具Michaelis-Menten型收获的Leslie-Gower捕食-食饵扩散模型的动力学和模式

数学物理学报 ›› 2022, Vol. 42 ›› Issue (5): 1575-1591.

具Michaelis-Menten型收获的Leslie-Gower捕食-食饵扩散模型的动力学和模式

马战平¹,霍海峰^2,*(),向红²

¹ 河南理工大学河南焦作 454003
² 兰州理工大学兰州 730050

收稿日期:2021-07-23 出版日期:2022-10-26 发布日期:2022-09-30
通讯作者: 霍海峰 E-mail:hfhuo@lut.edu.cn
基金资助:
国家自然科学基金(11861044);国家自然科学基金(11661050);甘肃省自然科学基金(21JR7RA212);甘肃省自然科学基金(21JR7RA535)

Dynamics and Patterns for a Diffusive Leslie-Gower Predator-Prey Model with Michaelis-Menten Type Harvesting in Prey

Zhanping Ma¹,Haifeng Huo^2,*(),Hong Xiang²

¹ School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan 454003
² Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050

Received:2021-07-23 Online:2022-10-26 Published:2022-09-30
Contact: Haifeng Huo E-mail:hfhuo@lut.edu.cn
Supported by:
the NSFC(11861044);the NSFC(11661050);the NSF of Gansu of China(21JR7RA212);the NSF of Gansu of China(21JR7RA535)

摘要/Abstract

摘要：

该文研究了食饵具有Michaelis-Menten型收获的Leslie-Gower捕食-食饵扩散模型的动力学行为和稳态模式. 首先证明了模型的一致持久性. 其次研究了非负常数平衡解及其稳定性, 分别利用Lyapunov函数和上下解两种方法证明得到了正常数平衡解全局渐近稳定的充分条件. 最后利用度理论研究了稳态模式. 研究结果表明: Michaelis-Menten型收项获对稳态模式的形成起着重要的作用, 这与模型没有收获项的结果形成了鲜明的对比.

关键词: Leslie-Gower扩散模型, Michaelis-Menten型收获, 一致持久, 全局渐近稳定, 稳态模式

Abstract:

This paper is devoted to study the dynamical properties and stationary patterns of a diffusive Leslie-Gower predator-prey model with Michaelis-Menten type harvesting in the prey population. We first prove the uniform persistence, and then study the nonnegative constant equilibrium solutions and their stabilities. Particularly, we obtain sufficient conditions of the global asymptotical stability of positive constant equilibrium solution by Lyapunov function method and the upper and lower solution method, respectively. Moreover, we investigate the stationary patterns induced (Turing pattern) by diffusion by degree theory. Our results show that Michaelis-Menten type harvesting in our model plays a crucial role in the formation of stationary patterns, which is a strong contrast to the case without harvesting.

Key words: Diffusive Leslie-Gower model, Michaelis-Menten type harvesting, Uniform persistence, Global asymptotical stability, Stationary patterns

中图分类号:

O175

马战平,霍海峰,向红. 具Michaelis-Menten型收获的Leslie-Gower捕食-食饵扩散模型的动力学和模式[J]. 数学物理学报, 2022, 42(5): 1575-1591.

Zhanping Ma,Haifeng Huo,Hong Xiang. Dynamics and Patterns for a Diffusive Leslie-Gower Predator-Prey Model with Michaelis-Menten Type Harvesting in Prey[J]. Acta mathematica scientia,Series A, 2022, 42(5): 1575-1591.

图/表 3

参考文献 30

1	Berryman A A . The orgins and evolution of predator-prey theory. Ecology, 1992, 73, 1530- 1535 doi: 10.2307/1940005
2	May R M . Limit cycles in predator-prey communities. Science, 1972, 177, 900- 902 doi: 10.1126/science.177.4052.900
3	Bacaër N . A Short History of Mathematical Population Dynamics. New York: Springer Verlag, 2011
4	Braza P A . The bifurcation structure of the Holling-Tanner model for predator-prey interactions using two-timing. SIAM J Appl Math, 2003, 63, 889- 904 doi: 10.1137/S0036139901393494
5	吴鑫, 袁荣, 马兆海. 非局部扩散Holling-Tanner捕食者-食饵系统的临界与非临界行波解分析. 数学物理学报, 2021, 41A (6): 1705- 1717 doi: 10.3969/j.issn.1003-3998.2021.06.011
	Wu X , Yuan R , Ma Z H . Analysis on critical waves and non-critical waves for Holling-Tanner predator-prey system with nonlocal diffusion. Acta Math Sci, 2021, 41A (6): 1705- 1717 doi: 10.3969/j.issn.1003-3998.2021.06.011
6	Guo S . Bifurcation and spatio-temporal patterns in a diffusive predator-prey system. Nonlinear Anal Real World Appl, 2018, 42, 448- 477 doi: 10.1016/j.nonrwa.2018.01.011
7	Huang J , Liu S , Ruan S , Xiao D . Bifurcations in a discrete predator-prey model with nonmonotonic functional response. J Math Anal Appl, 2018, 464, 201- 230 doi: 10.1016/j.jmaa.2018.03.074
8	Wang J , Wei J , Shi J . Global bifurcation analysis and pattern formation in homogeneous diffusive predator-prey systems. J Differential Equations, 2016, 260, 3495- 3523 doi: 10.1016/j.jde.2015.10.036
9	王雪臣, 魏俊杰. 时滞在具有Ivlev型功能反应函数的捕食者-食饵扩散系统中的作用. 数学物理学报, 2014, 34A (2): 234- 250
	Wang X C , Wei J J . The effect of delay on a diffusive predator-prey system with Ivlev-type functional response. Acta Math Sci, 2014, 34A (2): 234- 250
10	Du Y , Hsu S B . A diffusive predator-prey model in heterogeneous environment. J Differential Equations, 2004, 203, 331- 364 doi: 10.1016/j.jde.2004.05.010
11	Leslie P H , Gower J C . The properties of a stochastic model for the predator-prey type of interaction between two species. Biometrica, 1960, 47, 219- 234 doi: 10.1093/biomet/47.3-4.219
12	Hsu S B , Hwang T W . Global stability for a class of predator-prey systems. SIAM J Appl Math, 1995, 55, 763- 783 doi: 10.1137/S0036139993253201
13	Xiao D , Jennings L S . Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting. SIAM J Appl Math, 2005, 65, 737- 753 doi: 10.1137/S0036139903428719
14	Chen J , Huang J , Ruan S , Wang J . Bifurcations of invariant tori in predator-prey models with seasonal prey harvesting. SIAM J Appl Math, 2013, 73, 1876- 1905 doi: 10.1137/120895858
15	May R M , Beddington J R , Clark C W , Holt S J , Laws R M . Management of multispecies fisheries. Science, 1979, 205, 267- 277 doi: 10.1126/science.205.4403.267
16	Huang J , Gong Y , Ruan S . Bifurcation analysis in a predator-prey model with constant-yield predator harvesting. Discrete Contin Dynam Syst Ser B, 2013, 18, 2101- 2121
17	Beddington J R , May R M . Maximum sustainable yields in systems subject to harvesting at more than one trophic level. Math Biosci, 1980, 51, 261- 281 doi: 10.1016/0025-5564(80)90103-0
18	Beddington J R , Cooke J G . Harvesting from a prey-predator complex. Ecol Model, 1982, 14, 155- 177 doi: 10.1016/0304-3800(82)90016-3
19	Gong Y , Huang J . Bogdanov-Takens bifurcation in a Leslie-Gower predator-prey model with prey harvesting. Acta Math Appl Sinica Eng Ser, 2014, 30, 239- 244 doi: 10.1007/s10255-014-0279-x
20	Zhu C R , Lan K Q . Phase portraits, Hopf-bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates. Discrete Contin Dyn Syst Ser B, 2010, 14, 289- 306
21	Gupta R P , Banerjee M , Chandra P . Bifurcation analysis and control of Leslie-Gower predator-prey model with Michaelis-Menten type prey-harvesting. Differ Equ Dyn Syst, 2012, 20, 339- 366 doi: 10.1007/s12591-012-0142-6
22	Kar T K , Chaudhuri K S . Regulation of a prey-predator fishery by taxation: a dynamic reaction model. J Biol Syst, 2003, 11, 173- 187 doi: 10.1142/S0218339003000828
23	Clark C W . Aggregation and fishery dynamics: a theoretical study of schooling and the purse seine tuna fisheries. Fish Bull, 1979, 77, 317- 337
24	Wu J . Theory and Applications of Partial Functional Differential Equations. New York: Springer-Verlag, 1996
25	Turing A . The chemical basis of morphogenesis. Philos Trans R Soc Ser B, 1952, 237, 37- 72
26	Pao C V . Nonlinear Parabolic and Elliptic Equations. New York: Plenum Publishing Corporation, 1992,
27	Lou Y , Ni W M . Diffusion, self-diffusion and cross-diffusion. J Differential Equations, 1996, 131, 79- 131 doi: 10.1006/jdeq.1996.0157
28	Lin C S , Ni W M , Takagi I . Large amplitude stationary solutions to a chemotais systems. J Differential Equations, 1988, 72, 1- 27 doi: 10.1016/0022-0396(88)90147-7
29	Nirenberg L . Topics in Nonlinear Functional Analysis. Providence: American Mathematical Society, 2001,
30	张晓. Matlab微分方程高效解法: 谱方法原理与实现. 北京: 机械工业出版社, 2015
	Zhang X . Matlab Efficient Method of Differential Equation: Principle and Implementation of Spectral Method. Beijing: China Machine Press, 2015

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