您的位置：山东大学 -> 科技期刊社 -> 《山东大学学报（理学版）》

《山东大学学报(理学版)》 ›› 2019, Vol. 54 ›› Issue (9): 54-61.doi: 10.6040/j.issn.1671-9352.0.2018.667

• • 上一篇    下一篇

一类具有密度制约的捕食-食饵扩散模型的定性分析

李海侠   

  1. 宝鸡文理学院数学与信息科学学院, 陕西 宝鸡 721013
  • 出版日期:2019-09-20 发布日期:2019-07-30
  • 作者简介:李海侠(1977— ),女,博士,副教授,研究方向为偏微分方程及计算可视化. E-mail:xiami0820@163.com
  • 基金资助:
    国家自然科学基金资助项目(11801013);陕西省自然科学基础研究计划项目(2014JQ2-1003);陕西省教育厅专项(16JK1046);宝鸡市科技计划项目(2018JH-20);宝鸡文理学院博士科研项目(ZK2018069);渭南师范学院科研项目(16ZRRC05)

Qualitative analysis of a diffusive predator-prey model with density dependence

LI Hai-xia   

  1. Institute of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji 721013, Shaanxi, China
  • Online:2019-09-20 Published:2019-07-30

PDF (PC)

569

摘要： 研究了一类具有密度制约的捕食-食饵模型共存解的唯一性和渐近行为。利用不动点指数理论给出了共存解存在的充分条件。然后采用线性算子的扰动理论讨论了共存解的唯一性和稳定性。最后运用抛物方程的比较原理得到了系统的持久性和灭绝性条件,并通过数值模拟对其理论结果进行了验证。结果表明在一定条件下两物种共存且系统存在稳定的唯一共存解。

关键词: 捕食-食饵扩散模型, 密度制约, 不动点指数, 扰动理论, 唯一性, 渐近行为

Abstract: The uniqueness of coexistence solutions and asymptotic behavior for a diffusive predator-prey model with density dependence are studied. The sufficient conditions of the existence of coexistence solutions are given by means of the fixed point index theory. Then, by making use of the perturbation theory for linear operators, we discuss the stability and uniqueness of coexistence solutions. Finally, the conditions of the extinction and permanence for the system by the comparison principle for parabolic equations, and the theoretical results of asymptotic behavior for the system are verified by some numerical simulations. The results show that the two species can coexist and the system has a unique coexistence solution under certain conditions.

Key words: a diffusive predator-prey model, density dependence, fixed point index, perturbation theory, uniqueness, asymptotic behaviour

中图分类号: 

  • O175.26
[1] WU J H. Stability and persistence for prey-predator model with saturation[J].Chinese Science Bulletin,1998, 43(24):2102-2103.
[2] DU Y H, SHI J P. A diffusive predator-prey model with a protection zone[J]. Journal of Differential Equations, 2006, 229(1):63-91.
[3] PENG R, WANG M X. Note on a ratio-dependent predator-prey system with diffusion[J]. Nonlinear Analysis: Real World Applications, 2006, 1(7):1-11.
[4] KO W, KYU K. Qualitative analysis of a predator-prey model with Holling type-II functional response incorporating a prey refuge[J]. Journal of Differential Equations, 2006, 231(2):534-550.
[5] WANG Z G, WU J H. Qualitative analysis for a ratio-dependent predator-prey model with stage structure and diffusion[J]. Nonlinear Analysis, 2008, 9(5):2270-2287.
[6] PENG R, SHI J P. Non-existence of non-constant positive steady states of two Holling type-II predator-prey systems: strong interaction case[J]. Journal of Differential Equations, 2009, 247(3):866-886.
[7] YI F Q, WEI J J, SHI J P. Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system[J]. Journal of Differential Equations, 2009, 246(5):1944-1977.
[8] JIA Y F, WU J H, NIE H. The coexistence states of a predator-prey model with non-monotonic functional response and diffusion[J]. Acta Applicandae Mathematicae, 2009, 108(2):413-428.
[9] GUO G H, WU J H. Multiplicity and uniqueness of positive solutions for a predator-prey model with B-D functional response[J]. Nonlinear Analysis, 2010, 72(3/4):1632-1646.
[10] WEI M H, WU J H, GUO G H. The effect of predator competition on positive solutions for a predator-prey model with diffusion[J]. Nonlinear Analysis, 2012, 75(13):5053-5068.
[11] YANG W S, LI Y Q. Dynamics of a diffusive predator-prey model with modified Leslie-Gower and Holling-type III schemes[J]. Computers and Mathematics with Applications, 2013, 65(11):1727-1737.
[12] ZHOU J, SHI J P. The existence, bifurcation and stability of positive stationary solutions of a diffusive Leslie-Gower predator-prey model with Holling-type II functional responses[J]. Journal of Mathematical Analysis and Applications, 2013, 405(2):618-630.
[13] LI H X. Asymptotic behavior and multiplicity for a diffusive Leslie-Gower predator-prey system with Crowley-Martin functional response[J]. Computers and Mathematics with Applications, 2014, 68(7):693-705.
[14] 李海侠.带有保护区域的加法Allee效应捕食-食饵模型的共存解[J].山东大学学报(理学版),2015,50(5):9-15. LI Haixia. Coexistence solutions for a predator-prey model with additive Allee effect and a protection zone[J]. Journal of Shandong University(Natural Science), 2015, 50(5):9-15.
[15] JIANG H L, WANG L J, YAO R F. Numerical simulation and qualitative analysis for a predator-prey model with B-D functional response[J]. Mathematics and Computers in Simulation, 2015, 117(2):39-53.
[16] LI S B, WU J H. Qualitative analysis of a predator-prey model with predator saturation and competition[J]. Acta Applicandae Mathematicae, 2016, 141(1):165-185.
[17] LI H X, LI Y L,YANG W B. Existence and asymptotic behavior of positive solutions for a one-prey and two-competing-predators system with diffusion[J]. Nonlinear Analysis: Real World Applications, 2016, 27:261-282.
[18] 李海侠. 一类食物链模型正解的稳定性和唯一性[J].数学物理学报A辑,2017,37(6):1094-1104. LI Haixia. Stability and uniqueness of positive solutions for a food-chain model[J]. Acta Mathematica Scientia-Series A, 2017, 37(6):1094-1104.
[19] HUANG K G, CAI L, FU S M. Positive steady states of a density-dependent predator-prey model with diffusion[J]. Discrete and Continuous Dynamical Systems-Series B, 2018, 23(8):3087-3107.
[20] DANCER E N. On the indices of fixed points of mapping in cones and applications [J]. Journal of Mathematical Analysis and Applications, 1983, 91(1):131-151.
[21] PAO C V. Quasisolutions and global attractor of reaction-diffusion systems[J]. Nonlinear Analysis, 1996, 26(12): 1889-1903.
[1] 任建龙. 利用内部温度史重构表面未知热通量[J]. 《山东大学学报(理学版)》, 2019, 54(9): 83-90.
[2] 张伟杰,王新利,王汉杰. 亚纯函数微分多项式IM分担值的唯一性[J]. 《山东大学学报(理学版)》, 2019, 54(8): 90-96.
[3] 桑彦彬,陈娟,任艳. 带有Hardy项的奇异p-重调和方程正解的唯一性[J]. 《山东大学学报(理学版)》, 2019, 54(6): 75-80.
[4] 章欢,李永祥. 含时滞导数项的高阶常微分方程的正周期解[J]. 《山东大学学报(理学版)》, 2019, 54(4): 29-36.
[5] 王素云,李永军. 带超越共振点非线性项的二阶常微分方程边值问题的可解性[J]. 山东大学学报（理学版）, 2018, 53(6): 53-56.
[6] 肖新玲. 由马氏链驱动的正倒向随机微分方程[J]. 山东大学学报（理学版）, 2018, 53(4): 46-54.
[7] 甄苇苇,曾剑,任建龙. 基于变分理论与时间相关的抛物型反源问题[J]. 山东大学学报（理学版）, 2018, 53(10): 61-71.
[8] 闫东亮. 带有导数项的二阶周期问题正解[J]. 山东大学学报（理学版）, 2017, 52(9): 69-75.
[9] 张莎,贾梅,李燕,李晓晨. 分数阶脉冲微分方程三点边值问题解的存在性和唯一性[J]. 山东大学学报（理学版）, 2017, 52(2): 66-72.
[10] 扈培础,吴爱迪. 关于L-函数的问题[J]. 山东大学学报（理学版）, 2016, 51(6): 1-15.
[11] 蔡超. 一类Kolmogorov型方程的系数反演问题[J]. 山东大学学报（理学版）, 2016, 51(4): 127-134.
[12] 董建伟,娄光谱,王艳萍. 一个半导体简化能量输运模型稳态解的唯一性[J]. 山东大学学报（理学版）, 2016, 51(2): 37-41.
[13] 郭丽君. 非线性微分方程三阶三点边值问题正解的存在性[J]. 山东大学学报（理学版）, 2016, 51(12): 47-53.
[14] 王先飞, 江龙, 马娇娇. 具有Osgood型生成元的多维倒向重随机微分方程[J]. 山东大学学报（理学版）, 2015, 50(08): 24-33.
[15] 徐嫚. 带双参数的脉冲泛函微分方程正周期解的存在性[J]. 山东大学学报（理学版）, 2015, 50(06): 69-74.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] 汤晓宏1，胡文效2*，魏彦锋2，蒋锡龙2，张晶莹2，. 葡萄酒野生酿酒酵母的筛选及其生物特性的研究[J]. 山东大学学报（理学版）, 2014, 49(03): 12 -17 .
[2] 曾文赋1,黄添强1,2,李凯1,余养强1,郭躬德1,2. 基于调和平均测地线核的局部线性嵌入算法[J]. J4, 2010, 45(7): 55 -59 .
[3] 郭文鹃,杨公平*,董晋利. 指纹图像分割方法综述[J]. J4, 2010, 45(7): 94 -101 .
[4] 何海伦， 陈秀兰*. 变性剂和缓冲系统对适冷蛋白酶MCP-01和中温蛋白酶BP-01构象影响的圆二色光谱分析何海伦， 陈秀兰*[J]. 山东大学学报（理学版）, 2013, 48(1): 23 -29 .
[5] 王碧玉,曹小红*. 算子矩阵的Browder定理的摄动[J]. 山东大学学报（理学版）, 2014, 49(03): 90 -95 .
[6] 丁超1,2, 元昌安1,3*, 覃晓1,3. 基于GEP的多数据流预测算法[J]. J4, 2010, 45(7): 50 -54 .
[7] 孟祥波1,张立东1,杜子平2. 均值-方差标准下带跳的保险公司投资与再保险策略[J]. 山东大学学报（理学版）, 2014, 49(05): 36 -40 .
[8] 彭振华,徐义红*,涂相求. 近似拟不变凸集值优化问题弱有效元的最优性条件[J]. 山东大学学报（理学版）, 2014, 49(05): 41 -44 .
[9] 胡明娣1,2,折延宏1,王敏3. L3*系统中逻辑度量空间的拓扑性质[J]. J4, 2010, 45(6): 86 -90 .
[10] 张京友,张培爱,钟海萍. 进化图论在知识型企业组织结构设计中的应用[J]. J4, 2013, 48(1): 107 -110 .
版权所有 © 2018 《山东大学学报(理学版)》编辑部　
地址: 山东省济南市山大南路27号山东大学中心校区（250100）　电话: +86-0531-88366917　E-mail: xblxb@sdu.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发