JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2018, Vol. 53 ›› Issue (12): 53-61.doi: 10.6040/j.issn.1671-9352.0.2018.262

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Coexistence solution of a predator-prey system with B-D functional response and toxin effects

FENG Xiao-zhou1, XU Min2, WANG Guo-hui3   

  1. 1. College of Science, Xian Technological University, Xian 710021, Shaanxi, China;
    2. Shaanxi Aerospace Electromechanical Environment Engineering Design Institute Co., Ltd., Xian 710100, Shaanxi, China;
    3. College of Optoelectronic Engineering, Xian Technological University, Xian 710021, Shaanxi, China
  • Online:2018-12-20 Published:2018-12-18

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Abstract: The steady state of a predator-prey system with B-D reaction term and toxin effect under the homogeneous Dirichlet boundary condition is investigated. First, using the principle of maximum value and the comparison principle of eigenvalues, we give some prior estimates of coexistence solution on the system and obtain the necessary condition of non-coexistence solution. Secondly, by using the Leray-Schauder degree theory, the calculation of the fixed point index, the maximum principle and the method of upper and lower solutions, the sufficient condition for the existence of the coexistence solution is established. Finally, the local asymptotic stability of the trivial solution and the semi-trivial solution of the steady state system is proved by using the linearization operator and the Riesz-Schauder theory.

Key words: B-D functional response, predator-prey system, coexistence, stability

CLC Number: 

  • O175.26
[1] WU J H. Coexistence states for cooperative model with diffusion[J]. Computers and Mathematical with Applications, 2002, 43(14):1277-1290.
[2] ZHU C R, LAN K Q. Phase portraits, Hopf bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates[J]. Discrete and Continuous Dynamical Systems: Series B, 2010, 14(1):289-306.
[3] ZHANG N, CHEN F D, SU Q Q. Dynamic behaviors of a harvesting Leslie-Gower predator-prey model[J]. Discrete Dynamics in Nature and Society, 2011, 15(1):309-323.
[4] 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.
[5] WU J H. Maximal attractor, stability and persistence for prey-predator model with saturation[J]. Mathematical and Computer Modelling, 1999, 30(11):7-16.
[6] DAS T, MUKHERJEE R N, CHAUDHURI K S. Harvesting of a prey-predator fishery in the presence of toxicity[J]. Applied Mathematics Modeling, 2009, 33(5):2282-2292.
[7] 霍海峰,姜慧敏,苏克所.霉素影响下的捕食-食饵模型最优控制问题[J].甘肃科学学报,2010,22(1):18-23. HUO Haifeng, JIANG Huimin, SU Kesuo. An optimal harvesting problem of a prey-predator model in the presence of toxieity[J]. Journal of Gansu Sciences, 2010, 22(1):18-23.
[8] 沈莉莉,赵维锐.一类具有时滞和毒素的功能性反应的植物-食草动物系统性态分析[J].湖北民族学院学报(自然科学版),2010,28(1):201-208. SHEN Lili, ZHAO Weirui. Analysis of a plant-herbivore model with time-delay and tox in determined functional response[J]. Journal of Hubei University for Nationalities(Natural Science Edition), 2010, 28(1):201-208.
[9] 范学良,雒志学,张宇功.毒素影响下具有阶段结构的食饵捕食种群系统生存研究[J].宁夏大学学报(自然科学版),2014,35(3):41-45. FAN Xueliang, LUO Zhixue, ZHANG Yugong. Analysis of prey-predator system dynamics behavior with a stage structure in the effects of toxicants[J]. Journal of Ningxia University(Natural Science Edition), 2014, 35(3):41-45.
[10] 陈显,王稳地,陈晓平.毒素对阶段结构的单种群持续生存的影响[J].西南师范大学学报(自然科学版),2011,36(3):54-59. CHEN Xian, WANG Wendi, CHEN Xiaoping. Effects on survival of stage structural single population with toxicant[J]. Journal of Southwest China Normal University(Natural Science Edition), 2011, 36(3):54-59.
[11] BEDDINGTON J R. Mutual interference between parasites or predators and its effect on searching efficiency[J]. J Animal Ecol, 1975, 44(1):331-340.
[12] DEANGELIS D T, GOLDSTEIN R A. A model for trophic interaction[J]. Ecology, 1975, 56(2):881-892.
[13] 叶其孝,李正元,王明新,等.反应扩散方程引论[M]. 北京: 科学出版社, 2013: 1-56. YE Qixiao, LI Zhengyuan, WANG Mingxin, et al. Introduction to reaction-diffusion equations[M]. Beijing: Science Press, 2013: 1-56.
[14] 姜洪领.一类蜘蛛-昆虫模型平衡态正解的存在性[J].中山大学学报(自然科学版),2016,55(3):64-68. JIANG Hongling. The existence of steady-state positive solutions for a spider-insect model[J]. Acta Scientiarum Naturalium Universitatis Sunyatseni, 2016, 55(3):64-68.
[15] RYU Kimun, AHN Inkyung. Positive solution for ratio-dependent predator-prey interaction systems[J]. Journal of Differential Equations, 2005, 218(1):117-135.
[16] 袁海龙,李艳玲.一类捕食-食饵模型共存解的存在性与稳定性[J].陕西师范大学学报(自然科学版),2014,42(1):15-18. YUAN Hailong, LI Yanling. Coexistence of existence and stability of a predator-prey model[J]. Journal of Shaanxi Normal University(Natural Science Edition), 2014, 42(1):15-18.
[17] KO W, RYU K. Coexistence states of a predator-prey system with non-monotonic functional response[J]. Nonlinear Anal RWA, 2007, 8(3):769-786.
[18] RUAN W H, FENG W. On the fixed point index and multiple steady-state solutions of reaction-diffusion systems[J]. Differential Integral Equations, 1995, 8(2):371-392.
[19] YAMADA Y. Stability of steady states for prey-predator diffusion eduations with homogeneous dirichlet conditions[J]. SIAM J Math Anal, 1990, 21(2):327-345.
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