Journal of Applied Science and Engineering

Published by Tamkang University Press

1.30

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2.10

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Zaowang Xiao1 and Zhong Li This email address is being protected from spambots. You need JavaScript enabled to view it.1

1College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350116, P.R. China.


 

Received: July 19, 2018
Accepted: January 15, 2019
Publication Date: June 1, 2019

Download Citation: ||https://doi.org/10.6180/jase.201906_22(2).0001  

ABSTRACT


In this paper, we consider a predator-prey system incorporating mutual interference into predator and the fear effect into prey. By using theories on exceptional directions and normal sectors, we show that both the two boundary equilibria are saddle points and the interior equilibrium is globally stable. Compared to the corresponding predator-prey model without mutual interference, we find that the mutual interference can stabilize the predator-prey system.


Keywords: Predator-prey, Fear Effect, Mutual Interference, Global Stability


REFERENCES


  1. [1] Yu, S. B. (2012) Global asymptotic stability of a predator-prey model with modified Leslie-Gower and Holling-type II schemes. Discrete Dynamics in Nature and Society 2012(1), 857868. doi: 10.1155/2012/ 208167
  2. [2] Chen, F. D., Z. Z. Ma, and H. Y. Zhang (2012) Global asymptotical stability of the positive equilibrium of the Lotka-Volterra prey-predator model incorporating a constant number of prey refuges. Nonlinear Analysis: Real World Applications 13(6), 27902793. doi: 10. 1016/j.nonrwa.2012.04.006
  3. [3] Chen, L. J., and F. D. Chen (2011) Dynamic behaviors of the periodic predator-prey system with distributed time delays and impulsive effect. Nonlinear Analysis: Real World Applications 12(4), 24672473. doi: 10. 1016/j.nonrwa.2011.03.002
  4. [4] Xie, X. D., Y. L. Xue, J. H. Chen, and T. T. Li (2016) Permanence and global attractivity of a non autonomous modified Leslie-Gower predator-prey model with Holling-type II schemes and a prey refuge. Advances in Difference Equations 2016(1), 184. doi: 10.1186/ s13662-016-0892-5
  5. [5] Haque, M. (2011) Adetailed study of the BeddingtonDeAngelis predator-prey model. Mathematical Biosciences 234(1), 116. doi: 10.1016/j.mbs.2011.07.003
  6. [6] Xu, C. Q., S. L. Yuan, and T. H. Zhang (2016) Global dynamics of a predator-prey model with defense mechanism for prey. Applied Mathematics Letters 62, 4248.
  7. [7] Al Basheer, A., R. D. Parshad, E. Quansah, S. B. Yu, and R. K. Upadhyay (2018) Exploring the dynamics of a Holling-Tanner model with cannibalism in both predator and prey population. International Journal of Biomathematics 11(01), 1850010. doi: 10.1142/ S1793524518500109
  8. [8] Lin, Y. H., X. D. Xie, F. D. Chen, and T. T. Li (2016) Convergences of a stage-structured predator-prey model with modified Leslie-Gower and Holling-type II schemes. Advances in Difference Equations 2016(1), 181. doi: 10.1186/s13662-016-0887-2
  9. [9] Zhang, L., J. Liu, and M. Banerjee (2017) Hopf and steady state bifurcation analysis in a ratio dependent predator-prey model. Communications in Nonlinear Science and Numerical Simulation 44, 5273. doi: 10.1016/j.cnsns.2016.07.027
  10. [10] Wu, R. X., and L. Li (2013) Permanence and global attractivity of the discrete predator-prey system with Hassell-Varley-Holling III type functional response. Discrete Dynamics in Nature and Society 2013(2), 295321. doi: 10.1155/2013/393729
  11. [11] Garay, J., R. Cressman, F. Xu, Z. Varga, and T. Cabello (2015) Optimal forager against ideal free distributed prey. American Naturalist 186, 111122. doi: 10. 1086/681638
  12. [12] Xu, F., R. Cressman, and V. Krivan (2014) Evolution of mobility in predator-prey systems. Discrete and Continuous Dynamical Systems-Series B 19, 3397 3432. doi: 10.3934/dcdsb.2014.19.3397
  13. [13] Sasmal, S. K. (2018) Population dynamics with multiple Allee effects induced by fear factors-A mathematical study on prey-predator interactions. Applied Mathematical Modelling 64, 114. doi: 10.1016/j.apm. 2018.07.021
  14. [14] Zanette, L. Y., A. F. White, M. C. Allen, and M. Clinchy (2011) Perceived predation risk reduces the number of offspring songbirds produce per year. Science 334(6061), 13981401. doi: 10.1126/science. 1210908
  15. [15] Sheriff, M. J., C. J. Krebs, and R. Boonstra (2009) The sensitive hare: sublethal effects of predator stress on reproduction in snowshoe hares. Journal of Animal Ecology 78(6), 12491258. doi: 10.1111/j.1365-2656. 2009.01552.x
  16. [16] Orrock, J. L., and R. J. Fletcher (2014) An island-wide predator manipulation reveals immediate and long lasting matching of risk by prey. Proceedings of the Royal Society of London B: Biological Sciences 281(1784), 20140391. doi: 10.1098/rspb.2014.0391
  17. [17] Wang, X. Y., and X. F. Zou (2017) Modeling the fear effect in predator-prey interactions with adaptive avoidance of predators. Bulletin of Mathematical Biology 79(6), 13251359. doi: 10.1007/s11538-017-0287-0
  18. [18] Wang, X. Y., L. Y. Zanette, and X. F. Zou (2016) Modelling the fear effect in predator-prey interactions. Journal of Mathematical Biology 73(5), 11791204. doi: 10.1007/s00285-016-0989-1
  19. [19] Freedman, H. I. (1979) Stability analysis of a predator-prey system with mutual interference and density dependent death rates. Bulletin of Mathematical Biology 41(1), 6778. doi: 10.1007/BF02547925
  20. [20] Ma, Z. Z., F. D. Chen, C. Q. Wu, and W. L. Chen (2013) Dynamic behaviors of a Lotka-Volterra predator-prey model incorporating a prey refuge and predator mutual interference. Applied Mathematics and Computation 219(15), 79457953. doi: 10.1016/j.amc. 2013.02.033
  21. [21] Chen, L. J., F. D. Chen, and Y. Q. Wang (2013) Influence of predator mutual interference and prey refuge on Lotka-Volterra predator-prey dynamics. Communications in Nonlinear Science and Numerical Simulation 18(11), 31743180. doi: 10.1016/j.cnsns.2013. 04.004
  22. [22] Yu, S. B., and F. D. Chen (2014) Almost periodic solution of a modified Leslie-Gower predator-prey model with Holling-type II schemes and mutual interference. International Journal of Biomathematics 7(03), 1450028. doi: 10.1142/S1793524514500284
  23. [23] Wu, R. X. (2010) Permanence of a discrete periodic Volterra model with mutual interference and Beddington-DeAngelis functional response. Discrete Dyna
    mics in Nature and Society 2010(1026), 10381045. doi: 10.1155/2010/246783
  24. [24] Li, Z., M. A. Han, and F. D. Chen (2014) Global stability of a predator-prey system with stage structure and mutual interference. Discrete Continuous Dynamical Systems-Series B 19(1), 173187.
  25. [25] Sansone, G., and R. Conti (1964) Non-Linear Differential Equations, Pergamon Press, London.