Journal of Guangxi Normal University(Natural Science Edition) ›› 2021, Vol. 39 ›› Issue (2): 125-131.doi: 10.16088/j.issn.1001-6600.2019050801

Previous Articles     Next Articles

Global Existence of Solutions for a Three Species Predator-prey Model with Cross-diffusion

YAN Sha   

  1. School of Mathematics and Computer Science, Shaanxi University of Technology, Hanzhong Shaanxi 723000, China
  • Received:2019-05-08 Revised:2019-09-23 Online:2021-03-25 Published:2021-04-15

PDF (PC)

377

Abstract: In this paper, using the energy estimates and the bootstrap arguments, the global existence of solutions for a three species predator-prey model with cross-diffusion is proved when the space dimension is less than 10. Under certain conditions for the coefficients of the reaction functions, the convergence of solutions for the system is discussed by constructing Lyapunov function.

Key words: predator-prey, cross-diffusion, global solutions, existence, convergence

CLC Number: 

  • O175.26
[1] 陈兰荪.数学生态学模型与研究方法[M].成都:四川科学技术出版社,2003.
[2] 叶其孝,李正元.反应扩散方程引论[M]. 北京:科学出版社,1999.
[3] 肖燕妮, 周义仓, 唐三一. 生物数学原理[M]. 西安: 西安交通大学出版社,2012.
[4] ARMSTRONG R A,MCGHEE R.Competitive exclution[J]. The American Naturalist,1980, 115(2): 151-170.DOI:10.1086/283553.
[5] ABRAMSP A, BRASSIL C E, Holt R D.Dynamics and responses to mortality rates of competing predators undergoing predator-prey cycles[J].Theoretical Population Biology, 2003, 64(2):163-176.
[6] 闫莎.含一个食饵和两个竞争捕食者模型解的整体性态[J].陕西理工学院学报(自然科学版),2012,28(2):59-62.DOI: 10.3969/j.issn.1673-2944.2012.02.013.
[7] 伏升茂, 闫莎.含一个食饵和两个竞争捕食者种群模型解的整体性态[J].西北师范大学学报(自然科学版), 2010,46(3):1-5.DOI: 10.3969/j.issn.1001-988X.2010.03.001.
[8] 闫莎.一类三种群捕食者-食饵交错扩散模型整体解的存在性[J].纯粹数学与应用数学,2011,27(2):226-235.DOI: 10.3969/j.issn.1008-5513.2011.02.015.
[9] 郭凌, 伏升茂.具有Holling III 类功能反应的捕食者-食饵扩散模型的稳定性[J].兰州大学学报(自然科学版), 2008, 44(2):107-110.DOI: 10.13885/j.issn.0455-2059.2008.02.012.
[10] 伏升茂, 屈菲.非线性密度制约的Holling-Ⅲ型捕食者-食饵扩散模型的稳定性[J].西北师范大学学报(自然科学版),2013,49(3):1-4,9. DOI: 10.16783/j.cnki.nwnuz.2013.03.001.
[11] 胡晓丽, 伏升茂.带 Lotka-Volterra 互惠源的多种群趋化模型的稳定性[J].系统科学与数学, 2017, 37(6): 1541-1554.
[12] 冯金明,李遵先.一类具扩散的传染病模型的稳定性分析[J].广西师范大学学报(自然科学版),2018,36(2):63-68.DOI: 10.16088/j.issn.1001-6600.2018.02.009.
[13] 薛晋栋,冯春华.一类时滞脉冲 Lotka-Volterra系统的概周期解[J].广西师范大学学报(自然科学版),2014,32(1):69-73.DOI: 10.16088/j.issn.1001-6600.2014.01.006.
[14] 罗颜涛, 张龙,滕志东.一类间歇时滞扩散的概周期捕食系统的持久性[J].广西师范大学学报(自然科学版),2017,35(2):50-57. DOI: 10.16088/j.issn.1001-6600.2017.02.008.
[15] FU S M, WEN Z J, CUI S B.Uniform boundedness and stability of global solutions in a strongly coupled three-species cooperating model[J].Nonlinear Analysis-Real Word Applications, 2008, 9(2): 272-289. DOI: 10.1016/j.nonrwa.2006.10.003.
[16] 许生虎, 许万银.具有性别结构的交错扩散捕食者-食饵模型整体解的存在性和稳定性[J].应用数学, 2010, 23(3):482-490.
[17] YANG F, FU S M.Global solution for a tritrophic food chain model with diffusion[J].Rocky Mountain Journal of Mathematics, 2008, 38(5):1785-1812.
[18] 伏升茂, 高海燕,崔尚斌.竞争-竞争-互惠交错扩散模型的整体解[J]. 数学学报,2008,51(1):153-164.DOI: 10.3321/j.issn:0583-1431.2008.01.019.
[19] AMANN H.Dynamic theory of quasilinear parabolic systems:Ⅲ global existence[J]. Mathematische Zeitschrift,1989,202:219-250.
[20] CHOI Y S,LUI R,YAMADA Y.Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion[J]. Discrete and Continuous Dynamical Systems, 2004,10(3):719-730.DOI: 10.3934/dcds.2004.10.719.
[21] TUOV P V.On global existence of solutions to a cross-diffusion system[J]. Journal of Mathematical Analysis and Applications, 2008, 343(2): 826-834.DOI: 10.1016/j.jmaa.2008.01.089.
[22] LADYZENSKAJA O A,SOLONNIKOV V A,Uralceva N N.Linear andquasilinear equations of parabolic type[M].Washington DC:American Mathematical Society,1968.
[1] ZUO Jiabin, YUN Yongzhen. Anti-periodic Boundary Value Problem for a Class of Fractional Differential Equations [J]. Journal of Guangxi Normal University(Natural Science Edition), 2020, 38(6): 56-64.
[2] ZHU Yaping, QU Guorong, FAN Jianghua. The Existence of Solutions for Quasi-variational Inequalities by Using the Fixed Point Index Approach [J]. Journal of Guangxi Normal University(Natural Science Edition), 2019, 37(4): 79-85.
[3] ZHANG Lisheng, ZHANG Zhiyong, MA Kaihua, LI Guofang. Studying Oscillations in Convection Cahn-Hilliard System with Improved Lattice Boltzmann Model [J]. Journal of Guangxi Normal University(Natural Science Edition), 2019, 37(2): 15-26.
[4] TANG Guoji. Solvability for Generalized Mixed Variational Inequalities with Perturbation [J]. Journal of Guangxi Normal University(Natural Science Edition), 2018, 36(1): 76-83.
[5] LEI Qingzhu,QIN Yongsong,LUO Min. Empirical Bayes Estimation and Test for Scale ExponentialFamilies under Strong Mixing Samples [J]. Journal of Guangxi Normal University(Natural Science Edition), 2017, 35(3): 63-74.
[6] LUO Yantao, ZHANG Long, TENG Zhidong. Permanence of an Almost Periodic Predator-prey System withIntermit-tent Dispersal and Dispersal Delays between Patches [J]. Journal of Guangxi Normal University(Natural Science Edition), 2017, 35(2): 50-57.
[7] HUANG Kaijiao, XIAO Feiyan. A Stochastic Predator-prey System with Lévy Jumps [J]. Journal of Guangxi Normal University(Natural Science Edition), 2017, 35(2): 66-72.
[8] HAN Cai-hong, LI Lüe, PANG Lin-na, HOU Xin-xin. Global Attractivity of the Max-Type Difference Equation xn=max${\frac{1}{x_{n-k}^{\alpha}},\frac{A_n}{x^{\beta}_{n-k-2}}}$ [J]. Journal of Guangxi Normal University(Natural Science Edition), 2015, 33(3): 71-74.
[9] ZHANG Mei-yue. Some New Results for the Electron Beams Focusing System Model [J]. Journal of Guangxi Normal University(Natural Science Edition), 2015, 33(1): 38-44.
[10] HAN Cai-hong, LI Lue, HUANG Rong-li. Dynamics of the Difference Equation xn+1=pn+xnxn-1 [J]. Journal of Guangxi Normal University(Natural Science Edition), 2013, 31(1): 44-47.
[11] ZHAO Wei-qi, LIANG Jia-rong, LI Xia. Finite-time Terminal Sliding Mode Control for Singular Systems [J]. Journal of Guangxi Normal University(Natural Science Edition), 2011, 29(4): 73-78.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] HU Jinming, WEI Duqu. Research on Generalized Sychronization of Fractional-order PMSM[J]. Journal of Guangxi Normal University(Natural Science Edition), 2020, 38(6): 14 -20 .
[2] ZHU Yongjian, LUO Jian, QIN Yunbai, QIN Guofeng, TANG Chuliu. A Method for Detecting Metal Surface Defects Based on Photometric Stereo and Series Expansion Methods[J]. Journal of Guangxi Normal University(Natural Science Edition), 2020, 38(6): 21 -31 .
[3] YANG Liting, LIU Xuecong, FAN Penglai, ZHOU Qihai. Research Progress in Vocal Communication of Nonhuman Primates in China[J]. Journal of Guangxi Normal University(Natural Science Edition), 2021, 39(1): 1 -9 .
[4] BIN Shiyu, LIAO Fang, DU Xuesong, XU Yilan, WANG Xin, WU Xia, LIN Yong. Research Progress on Cold Tolerance of Tilapia[J]. Journal of Guangxi Normal University(Natural Science Edition), 2021, 39(1): 10 -16 .
[5] LIU Jing, BIAN Xun. Characteristics of the Orthoptera Mitogenome and Its Application[J]. Journal of Guangxi Normal University(Natural Science Edition), 2021, 39(1): 17 -28 .
[6] LI Xingkang, ZHONG Enzhu, CUI Chunyan, ZHOU Jia, LI Xiaoping, GUAN Zhenhua. Monitoring Singing Behavior of Western Black Crested Gibbon (Nomascus concolor furvogaster)[J]. Journal of Guangxi Normal University(Natural Science Edition), 2021, 39(1): 29 -37 .
[7] HE Xinming, XIA Wancai, BA Sang, LONG Xiaobin, LAI Jiandong, YANG Chan, WANG Fan, LI Dayong. Grooming Strategies of Resident Males with Different Number of Mates in Yunnan Snub-nosed Monkeys (Rhinopithecus bieti)[J]. Journal of Guangxi Normal University(Natural Science Edition), 2021, 39(1): 38 -44 .
[8] FU Wen, REN Baoping, LIN Jianzhong, LUAN Ke, WANG Pengcheng, WANG Bing, LI Dayong, ZHOU Qihai. Jiyuan Taihang Mountain Macaque Population and Conservation Status[J]. Journal of Guangxi Normal University(Natural Science Edition), 2021, 39(1): 45 -52 .
[9] ZHENG Jingjin, LIANG Jipeng, ZHANG Kechu, HUANG Aimian, LU Qian, LI Youbang, HUANG Zhonghao. White-headed Langurs Select Foods Based on Woody Plants' Dominances[J]. Journal of Guangxi Normal University(Natural Science Edition), 2021, 39(1): 53 -64 .
[10] YANG Chan, WAN Yaqiong, HUANG Xiaofu, YUAN Xudong, ZHOU Hongyan, FANG Haocun, LI Dayong, LI Jiaqi. Activity Rhythm of Muntiacus reevesi Based on Infrared Camera Technology[J]. Journal of Guangxi Normal University(Natural Science Edition), 2021, 39(1): 65 -70 .
Copyright © Journal of Guangxi Normal University(Natural Science Edition), All Rights Reserved.
Tel: 0773-5857325　E-mail: gxsdzkb@mailbox.gxnu.edu.cn
Powered by Beijing Magtech Co. Ltd