\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Dynamics of a prey-predator system with modified Leslie-Gower and Holling type Ⅱ schemes incorporating a prey refuge

  • * Corresponding author: P. Raynaud de Fitte

    * Corresponding author: P. Raynaud de Fitte 

The first author is supported by TASSILI research program 16MDU972 between the University of Annaba (Algeria) and the University of Rouen (France)

Abstract Full Text(HTML) Figure(4) Related Papers Cited by
  • We study a modified version of a prey-predator system with modified Leslie-Gower and Holling type Ⅱ functional responses studied by M.A. Aziz-Alaoui and M. Daher-Okiye. The modification consists in incorporating a refuge for preys, and substantially complicates the dynamics of the system. We study the local and global dynamics and the existence of cycles. We also investigate conditions for extinction or existence of a stationary distribution, in the case of a stochastic perturbation of the system.

    Mathematics Subject Classification: Primary: 92D25, 34D23, Secondary: 60H10.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  A phase portrait of (1.2) with three equilibrium points and a cycle in the interior of $ \mathcal{A} $. The dashed lines are isoclines $ y = \frac{x(1-x)(k_1+x-m)}{a(x-m)} $ and $ y = k_2+x-m $. The grey region is the invariant attracting domain $ \mathcal{A} $

    Figure 2.  A phase portrait of (1.2) with an unstable equilibrium and a stable limit cycle

    Figure 3.  Hopf bifurcation of the system (1.2)

    Figure 4.  Solutions to the stochastic system (1.3) and the corresponding deterministic system, represented respectively by the blue line and the red line

  • [1] W. Abid, R. Yafia, M. A. Aziz-Alaoui and A. Aghriche, Turing Instability and Hopf Bifurcation in a Modified Leslie–Gower Predator–Prey Model with Cross-Diffusion, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850089, 17pp. doi: 10.1142/S021812741850089X.
    [2] W. AbidR. YafiaM. A. Aziz-AlaouiH. Bouhafa and A. Abichou, Diffusion driven instability and Hopf bifurcation in spatial predator-prey model on a circular domain, Appl. Math. Comput., 260 (2015), 292-313.  doi: 10.1016/j.amc.2015.03.070.
    [3] M. A. Aziz-Alaoui and M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, Appl. Math. Lett., 16 (2003), 1069-1075.  doi: 10.1016/S0893-9659(03)90096-6.
    [4] M. Bandyopadhyay and J. Chattopadhyay, Ratio-dependent predator-prey model: Effect of environmental fluctuation and stability, Nonlinearity, 18 (2005), 913-936.  doi: 10.1088/0951-7715/18/2/022.
    [5] N. P. Bhatia and G. P. Szegö, Stability Theory of Dynamical Systems, Die Grundlehren der mathematischen Wissenschaften, Band 161, Springer-Verlag, New York-Berlin, 1970.
    [6] B. I. Camara, Waves analysis and spatiotemporal pattern formation of an ecosystem model, Nonlinear Anal. Real World Appl., 12 (2011), 2511-2528.  doi: 10.1016/j.nonrwa.2011.02.020.
    [7] F. ChenL. Chen and X. Xie, On a Leslie-Gower predator-prey model incorporating a prey refuge, Nonlinear Anal. Real World Appl., 10 (2009), 2905-2908.  doi: 10.1016/j.nonrwa.2008.09.009.
    [8] G. Da Prato and H. Frankowska, Stochastic viability of convex sets, J. Math. Anal. Appl., 333 (2007), 151-163.  doi: 10.1016/j.jmaa.2006.08.057.
    [9] M. Daher Okiye and M. A. Aziz-Alaoui, On the dynamics of a predator-prey model with the Holling-Tanner functional response, in Mathematical modelling & computing in biology and medicine, vol. 1 of Milan Res. Cent. Ind. Appl. Math. MIRIAM Proj., Esculapio, Bologna, 2003, 270–278.
    [10] N. DalalD. Greenhalgh and X. Mao, A stochastic model for internal HIV dynamics, J. Math. Anal. Appl., 341 (2008), 1084-1101.  doi: 10.1016/j.jmaa.2007.11.005.
    [11] F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, Berlin, 2006.
    [12] G. Ferreyra and P. Sundar, Comparison of solutions of stochastic equations and applications, Stochastic Anal. Appl., 18 (2000), 211-229.  doi: 10.1080/07362990008809665.
    [13] J. FuD. JiangN. ShiT. Hayat and A. Alsaedi, Qualitative analysis of a stochastic ratio-dependent Holling-Tanner system, Acta Math. Sci. Ser. B (Engl. Ed.), 38 (2018), 429-440.  doi: 10.1016/S0252-9602(18)30758-6.
    [14] F. R. Gantmacher, The Theory of Matrices. Vols. 1, 2, Translated by K. A. Hirsch, Chelsea Publishing Co., New York, 1959.
    [15] D. H. Gottlieb, A de Moivre like formula for fixed point theory, in Fixed Point Theory and Its Applications (Berkeley, CA, 1986), vol. 72 of Contemp. Math., Amer. Math. Soc., Providence, RI, 1988, 99–105. doi: 10.1090/conm/072/956481.
    [16] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.
    [17] C. JiD. Jiang and N. Shi, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with stochastic perturbation, J. Math. Anal. Appl., 359 (2009), 482-498.  doi: 10.1016/j.jmaa.2009.05.039.
    [18] R. Khasminskii, Stochastic Stability of Differential Equations, vol. 66 of Stochastic Modelling and Applied Probability, 2nd edition, Springer, Heidelberg, 2012, With contributions by G. N. Milstein and M. B. Nevelson. doi: 10.1007/978-3-642-23280-0.
    [19] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, vol. 23 of Applications of Mathematics (New York), Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.
    [20] P. H. Leslie and J. C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219-234.  doi: 10.1093/biomet/47.3-4.219.
    [21] L. Liu and Y. Shen, Sufficient and necessary conditions on the existence of stationary distribution and extinction for stochastic generalized logistic system, Adv. Difference Equ., 2015 (2015), 13pp. doi: 10.1186/s13662-014-0345-y.
    [22] Z. Liu, Stochastic dynamics for the solutions of a modified Holling-Tanner model with random perturbation, Internat. J. Math., 25 (2014), 1450105, 23pp. doi: 10.1142/S0129167X14501055.
    [23] J. Llibre and J. Villadelprat, A Poincaré index formula for surfaces with boundary, Differential Integral Equations, 11 (1998), 191-199. 
    [24] J. Lv and K. Wang, Analysis on a stochastic predator-prey model with modified Leslie-Gower response, Abstr. Appl. Anal., 2011 (2011), Art. ID 518719, 16pp. doi: 10.1155/2011/518719.
    [25] J. Lv and K. Wang, Asymptotic properties of a stochastic predator-prey system with Holling Ⅱ functional response, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 4037-4048.  doi: 10.1016/j.cnsns.2011.01.015.
    [26] T. Ma and S. Wang, A generalized Poincaré-Hopf index formula and its applications to 2-D incompressible flows, Nonlinear Anal. Real World Appl., 2 (2001), 467-482.  doi: 10.1016/S1468-1218(01)00004-9.
    [27] P. S. Mandal and M. Banerjee, Stochastic persistence and stability analysis of a modified Holling-Tanner model, Math. Methods Appl. Sci., 36 (2013), 1263-1280.  doi: 10.1002/mma.2680.
    [28] R. M. MayStability and Complexity in Model Ecosystems, Princeton University Press, Princeton, New Jersey, 1973. 
    [29] A. F. NindjinM. A. Aziz-Alaoui and M. Cadivel, Analysis of a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with time delay, Nonlinear Anal. Real World Appl., 7 (2006), 1104-1118.  doi: 10.1016/j.nonrwa.2005.10.003.
    [30] E. C. Pielou, Mathematical Ecology, 2nd edition, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977.
    [31] C. C. Pugh, A generalized Poincaré index formula, Topology, 7 (1968), 217-226.  doi: 10.1016/0040-9383(68)90002-5.
    [32] J. Tong, $b^2-4ac$ and $b^2-3ac$, Math. Gaz., 88 (2004), 511-513. 
    [33] R. Yafia and M. A. Aziz-Alaoui, Existence of periodic travelling waves solutions in predator prey model with diffusion, Appl. Math. Model., 37 (2013), 3635-3644.  doi: 10.1016/j.apm.2012.08.003.
    [34] R. YafiaF. El Adnani and H. T. Alaoui, Limit cycle and numerical similations for small and large delays in a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes, Nonlinear Anal. Real World Appl., 9 (2008), 2055-2067.  doi: 10.1016/j.nonrwa.2006.12.017.
    [35] R. YafiaF. El Adnani and H. Talibi Alaoui, Stability of limit cycle in a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with time delay., Appl. Math. Sci., Ruse, 1 (2007), 119-131. 
  • 加载中

Figures(4)

SHARE

Article Metrics

HTML views(745) PDF downloads(246) Cited by(0)

Access History

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return