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Preface
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 |
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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