Permanence of a Lotka-Volterra Predator-Prey Model with Feedback Controls and Prey Diffusion
Mathematics Letters
Volume 4, Issue 1, March 2018, Pages: 6-13
Received: Jan. 19, 2018; Accepted: Feb. 6, 2018; Published: Feb. 28, 2018
Views 1120      Downloads 96
Authors
Shuang Pan, College of Automation, Chongqing University of Posts and Telecommunications, Chongqing, P. R. China; Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing, P. R. China
Yonghong Li, Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing, P. R. China
Changyou Wang, Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing, P. R. China
Article Tools
Follow on us
Abstract
This paper is concerned with a multi-delay three-species predator-prey model with feedback controls and prey diffusion. By developing some new analysis techniques and using the comparison principle of differential equations, we obtained some new sufficient conditions which ensure the system to be permanent.
Keywords
Predator-Prey Model, Feedback Control, Time Delay, Diffusion, Permanence
To cite this article
Shuang Pan, Yonghong Li, Changyou Wang, Permanence of a Lotka-Volterra Predator-Prey Model with Feedback Controls and Prey Diffusion, Mathematics Letters. Vol. 4, No. 1, 2018, pp. 6-13. doi: 10.11648/j.ml.20180401.12
Copyright
Copyright © 2018 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
[1]
SONG X, CHEN L. Persistence and periodic orbits for two-species predator-prey system with diffusion. Canadian Applied Mathematics Quarterly, 1998, 6 (3): 233-244.
[2]
SONG X, CHEN L. Persistence and global stability for nonautonomous predator-prey system with diffusion and time delay. Computers & Mathematics with Applications, 1998, 35 (6): 33-40.
[3]
CUI J. The Effect of Dispersal on Permanence in a Predator-Prey Population Growth Model. Computers and Mathematics with Applications, 2002, 44 (8):1085-1097.
[4]
CHEN F, XIE X. Permanence and Extinction in Nonlinear Single and Multiple Species System with Diffusion. Applied Mathematics and Computation, 2006, 177 (1): 410-426.
[5]
ZHANG F, ZHAO X. Global Dynamics of a Nonautonomous Predator-Prey System with Dispersion. Mathematical Analysis, 2007, 14 (1): 81-87.
[6]
WEI F, LIN Y, QUE L, et al. Periodic Solution and Global Stability for a Nonautonomous Competitive Lotka–Volterra Diffusion System. Applied Mathematics and Computation, 2010, 216 (10): 3097-3104.
[7]
MUHAMMADHAJI A, TENG Z, REHIM M. Dynamical Behavior for a Class of Delayed Competitive-Mutulism Systems. Differential Equations and Dynamical Systems, 2015, 23 (3): 281-301.
[8]
XU R, CHAPLAIN M, DAVIDSON F A. Periodic Solution of a Lotka–Volterra Predator–Prey Model with Dispersion and Time Delays. Applied Mathematics and Computation, 2004, 148 (2): 537-560.
[9]
ZHOU X, SHI X, SONG X. Analysis of Nonautonomous Predator-Prey Model with Nonlinear Diffusion and Time Delay. Applied Mathematics and Computation, 2008, 196(1): 129-136.
[10]
ZHANG Z, WANG Z. Periodic Solutions of a Two-Species Ratio-Dependent Predator-Prey System With TimeDelay in a Two-Patch Environment. Anziam Journal, 2003, 45 (2): 233-244.
[11]
LIANG R, SHEN J. Positive Periodic Solutions for Impulsive Predator–Prey Model with Dispersion and Time Delays. Applied Mathematics and Computation, 2010, 217 (2): 661-676.
[12]
MUHAMMADHAJI A, MAHEMUTI R, TENG Z. On a Periodic Predator-Prey System with Nonlinear Diffusion and Delays. Afrika Matematika, 2016, 27 (7-8): 1179-1197.
[13]
GOPALSAMY K, WENG P. Global Attractivity in a Competition System with Feedback Controls. Computers and Mathematics with Applications, 2003, 45 (4-5): 665-676.
[14]
CHEN F. The Permanence and Global Attractivity of Lotka–Volterra Competition System with Feedback Controls. Nonlinear Analysis: Real World Applications, 2006, 7 (1): 133-143.
[15]
NIE L, TENGA Z, HU L, et al. Permanence and Stability in Nonautonomous Predator–Prey Lotka–Volterra Systems with Feedback Controls. Computers and Mathematics with Applications, 2009, 58 (3): 436-448.
[16]
CHEN F, GONG X, PU L, et al. Dynamic Behaviors of a Lotka-Volterra Predator-Prey System with Feedback Controls. Journal of Biomathematics, 2015, 30 (2): 328-332. (In chinese).
[17]
DING X, FANGFANGWANG. Positive Periodic Solution for a Semi-Ratio-Dependent Predator–Prey System with Diffusion and Time Delays. Nonlinear Analysis: RealWorld Applications, 2008, 9 (2): 239-249.
[18]
GOPALSAMY K, WENG P. Feedback Regulation of Logistic Growth. International Journal of Mathematics & Mathematical Sciences, 1993, 16 (1): 177-192.
[19]
XU J, CHEN F. Permanence of a Lotka-Volterra Cooperative System with Time Delays and Feedback Controls. Communications in Mathematical Biology & Neuroscience, 2015, 18. (2): 226-237.
[20]
XIE W, WENG P. Existence of Periodic Solution for a Predator - Prey Model with Patch - Diffusion and Feedback Control. Journal of South China Normal University (Natural Science Edition), 2012, 44 (1): 42-47. (In Chinese).
[21]
CHEN F. On a Nonlinear Nonautonomous Predator–Prey Model with Diffusion and Distributed Delay. Journal of Computational and Applied Mathematics, 2005, 180 (1): 33-49.
[22]
NAKATA Y, MUROYA Y. Permanence for Nonautonomous Lotka–Volterra Cooperative Systems with Delays. Nonlinear Analysis: Real World Applications, 2010, 11(1): 528-534.
ADDRESS
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
U.S.A.
Tel: (001)347-983-5186