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June  2015, 4(2): 115-129. doi: 10.3934/eect.2015.4.115

Global dynamics on a circular domain of a diffusion predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional type

Walid Abid 1, , Radouane Yafia 2, , M.A. Aziz-Alaoui 3, , Habib Bouhafa 1, and  Azgal Abichou 1,

1. 

University of Tunis El Manar, National Engineering School of Tunis, Laboratory of Engineering Mathematics EPT, B.P 743, La Marsa 2078, Tunisia, Tunisia, Tunisia
2. 

Ibn Zohr University, Polydisciplinary Faculty of Ouarzazate, B.P: 638, Ouarzazate, Morocco
3. 

Normandie Univ, France; ULH, LMAH, F-76600 Le Havre; FR CNRS 3335, ISCN, 25 rue Philippe Lebon 76600 Le Havre, France

Received  May 2014 Revised  October 2014 Published  May 2015

In this paper, we present a predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional response. The model is governed by a two dimensional reaction diffusion system defined on a disc domain. The conditions of boundedness, existence of a positively invariant and attracting set are proved. Sufficient conditions of local and global stability of the positive steady state are established. In the end, we carry out some numerical simulations in order to illustrate our theoretical results and to interpret how biological processes affect spatio-temporal pattern formation.
Keywords: local and global stability, Beddington-DeAngelis functional response, chaos, disc domain., pattern formation.
Mathematics Subject Classification: Primary: 35Kxx, 35Pxx, 35Exx; Secondary: 35B32, 35B35, 35B36, 35B5.
Citation: Walid Abid, Radouane Yafia, M.A. Aziz-Alaoui, Habib Bouhafa, Azgal Abichou. Global dynamics on a circular domain of a diffusion predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional type. Evolution Equations and Control Theory, 2015, 4 (2) : 115-129. doi: 10.3934/eect.2015.4.115
References:
[1]

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 shemes, Applied Math. Let., 16 (2003), 1069-1075. doi: 10.1016/S0893-9659(03)90096-6.

[2]

J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 44 (1975), 331-340. doi: 10.2307/3866.

[3]

A. D. Bazykin, Nonlinear Dynamics of Interacting Populations, World Scientific Series on Nonlinear Science Series A, 11, Singapore, 1998.

[4]

B. I. Camara and M. A. Aziz-Alaoui, Dynamics of predator-prey model with diffusion, Dynamics of Continuous, Discrete and Impulsive System, Series A, 15 (2008), 897-906.

[5]

B. I. Camara, Complexity of Dynamic Models of Predator-Prey with Diffusion and Applications, Ph.D thesis, Le Havre University, 2009.

[6]

F. D. Chen, On a nonlinear non-autonomous predator-prey model with diffusion and distributed delay, J. Comput. Appl. Math., 180 (2005), 33-49.

[7]

W. Y. Chen and M. X. Wang, Qualitative analysis of predator-prey models with Beddington-DeAngelis functional response and diffusion, Math. Comput. Model., 42 (2005), 31-44. doi: 10.1016/j.mcm.2005.05.013.

[8]

M. O. Daher and M. A. Aziz-Alaoui, On the dynamics of a predator-prey model with the Holling-Tanner functional, in Proc. ESMTB Conf. (ed. V. Capasso), 2002, 270-278.

[9]

D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892.

[10]

M. P. Hassell and C. C. Varley, New inductive population model for insect parasites and its bearing on biological control, Nature, 223 (1969), 1133-1137. doi: 10.1038/2231133a0.

[11]

S. B. Hsu, Constructing lyapunov functions for mathematical models in population biology, Taiwanese Journal of Math., 9 (2005), 151-173.

[12]

Z. Li, M. Han and F. Chen, Global stability of a stage-structured predator-prey model with modified Leslie-Gower and Holling-type II schemes, Int. J. Biomath., 5 (2012), 1250057.

[13]

A. J. Lotka, Elements of Physical Biology, Williams and Wilkins, Baltimore, 1925.

[14]

R. M. May, Stability and Complexity in Model Ecosystem, Princeton Univ. Press, Princeton, 1976. doi: 10.1109/TSMC.1976.4309488.

[15]

J. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993.

[16]

J. Murray, Mathematical Biology: II. Spatial Models and Biomedical Applications, Springer-Verlag, Berlin Heidelberg, 2003.

[17]

M. L. Rosenzweig and R. H. MacArthue, Graphical representation and stability conditions of predator-prey interactions, Am. Nat., 97 (1963), 209-223. doi: 10.1086/282272.

[18]

V. Volterra, Variations and fluctuations of the number of individuals in animal species living together, J. Cons. int. Explor. Mer., 3 (1928), 3-51. doi: 10.1093/icesjms/3.1.3.

[19]

S. Yu, Global stability of a modified Leslie-Gower model with Beddington-DeAngelis functional response, Advances in Difference Equations, 2014 (2014), p84. doi: 10.1186/1687-1847-2014-84.

[20]

R. Yafia, F. 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., 1 (2007), 119-131.

[21]

R. Yafia, F. El Adnani and H. Talibi 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 Analysis Real World Appl., 9 (2008), 2055-2067. doi: 10.1016/j.nonrwa.2006.12.017.

[22]

R. Yafia and M. A. Aziz Alaoui, Existence of periodic travelling waves solutions in predator prey model with diffusion, Applied Mathematical Modelling, 37 (2013), 3635-3644. doi: 10.1016/j.apm.2012.08.003.

[23]

S. B. Yu, Global asymptotic stability of a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Discrete Dyn. Nat. Soc., Article ID 208167, (2012).

[24]

Z. Yao, S. Xie and N. Yu, Dynamics of cooperative predator-prey system with impulsive effects and Beddington-DeAngelis functional response, J. Egypt. Math. Soc., 21 (2013), 213-223. doi: 10.1016/j.joems.2013.04.008.

show all references

References:
[1]

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 shemes, Applied Math. Let., 16 (2003), 1069-1075. doi: 10.1016/S0893-9659(03)90096-6.

[2]

J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 44 (1975), 331-340. doi: 10.2307/3866.

[3]

A. D. Bazykin, Nonlinear Dynamics of Interacting Populations, World Scientific Series on Nonlinear Science Series A, 11, Singapore, 1998.

[4]

B. I. Camara and M. A. Aziz-Alaoui, Dynamics of predator-prey model with diffusion, Dynamics of Continuous, Discrete and Impulsive System, Series A, 15 (2008), 897-906.

[5]

B. I. Camara, Complexity of Dynamic Models of Predator-Prey with Diffusion and Applications, Ph.D thesis, Le Havre University, 2009.

[6]

F. D. Chen, On a nonlinear non-autonomous predator-prey model with diffusion and distributed delay, J. Comput. Appl. Math., 180 (2005), 33-49.

[7]

W. Y. Chen and M. X. Wang, Qualitative analysis of predator-prey models with Beddington-DeAngelis functional response and diffusion, Math. Comput. Model., 42 (2005), 31-44. doi: 10.1016/j.mcm.2005.05.013.

[8]

M. O. Daher and M. A. Aziz-Alaoui, On the dynamics of a predator-prey model with the Holling-Tanner functional, in Proc. ESMTB Conf. (ed. V. Capasso), 2002, 270-278.

[9]

D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892.

[10]

M. P. Hassell and C. C. Varley, New inductive population model for insect parasites and its bearing on biological control, Nature, 223 (1969), 1133-1137. doi: 10.1038/2231133a0.

[11]

S. B. Hsu, Constructing lyapunov functions for mathematical models in population biology, Taiwanese Journal of Math., 9 (2005), 151-173.

[12]

Z. Li, M. Han and F. Chen, Global stability of a stage-structured predator-prey model with modified Leslie-Gower and Holling-type II schemes, Int. J. Biomath., 5 (2012), 1250057.

[13]

A. J. Lotka, Elements of Physical Biology, Williams and Wilkins, Baltimore, 1925.

[14]

R. M. May, Stability and Complexity in Model Ecosystem, Princeton Univ. Press, Princeton, 1976. doi: 10.1109/TSMC.1976.4309488.

[15]

J. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993.

[16]

J. Murray, Mathematical Biology: II. Spatial Models and Biomedical Applications, Springer-Verlag, Berlin Heidelberg, 2003.

[17]

M. L. Rosenzweig and R. H. MacArthue, Graphical representation and stability conditions of predator-prey interactions, Am. Nat., 97 (1963), 209-223. doi: 10.1086/282272.

[18]

V. Volterra, Variations and fluctuations of the number of individuals in animal species living together, J. Cons. int. Explor. Mer., 3 (1928), 3-51. doi: 10.1093/icesjms/3.1.3.

[19]

S. Yu, Global stability of a modified Leslie-Gower model with Beddington-DeAngelis functional response, Advances in Difference Equations, 2014 (2014), p84. doi: 10.1186/1687-1847-2014-84.

[20]

R. Yafia, F. 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., 1 (2007), 119-131.

[21]

R. Yafia, F. El Adnani and H. Talibi 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 Analysis Real World Appl., 9 (2008), 2055-2067. doi: 10.1016/j.nonrwa.2006.12.017.

[22]

R. Yafia and M. A. Aziz Alaoui, Existence of periodic travelling waves solutions in predator prey model with diffusion, Applied Mathematical Modelling, 37 (2013), 3635-3644. doi: 10.1016/j.apm.2012.08.003.

[23]

S. B. Yu, Global asymptotic stability of a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Discrete Dyn. Nat. Soc., Article ID 208167, (2012).

[24]

Z. Yao, S. Xie and N. Yu, Dynamics of cooperative predator-prey system with impulsive effects and Beddington-DeAngelis functional response, J. Egypt. Math. Soc., 21 (2013), 213-223. doi: 10.1016/j.joems.2013.04.008.

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