. All the results indicate that the harvesting rate should be maintained at a reasonable range to keep the sustainable development of ecological systems.
Citation: Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy[J]. Mathematical Biosciences and Engineering, 2015, 12(5): 1065-1081. doi: 10.3934/mbe.2015.12.1065
Related Papers:
| [1] |
Fang Liu, Yanfei Du .
Spatiotemporal dynamics of a diffusive predator-prey model with delay and Allee effect in predator. Mathematical Biosciences and Engineering, 2023, 20(11): 19372-19400.
doi: 10.3934/mbe.2023857
|
| [2] |
Yongli Cai, Malay Banerjee, Yun Kang, Weiming Wang .
Spatiotemporal complexity in a predator--prey model with weak Allee effects. Mathematical Biosciences and Engineering, 2014, 11(6): 1247-1274.
doi: 10.3934/mbe.2014.11.1247
|
| [3] |
Moitri Sen, Malay Banerjee, Yasuhiro Takeuchi .
Influence of Allee effect in prey populations on the dynamics of two-prey-one-predator model. Mathematical Biosciences and Engineering, 2018, 15(4): 883-904.
doi: 10.3934/mbe.2018040
|
| [4] |
Kawkab Al Amri, Qamar J. A Khan, David Greenhalgh .
Combined impact of fear and Allee effect in predator-prey interaction models on their growth. Mathematical Biosciences and Engineering, 2024, 21(10): 7211-7252.
doi: 10.3934/mbe.2024319
|
| [5] |
Mengyun Xing, Mengxin He, Zhong Li .
Dynamics of a modified Leslie-Gower predator-prey model with double Allee effects. Mathematical Biosciences and Engineering, 2024, 21(1): 792-831.
doi: 10.3934/mbe.2024034
|
| [6] |
Juan Ye, Yi Wang, Zhan Jin, Chuanjun Dai, Min Zhao .
Dynamics of a predator-prey model with strong Allee effect and nonconstant mortality rate. Mathematical Biosciences and Engineering, 2022, 19(4): 3402-3426.
doi: 10.3934/mbe.2022157
|
| [7] |
A. Q. Khan, I. Ahmad, H. S. Alayachi, M. S. M. Noorani, A. Khaliq .
Discrete-time predator-prey model with flip bifurcation and chaos control. Mathematical Biosciences and Engineering, 2020, 17(5): 5944-5960.
doi: 10.3934/mbe.2020317
|
| [8] |
Parvaiz Ahmad Naik, Muhammad Amer, Rizwan Ahmed, Sania Qureshi, Zhengxin Huang .
Stability and bifurcation analysis of a discrete predator-prey system of Ricker type with refuge effect. Mathematical Biosciences and Engineering, 2024, 21(3): 4554-4586.
doi: 10.3934/mbe.2024201
|
| [9] |
Yuhong Huo, Gourav Mandal, Lakshmi Narayan Guin, Santabrata Chakravarty, Renji Han .
Allee effect-driven complexity in a spatiotemporal predator-prey system with fear factor. Mathematical Biosciences and Engineering, 2023, 20(10): 18820-18860.
doi: 10.3934/mbe.2023834
|
| [10] |
Xin-You Meng, Yu-Qian Wu .
Bifurcation analysis in a singular Beddington-DeAngelis predator-prey model with two delays and nonlinear predator harvesting. Mathematical Biosciences and Engineering, 2019, 16(4): 2668-2696.
doi: 10.3934/mbe.2019133
|
Abstract
In this paper, we analyze a general predator-prey modelwith state feedback impulsive harvesting strategies in which the prey species displays a strongAllee effect. We firstly show the existence of order-$1$ heteroclinic cycle and order-$1$ positive periodic solutions by using the geometric theory of differential equations for the unperturbed system. Based on the theory of rotated vector fields, the order-$1$ positive periodic solutions and heteroclinic bifurcation are studied for the perturbed system. Finally, some numerical simulations are provided to illustrate our main results. All the results indicate that the harvesting rate should be maintained at a reasonable range to keep the sustainable development of ecological systems.
References
|
[1]
|
University of Chicago Press, Chicago, 1931.
|
|
[2]
|
Journal of Beihua University (Natural Science), 12 (2011), 1-9. (in Chinese)
|
|
[3]
|
Ecol. Model., 213 (2008), 356-364.
|
|
[4]
|
Proc. R. Soc. Lond. B., 266 (1999), 557-563.
|
|
[5]
|
Theor. Popul. Biol., 64 (2003), 1-10.
|
|
[6]
|
Int. J. Biomath., 5 (2012), 1250059, 19pp.
|
|
[7]
|
Hifr Co, Edmonton, 1980.
|
|
[8]
|
Amer. Naturalist, 151 (1998), 487-496.
|
|
[9]
|
Nonlinear Anal. Hybrid Syst., 15 (2015), 98-111.
|
|
[10]
|
SIAM J. Appl. Math., 72 (2012), 1524-1548.
|
|
[11]
|
Nonlinear Dynam., 73 (2013), 815-826.
|
|
[12]
|
Int. J. Biomath., 7 (2014), 1450035, 21pp.
|
|
[13]
|
Oikos, 82 (1998), 384-392, http://www.jstor.org/stable/3546980.
|
|
[14]
|
Ecol. Model., 154 (2002), 1-7.
|
|
[15]
|
Commun. Nonlinear Sci. Numer. Simulat., 23 (2015), 78-88.
|
|
[16]
|
Biosystems, 104 (2011), 77-86.
|
|
[17]
|
Nonlinear Dynam., 65 (2011), 1-10.
|
|
[18]
|
J. Math. Biol., 62 (2011), 291-331.
|
|
[19]
|
Int. J. Biomath., 6 (2013), 1350031, 15pp.
|
|
[20]
|
Appl. Math. Comput., 237 (2014), 282-292.
|
|
[21]
|
Nonlinear Dynam., 76 (2014), 1109-1117.
|
|
[22]
|
Shanghai Science and Technology Press, Shanghai, 1984. (in Chinese)
|
|
[23]
|
Math. Biosci., 238 (2012), 55-64.
|
|
[24]
|
Theor. Popul. Biol., 67 (2005), 23-31.
|
|
[25]
|
Math. Biosci., 189 (2004), 103-113.
|
DownLoad: