Spatial Turing pattern of a predator-prey system with nonlinear harvesting effect

ZHANG Dao-xiang; SUN Guang-xun; HU Wei; KAI Ge

doi:10.3969/j.issn.1000-5641.2018.04.002

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ZHANG Dao-xiang, SUN Guang-xun, HU Wei, KAI Ge. Spatial Turing pattern of a predator-prey system with nonlinear harvesting effect[J]. Journal of East China Normal University (Natural Sciences), 2018, (4): 9-22, 31. doi: 10.3969/j.issn.1000-5641.2018.04.002

Citation:

ZHANG Dao-xiang, SUN Guang-xun, HU Wei, KAI Ge. Spatial Turing pattern of a predator-prey system with nonlinear harvesting effect[J]. Journal of East China Normal University (Natural Sciences), 2018, (4): 9-22, 31. doi: 10.3969/j.issn.1000-5641.2018.04.002

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Spatial Turing pattern of a predator-prey system with nonlinear harvesting effect

doi: 10.3969/j.issn.1000-5641.2018.04.002

1.
School of Mathematics and Statistics, Anhui Normal University, Wuhu Anhui 241002, China
2.
College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing 100124, China

Received Date: 2017-09-20
Publish Date: 2018-07-25

Abstract

Abstract

We study the formation and selection of Turing patterns for a class of predator-prey systems with nonlinear harvesting effect. Firstly, the conditions of Turing instability induced by cross-diffusion terms are given by stability theory, and the existence region of Turing patterns of the system are obtained by bifurcation theory. Secondly, the amplitude equations of the system are derived using the multi-scales analysis method, and the selection results of Turing patterns are given. Finally, Matlab is used to simulate the pattern formation and selection results of the system. The results show that the system has rich Turing patterns, such as spot, stripe, and coexistence of the two types.
- predator-prey system,
- Hopf bifurcation,
- Turing patterns,
- amplitude equation

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References(16)

References

[1]	ZADORIN A S, RONDELEZ Y, GINES G, et al. Synthesis and materialization of a reaction-diffusion French flag pattern[J]. Nature Chemistry, 2017, 9(10):990-996. doi: 10.1038/nchem.2770
[2]	李新政, 白占国, 李燕, 等.双层非线性耦合反应扩散系统中复杂Turing斑图[J].物理学报, 2013, 62(22):45-51. http://www.cqvip.com/QK/94684X/201322/47915498.html
[3]	ZHAO H Y, HUANG X X, ZHANG X B. Turing instability and pattern formation of neural networks with reaction-diffusion terms[J]. Nonlinear Dynamics, 2014, 76(1):115-124. doi: 10.1007/s11071-013-1114-2
[4]	SONG Y L, YANG R, SUN G Q. Pattern dynamics in a Gierer-Meinhardt model with a saturating term[J]. Applied Mathematical Modelling, 2017, 46:476-491. doi: 10.1016/j.apm.2017.01.081
[5]	SUN G Q. Mathematical modeling of population dynamics with Allee effect[J]. Nonlinear Dynamics, 2016, 85(1):1-12. doi: 10.1007/s11071-016-2671-y
[6]	WANG X L, WANG W D, ZHANG G H. Vegetation pattern formation of a water-biomass model[J]. Commun Nonlinear Sci Numer Simulat, 2017, 42:571-584. doi: 10.1016/j.cnsns.2016.06.008
[7]	张道祥, 赵李鲜, 孙光讯, 等.一类带负交叉扩散项二维系统的空间Turing斑图[J].吉林大学学报(理学版), 2017, 55(3):537-546. http://www.cnki.com.cn/Article/CJFDTOTAL-XBDZ200504000.htm
[8]	SUN G Q, WANG C H, WU Z Y. Pattern dynamics of aGierer-Meinhardt model with spatial effects. Nonlinear Dynamics, 2017, 88(2):1385-1396. doi: 10.1007/s11071-016-3317-9
[9]	GUIN L N. Existence of spatial patterns in a predator-preymodel with self-and cross-diffusion. Applied Mathematics andComputation, 2014, 226:320-335. doi: 10.1016/j.amc.2013.10.005
[10]	张道祥, 赵李鲜, 胡伟.一类三种群食物链模型中交错扩散引起的Turing不稳定.山东大学学报(理学版), 2017, 52(1):88-97. doi: 10.6040/j.issn.1671-9352.0.2016.286
[11]	SAMBATH M, BALACHANRAN K, SUVINTHRA M. Stability and Hopfbifurcation of a diffusive predator-prey model with hyperbolicmortality. Complexity, 2016, 21:34-43. doi: 10.1002/cplx.21708
[12]	XIAO M, CAO J D. Hopf bifurcation and non-hyperbolic equilibrium ina ratio-dependent predator-prey model with linear harvesting rate:Analysis and computation. Mathematical and Computer Modelling, 2009, 50:360-379. doi: 10.1016/j.mcm.2009.04.018
[13]	ZHANG F R, LI Y. Stability and Hopf bifurcation of adelayed-diffusive predator-prey model with hyperbolic mortality andnonlinear prey harvesting. Nonlinear Dynamics, 2017, 88(2):1397-1412. doi: 10.1007/s11071-016-3318-8
[14]	LI Y. Dynamics of a delayed diffusive predator-prey modelwith hyperbolic mortality. Nonlinear Dynamics, 2016, 85(4):2425-2436. doi: 10.1007/s11071-016-2835-9
[15]	欧阳颀.非线性科学与斑图动力学导论.北京:北京大学出版社, 2010.
[16]	KUZNETSOV Y A. Elements of applied bifurcation theory. 2nd ed. New York: Springer, 1995.