GLOBAL ANALYSIS IN DELAYED RATIO-DEPENDENT GAUSE-TYPE PREDATOR-PREY MODELS

Shuang Guo; Weihua Jiang; Hongbin Wang

doi:10.11948/2017068

2017 Volume 7 Issue 3

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Shuang Guo, Weihua Jiang, Hongbin Wang. GLOBAL ANALYSIS IN DELAYED RATIO-DEPENDENT GAUSE-TYPE PREDATOR-PREY MODELS[J]. Journal of Applied Analysis & Computation, 2017, 7(3): 1095-1111. doi: 10.11948/2017068

Citation:

Shuang Guo, Weihua Jiang, Hongbin Wang. GLOBAL ANALYSIS IN DELAYED RATIO-DEPENDENT GAUSE-TYPE PREDATOR-PREY MODELS[J]. Journal of Applied Analysis & Computation, 2017, 7(3): 1095-1111. doi: 10.11948/2017068

GLOBAL ANALYSIS IN DELAYED RATIO-DEPENDENT GAUSE-TYPE PREDATOR-PREY MODELS

1 College of Teacher Education, Daqing Normal University, Daqing, 163712, Heilongjiang, China;
2 Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, Heilongjiang, China

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Abstract

Abstract

A class of three-dimensional delayed Gause-type predator-prey model with ratio-dependent is considered. Firstly, we present some results, including the boundedness of solutions and the permanence of system. Secondly, we construct a Lyapunov function to give the global asymptotically stable of the positive equilibrium under some parameter conditions. Finally, we analyed the influence of the time delay on the system and showed that the occurrence of small range of periodic motion.
- Gause-type model /
- ratio-dependent /
- permanence /
- global asymptotically stable /
- delay
MSC: 34A34;34D23;34C25

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References

[1]	R. Arditi and H. R. Akcakaya, Underestimation of mutual interference of predators, Oecologia, 1990, 83(3), 358-361. Google Scholar
[2]	R. Arditi and A. A. Berryman, The biological control paradox, Tree, 1991, 6(7), 32-32. Google Scholar
[3]	R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics:ratiodependence, J. Theor. Biol., 1989, 139(3), 311-326. Google Scholar
[4]	R. Arditi, L. R. Ginzburg and H. R. Akcakaya, Variation in plankton densities among lakes:a case for ratio-dependent predation models, Am. Nat., 1991, 138(5), 1287-1296. Google Scholar
[5]	J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 1975, 44(1), 331-340. Google Scholar
[6]	E. Beretta and Y. Kuang, Global analyses in some delayed ratio-dependent predator-prey systems, Nonlinear Anal-theor., 1998, 32(3), 381-408. Google Scholar
[7]	A. A. Berryman, The origins and evolution of predator-prey theory, Ecology, 1992, 73(5), 1530-1535. Google Scholar
[8]	H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York, 1980. Google Scholar
[9]	M. Freeze, Y. Chang, and W. Feng, Analysis of dynamics in a complex food chain with ratio-dependent functional response, J. Appl. Anal. Comput., 2014, 4(1), 69-87. Google Scholar
[10]	W. M. Getz, Population dynamics:a per capita resource approach, J. Theor. Biol., 1984, 108(4), 623-643. Google Scholar
[11]	J. M. Ginoux, B. Rossetto and J. L. Jamet, Chaos in a three-dimensional Volterra-Gause model of predator-prey type, Int. J. Bifurcat. Chaos., 2005, 15(5), 1689-1708. Google Scholar
[12]	A. A. Gomes, E. Manica and M. C. Varriale, Applications of chaos control techniques to a three-species food chain, Chaos, Soliton. Fract., 2008, 35(4), 432-441. Google Scholar
[13]	H. B. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Eng., 2006, 3(3), 513-525. Google Scholar
[14]	I. Hanski, The functional response of predators worries about scale, TREE, 1991, 6(5), 141-142. Google Scholar
[15]	S. B. Hsu, T. W. Hwang and Y. Kuang, A ratio-dependent food chain and its applications to biological control, Math. Biosci., 2003, 181(1), 55-83. Google Scholar
[16]	G. P. Hu and X. L. Li, Stability and Hopf bifurcation for a delayed predator-prey model disease in the prey. Chaos, Soliton. Fract., 2012, 45(3), 229-237. Google Scholar
[17]	J. H. Huang and X. F. Zou, Traveling wavefronts in diffusive and cooperative Lotka-Volterra system with delays. J. Math. Anal. Appl., 2002, 271(2), 455-466. Google Scholar
[18]	W. H. Jiang and J. J. Wei, Bifurcation analysis in a limit cycle oscillator with delayed feedback, Chaos, Soliton. Fract., 2005, 23(3), 817-831. Google Scholar
[19]	W. H. Jiang and J. J. Wei, Bifurcation analysis in van der Pol's oscillator with delayed feedback, J. Comput. Appl. Math., 2008, 213(2), 604-615. Google Scholar
[20]	C. Jost, O. Arino and R. Arditi, About deterministic extinction in ratiodependent predator-prey models, B. Math. Biol., 1999, 61(1), 19-32. Google Scholar
[21]	Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 1998, 36(4), 389-406. Google Scholar
[22]	M. Y. Li and H. Y. Shu, Global dynamics of an in-host viral model with intracellular delay, B. Math. Biol., 2010, 72(6), 1492-1505. Google Scholar
[23]	H. S. Mahato and M. Bohm, Global existence and uniqueness of solution for a system of semilinear diffusion-reaction equations, J. Appl. Anal. Comp., 2013, 3(4), 357-376. Google Scholar
[24]	M. C. Varriale and A. A. Gomes, A study of a three species food chain, Ecol. Model., 1998, 110(2), 119-133. Google Scholar
[25]	H. B. Wang and W. H. Jiang, Hopf-pitchfork bifurcation in van der Pol's oscillator with nonlinear delayed feedback, J. Math. Anal. Appl., 2010, 368(1), 9-18. Google Scholar
[26]	J. N. Wang and W. H. Jiang. Bogdanov-Takens singularity in the comprehensive national power model with time delays, J. Appl. Anal. Comp., 2013, 3(1), 81-94. Google Scholar
[27]	J. J. Wei and M. Y. Li, Global existence of periodic solutions in a tri-neuron network model with delays, Physica D., 2004, 198(1-2), 106-119. Google Scholar
[28]	D. M. Xiao and W. X. Li, Stability and bifurcation in a delayed ratio-dependent predator-prey system, P. Edinburgh Math. Soc., 2003, 46(1), 205-220. Google Scholar
[29]	R. Xu and M. A. J. Chaplain, Persistence and global stability in a delayed Gause-type predator-prey system without dominating instantaneous negative feedbacks, J. Math. Anal. Appl., 2002, 265(1), 148-162. Google Scholar
[30]	T. S. Yi and X. F. Zou, Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain, J. Diff. Eqs., 2011, 251(9), 2598-2611. Google Scholar