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The basins of attraction in a modified May-Holling-Tanner predator-prey model with Allee affect. (English) Zbl 1421.34032
Summary: I analyse a modified May-Holling-Tanner predator-prey model considering an Allee effect in the prey and alternative food sources for predator. Additionally, the predation functional response or predation consumption rate is linear. The extended model exhibits rich dynamics and we prove the existence of separatrices in the phase plane separating basins of attraction related to oscillation, co-existence and extinction of the predator-prey population. We also show the existence of a homoclinic curve that degenerates to form a limit cycle and discuss numerous potential bifurcations such as saddle-node, Hopf, and Bogdanov-Takens bifurcations. We use simulations to illustrate the behaviour of the model.

34C60 Qualitative investigation and simulation of ordinary differential equation models
92D25 Population dynamics (general)
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
Full Text: DOI
[1] Yu, S., Global asymptotic stability of a predator-prey model with modified Leslie-Gower and Holling-Type II schemes, Discrete Dyn. Nat. Soc., 2012, (2012) · Zbl 1248.34050
[2] Zhao, Z.; Yang, L.; Chen, L., Impulsive perturbations of a predator-prey system with modified Leslie-Gower and Holling type II schemes, J. Appl. Math. Comput., 35, 119-134, (2011) · Zbl 1222.34057
[3] Arrowsmith, D.; Place, C., Dynamical Systems: Differential Equations, Maps and Chaotic Behaviour, 1-330, (2017), CRC Press · Zbl 0754.34001
[4] Saez, E.; Gonzalez-Olivares, E., Dynamics on a predator-prey model, SIAM J. Appl. Math., 59, 1867-1878, (1999) · Zbl 0934.92027
[5] Banerjee, M., Turing and non-Turing patterns in two-dimensional prey-predator models, (Applications of Chaos and Nonlinear Dynamics in Science and Engineering, vol. 4, (2015), Springer), 257-280
[6] Ghazaryan, A.; Manukian, V.; Schecter, S., Travelling waves in the Holling-Tanner model with weak diffusion, Proc. R. Soc. A, 471, Article 20150045 pp., (2015) · Zbl 1371.35302
[7] Martínez-Jeraldo, N.; Aguirre, P., Allee effect acting on the prey species in a Leslie-Gower predation model, Nonlinear Anal. Real World Appl., 45, 895-917, (2019) · Zbl 1408.34039
[8] Hanski, I.; Henttonen, H.; Korpimäki, E.; Oksanen, L.; Turchin, P., Small-rodent dynamics and predation, Ecology, 82, 1505-1520, (2001)
[9] Hanski, I.; Hansson, L.; Henttonen, H., Specialist predators, generalist predators, and the microtine rodent cycle, J. Anim. Ecol., 353-367, (1991)
[10] Hanski, I.; Turchin, P.; Korpimaki, E.; Henttonen, H., Population oscillations of boreal rodents: regulation by mustelid predators leads to chaos, Nature, 364, 232, (1993)
[11] Turchin, P.; Hanski, I., An empirically based model for latitudinal gradient in vole population dynamics, Amer. Nat., 149, 842-874, (1997)
[12] Turchin, P., Complex population dynamics: a theoretical/empirical synthesis, (Monographs in Population Biology, vol. 35, (2003), Princeton University Press: Princeton University Press Princeton, N.J.) · Zbl 1062.92077
[13] May, R., Stability and complexity in model ecosystems, (Monographs in Population Biology, vol. 6, (1974), Princeton University Press: Princeton University Press Princeton, N.J.)
[14] Freedman, H., Deterministic mathematical models in population ecology, (Pure and Applied Mathematics (Dekker), vol. 57, (1980), Wiley: Wiley New York) · Zbl 0448.92023
[15] Aziz-Alaoui, M.; Daher, M., Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Appl. Math. Lett., 16, 1069-1075, (2003) · Zbl 1063.34044
[16] Arancibia-Ibarra, C.; Gonzalez-Olivares, E., A modified Leslie-Gower predator-prey model with hyperbolic functional response and allee effect on prey, (BIOMAT 2010 International Symposium on Mathematical and Computational Biology, (2011), World Scientific Co. Pty. Ltd., Singapore), 146-162
[17] Feng, P.; Kang, Y., Dynamics of a modified Leslie-Gower model with double Allee effects, Nonlinear Dynam., 80, 1051-1062, (2015) · Zbl 1345.92115
[18] Singh, A.; Gakkhar, S., Stabilization of modified Leslie-Gower prey-predator model, Differ. Equ. Dyn. Syst., 22, 239-249, (2014) · Zbl 1300.34115
[19] Korobeinikov, A., A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14, 697-699, (2001) · Zbl 0999.92036
[20] Kramer, A.; Berec, L.; Drake, J., Allee effects in ecology and evolution, J. Anim. Ecol., 87, 7-10, (2018)
[21] Allee, W.; Park, O.; Emerson, A.; Park, T.; Schmidt, K., Principles of Animal Ecology, (1949), WB Saundere Co. Ltd.: WB Saundere Co. Ltd. Philadelphia
[22] Berec, L.; Angulo, E.; Courchamp, F., Multiple Allee effects and population management, Trends Ecol. Evol., 22, 185-191, (2007)
[23] Courchamp, F.; Clutton-Brock, T.; Grenfell, B., Inverse density dependence and the Allee effect, Trends Ecol. Evol., 14, 405-410, (1999)
[24] Stephens, P.; Sutherland, W., Consequences of the Allee effect for behaviour, ecology and conservation, Trends Ecol. Evol., 14, 401-405, (1999)
[25] Liermann, M.; Hilborn, R., Depensation: evidence, models and implications, Fish Fish., 2, 33-58, (2001)
[26] Courchamp, F.; Grenfell, B.; Clutton-Brock, T., Impact of natural enemies on obligately cooperative breeders, Oikos, 91, 311-322, (2000)
[27] Courchamp, F.; Berec, L.; Gascoigne, J., Allee Effects in Ecology and Conservation, (2008), Oxford University Press
[28] Stephens, P.; Sutherland, W.; Freckleton, R., What is the Allee effect?, Oikos, 87, 185-190, (1999)
[29] Yue, Z.; Wang, X.; Liu, H., Complex dynamics of a diffusive Holling-Tanner predator-prey model with the Allee effect, Abstr. Appl. Anal., 2013, (2013) · Zbl 1267.37104
[30] C. Arancibia-Ibarra, E. Gonzalez-Olivares, The Holling-Tanner model considering an alternative food for predator, in: Proceedings of the 2015 International Conference on Computational and Mathematical Methods in Science and Engineering CMMSE 2015, 2015, pp. 130-141.
[31] Blows, T.; Lloyd, N., The number of limit cycles of certain polynomial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 98, 215-239, (1984) · Zbl 0603.34020
[32] Chicone, C., Ordinary differential equations with applications, (Texts in Applied Mathematics, vol. 34, (2006), World Scientific: World Scientific Springer-Verlag New York) · Zbl 1120.34001
[33] Dumortier, F.; Llibre, J.; Artés, J., Qualitative theory of planar differential systems, (2006), Springer Berlin Heidelberg: Springer Berlin Heidelberg Springer-Verlag Berlin Heidelberg · Zbl 1110.34002
[34] Perko, L., Differential Equations and Dynamical Systems, (2001), Springer New York · Zbl 0973.34001
[35] Xiao, D.; Ruan, S., Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, Fields Institute Communications, 21, 493-506, (1999) · Zbl 0917.34029
[36] Dhooge, A.; Govaerts, W.; Kuznetsov, Y., MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs, ACM Trans. Math. Software, 29, 141-164, (2003) · Zbl 1070.65574
[37] Gaiko, V., Global bifurcation theory and Hilbert’s sixteenth problem, (Mathematics and Its Applications, vol. 562, (2013), Springer Science & Business Media)
[38] González-Olivares, E.; Mena-Lorca, J.; Rojas-Palma, A.; Flores, J., Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey, Appl. Math. Model., 35, 366-381, (2011) · Zbl 1202.34079
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