September  2019, 24(9): 5003-5039. doi: 10.3934/dcdsb.2019042

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

1. 

Normandie Univ, Laboratoire Raphaël Salem, UMR CNRS 6085, Rouen, France

2. 

PSA & Inria DISCO & Laboratoire des Signaux et Systèmes, Université Paris Saclay, CNRS-CentraleSupélec-Université Paris Sud, 3 rue Joliot-Curie, 91192 Gif-sur-Yvette cedex, France

* Corresponding author: P. Raynaud de Fitte

Received  June 2018 Revised  September 2018 Published  February 2019

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

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.

Citation: Safia Slimani, Paul Raynaud de Fitte, Islam Boussaada. Dynamics of a prey-predator system with modified Leslie-Gower and Holling type Ⅱ schemes incorporating a prey refuge. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5003-5039. doi: 10.3934/dcdsb.2019042
References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, Berlin, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

F. R. Gantmacher, The Theory of Matrices. Vols. 1, 2, Translated by K. A. Hirsch, Chelsea Publishing Co., New York, 1959.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. Llibre and J. Villadelprat, A Poincaré index formula for surfaces with boundary, Differential Integral Equations, 11 (1998), 191-199.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, New Jersey, 1973.   Google Scholar
[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.  Google Scholar

[30]

E. C. Pielou, Mathematical Ecology, 2nd edition, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977.  Google Scholar

[31]

C. C. Pugh, A generalized Poincaré index formula, Topology, 7 (1968), 217-226.  doi: 10.1016/0040-9383(68)90002-5.  Google Scholar

[32]

J. Tong, $b^2-4ac$ and $b^2-3ac$, Math. Gaz., 88 (2004), 511-513.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, Berlin, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

F. R. Gantmacher, The Theory of Matrices. Vols. 1, 2, Translated by K. A. Hirsch, Chelsea Publishing Co., New York, 1959.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. Llibre and J. Villadelprat, A Poincaré index formula for surfaces with boundary, Differential Integral Equations, 11 (1998), 191-199.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28] R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, New Jersey, 1973.   Google Scholar
[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.  Google Scholar

[30]

E. C. Pielou, Mathematical Ecology, 2nd edition, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977.  Google Scholar

[31]

C. C. Pugh, A generalized Poincaré index formula, Topology, 7 (1968), 217-226.  doi: 10.1016/0040-9383(68)90002-5.  Google Scholar

[32]

J. Tong, $b^2-4ac$ and $b^2-3ac$, Math. Gaz., 88 (2004), 511-513.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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} $

$ m = 0.0025 $, $ a = 0.5 $, $ k_1 = 0.08 $, $ k_2 = 0.2 $, $ b = 0.1 $

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

m = 0.01, a = 1, k1 = 0.1, k2 = 0.1, b = 0.05

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

$a=0.4$, $k_{1}=0.08$, $k_{2}=0.2$, $b=0.1$, $m=0.0025$, the initial value $(x(0), y(0))=(0.55, 0.6), $ and the time step $h=0.01.$ The deterministic model has a globally stable equilibrium point $(x^*, y^*)=(0.55, 0.75)$

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