DYNAMIC ANALYSIS OF A NON-AUTONOMOUS RATIO-DEPENDENT PREDATOR-PREY MODEL WITH ADDITIONAL FOOD

Jie Song; Mi Hu; Yuzhen Bai; Yonghui Xia

doi:10.11948/2018.1893

2018 Volume 8 Issue 6

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2018 > 8(6): 1893-1909

Next Article Previous Article

Jie Song, Mi Hu, Yuzhen Bai, Yonghui Xia. DYNAMIC ANALYSIS OF A NON-AUTONOMOUS RATIO-DEPENDENT PREDATOR-PREY MODEL WITH ADDITIONAL FOOD[J]. Journal of Applied Analysis & Computation, 2018, 8(6): 1893-1909. doi: 10.11948/2018.1893

Citation:

Jie Song, Mi Hu, Yuzhen Bai, Yonghui Xia. DYNAMIC ANALYSIS OF A NON-AUTONOMOUS RATIO-DEPENDENT PREDATOR-PREY MODEL WITH ADDITIONAL FOOD[J]. Journal of Applied Analysis & Computation, 2018, 8(6): 1893-1909. doi: 10.11948/2018.1893

DYNAMIC ANALYSIS OF A NON-AUTONOMOUS RATIO-DEPENDENT PREDATOR-PREY MODEL WITH ADDITIONAL FOOD

1 School of Mathematics Science, Huaqiao University, Quanzhou 362000, China;
2 Faculty of Mathematics and Physics, University of Ljubljana, 1000, Slovenia;
3 School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, China;
4 Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

Fund Project:

Abstract

Abstract

In this paper, a non-autonomous ratio-dependent three species predator-prey system with additional food to top predator was proposed. The permanence of the model is obtained. Based on the continuation theorem, the sufficient conditions for the existence of a periodic solution are obtained. By using the method of Lyapunov function, we prove that the system exists a unique positive almost periodic solution under some certain conditions.
- Non-autonomous predator-prey model /
- permanence /
- periodic solutions /
- almost periodic solutions
MSC: 34C27;92D25;34D20

FullText(HTML)

References

[1]	R. Arditi, L. R. Ginzburg, Coupling in predator-prey dynamics:ratiodependence, Journal of Theoretical Biology, 1989, 311-326. Google Scholar
[2]	Y. Bai, C. Guo, New results on stability and boundedness of third order nonlinear delay differential equations, Dynamic Systems and Applications, 2013, 22(1), 95-104. Google Scholar
[3]	Y. Bai, X. Mu, Global asymptotic stability of a generalized SIRS epidemic model with transfer from infectious to susceptible, Journal of Applied Analysis and Computation, 2018, 8(2), 402-412. Google Scholar
[4]	H. I. Freedman, P. Waltman, Persistence in models of three interacting predator-prey populations, Math. Biosci, 1994, 68(2), 213-231. Google Scholar
[5]	M. Fan, Q. Wang, X. Zou, Dynamics of a non-autonomous ratio-dependent predator-prey system, Proceedings of the Royal Society of Edinburgh, 2003, 133(1), 97-118. Google Scholar
[6]	R. E. Gaines, J. L. Mawhin, Coincidence degree and nonlinear differential equations, Springer-Verlag, Berlin, 1977. Google Scholar
[7]	A. Hastings, T. Powell, Chaos in a three species food chain, Ecology, 1991, 896-903. Google Scholar
[8]	M. Han, X. Hou, L. Sheng, C. Wang, Theory of rotated equations and applications to a population model, Discrete and Continuous Dynamical Systems-A, 2018, 2171-2185. Google Scholar
[9]	M. Han, L. Sheng, X. Zhang, Bifurcation theory for finitely smooth planar autonomous differential systems, Journal of Differential Equations, 2018, 264(5), 3596-3618. Google Scholar
[10]	M. Han, L. Sheng, Bifurcation of limit cycles in piecewise smooth systems via Melnikov function, Journal of Applied Analysis and Computation, 2015, 5(4), 809-815. Google Scholar
[11]	Y. Kuang, E. Beretta, Global qualitative analysis of a ratio-dependent predatorprey system, Journal of Mathematical Biology, 1998, 389-406. Google Scholar
[12]	P. Lundberg, J. M. Fryxell, Expected population density versus productivity in ratio-dependent and prey-dependent models, The American Naturalist, 1995, 153-161. Google Scholar
[13]	N. Pal, S. Samanta, J. Chattopadhyay, Revisited hastings and powell model with omnivory and predator switching, Chaos, Solit and Fract, 2014, 58-73. Google Scholar
[14]	P. Panja, S. K. Mondal, Stability analysis of coexistence of three species preypredator model, Nonlinear Dynamics, 2015, 373-382. Google Scholar
[15]	P. Panja, S. Poria, S. K. Mondal, Analysis of a harvested tritrophic food chain model in the presence of additional food for top predator, International Journal of Biomathematics, 2018, 11, 1-29. Google Scholar
[16]	V. G. Romanovski, M. Han, W. Huang, Bifurcation of critical periods of a quintic system, Electronic Journal of Differential Equations, 2018, 2018(66), 1-11. Google Scholar
[17]	B. Sahoo, S. Poria, Effects of additional food in a delayed predator prey model, Mathematical Biosciences, 2015, 62-73. Google Scholar
[18]	B. Sahoo, S. Poria, Effects of supplying alternative food in a predator-prey model with harvesting, Applied Mathematics and Computation, 2014, 150-166. Google Scholar
[19]	B. Sahoo, S. Poria, The chaos and control of a food chain model supplying additional food to top-predator, Chaos, Solitons and Fractals, 2014, 52-64. Google Scholar
[20]	Y. L. Song, S. L. Yuan, Bifurcation analysis for a regulated logistic growth model, Applied mathematical modelling, 2007, 31(9), 1729-1738. Google Scholar
[21]	H. Tian, M. Han, Bifurcation of periodic orbits by perturbing high-dimensional piecewise smooth integrable systems, Journal of Differential Equations, 2017, 263(11), 7448-7474. Google Scholar
[22]	Y. Xia, J. Cao, S. Cheng, Multiple periodic solutions of a delayed stagestructured predator-prey model with non-monotone functional responses, Applied Mathematical Modelling, 2007, 1947-1959. Google Scholar
[23]	Y. Xia, H. Wang, K. Kou, Z. Hu, Periodic solution of a higher dimensional ecological system with feedback control, Journal of Applied Analysis and Computation, 2016, 6(3), 893-906. Google Scholar
[24]	J. Yang, F. Liang, Limit cycle bifurcations of a kind of Lienard system with a hypobolic saddle and a nilpotent cusp, Journal of Applied Analysis and Computation, 2015, 5(3), 515-526. Google Scholar
[25]	T. Yoshizawa, Stability theory and the existence of periodic solutions and almost periodic solutions, IEEE Transactions on Systems Man and Cybernetic, 1975, 210-223. Google Scholar
[26]	S. L. Yuan, Y. L. Song, Stability and hopf bifurcations in a delayed LeslieGower predator-prey system, Journal of Applied Analysis and Computation, 2009, 355, 82-100. Google Scholar
[27]	T. Zhang, J. Liu, and Z. Teng, Existence of positive periodic solutions of an SEIR model with periodic coefficients, Applications of Mathematics, 2012, 57(6), 601-616. Google Scholar