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

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.
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.

[1]

Jinliang Wang, Jiying Lang, Xianning Liu. Global dynamics for viral infection model with Beddington-DeAngelis functional response and an eclipse stage of infected cells. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3215-3233. doi: 10.3934/dcdsb.2015.20.3215

[2]

Xiao He, Sining Zheng. Bifurcation analysis and dynamic behavior to a predator-prey model with Beddington-DeAngelis functional response and protection zone. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4641-4657. doi: 10.3934/dcdsb.2020117

[3]

Renji Han, Binxiang Dai, Lin Wang. Delay induced spatiotemporal patterns in a diffusive intraguild predation model with Beddington-DeAngelis functional response. Mathematical Biosciences & Engineering, 2018, 15 (3) : 595-627. doi: 10.3934/mbe.2018027

[4]

Sze-Bi Hsu, Shigui Ruan, Ting-Hui Yang. On the dynamics of two-consumers-one-resource competing systems with Beddington-DeAngelis functional response. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2331-2353. doi: 10.3934/dcdsb.2013.18.2331

[5]

Haiyin Li, Yasuhiro Takeuchi. Dynamics of the density dependent and nonautonomous predator-prey system with Beddington-DeAngelis functional response. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1117-1134. doi: 10.3934/dcdsb.2015.20.1117

[6]

Biruk Tafesse Mulugeta, Liping Yu, Qigang Yuan, Jingli Ren. Bifurcation analysis of a predator-prey model with strong Allee effect and Beddington-DeAngelis functional response. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022153

[7]

Qi Wang, Ling Jin, Zengyan Zhang. Global well-posedness, pattern formation and spiky stationary solutions in a Beddington–DeAngelis competition system. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2105-2134. doi: 10.3934/dcds.2020108

[8]

Yu Yang, Shigui Ruan, Dongmei Xiao. Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function. Mathematical Biosciences & Engineering, 2015, 12 (4) : 859-877. doi: 10.3934/mbe.2015.12.859

[9]

Eric Avila-Vales, Gerardo García-Almeida, Erika Rivero-Esquivel. Bifurcation and spatiotemporal patterns in a Bazykin predator-prey model with self and cross diffusion and Beddington-DeAngelis response. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 717-740. doi: 10.3934/dcdsb.2017035

[10]

Seong Lee, Inkyung Ahn. Diffusive predator-prey models with stage structure on prey and beddington-deangelis functional responses. Communications on Pure and Applied Analysis, 2017, 16 (2) : 427-442. doi: 10.3934/cpaa.2017022

[11]

Tao Zheng, Yantao Luo, Xinran Zhou, Long Zhang, Zhidong Teng. Spatial dynamic analysis for COVID-19 epidemic model with diffusion and Beddington-DeAngelis type incidence. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2021154

[12]

Peng Yang, Yuanshi Wang. On oscillations to a 2D age-dependent predation equations characterizing Beddington-DeAngelis type schemes. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3845-3895. doi: 10.3934/dcdsb.2021209

[13]

Yinshu Wu, Wenzhang Huang. Global stability of the predator-prey model with a sigmoid functional response. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1159-1167. doi: 10.3934/dcdsb.2019214

[14]

Jian-Jun Xu, Junichiro Shimizu. Asymptotic theory for disc-like crystal growth (II): interfacial instability and pattern formation at early stage of growth. Communications on Pure and Applied Analysis, 2004, 3 (3) : 527-543. doi: 10.3934/cpaa.2004.3.527

[15]

Qiumei Zhang, Daqing Jiang, Li Zu. The stability of a perturbed eco-epidemiological model with Holling type II functional response by white noise. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 295-321. doi: 10.3934/dcdsb.2015.20.295

[16]

Sze-Bi Hsu, Tzy-Wei Hwang, Yang Kuang. Global dynamics of a Predator-Prey model with Hassell-Varley Type functional response. Discrete and Continuous Dynamical Systems - B, 2008, 10 (4) : 857-871. doi: 10.3934/dcdsb.2008.10.857

[17]

Xin Jiang, Zhikun She, Shigui Ruan. Global dynamics of a predator-prey system with density-dependent mortality and ratio-dependent functional response. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 1967-1990. doi: 10.3934/dcdsb.2020041

[18]

Ke Guo, Wanbiao Ma. Global dynamics of a delayed air pollution dynamic model with saturated functional response and backward bifurcation. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022124

[19]

Julien Barré, Pierre Degond, Diane Peurichard, Ewelina Zatorska. Modelling pattern formation through differential repulsion. Networks and Heterogeneous Media, 2020, 15 (3) : 307-352. doi: 10.3934/nhm.2020021

[20]

Julien Cividini. Pattern formation in 2D traffic flows. Discrete and Continuous Dynamical Systems - S, 2014, 7 (3) : 395-409. doi: 10.3934/dcdss.2014.7.395

2021 Impact Factor: 1.169

Metrics

  • PDF downloads (128)
  • HTML views (0)
  • Cited by (2)

[Back to Top]