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Bifurcation analysis of a Leslie-type predator-prey system with simplified Holling type IV functional response and strong Allee effect on prey. (English) Zbl 1484.34119

Summary: In this paper, a Leslie-type predator-prey system with simplified Holling type IV functional response and strong Allee effect on prey is proposed. The dissipativity of the system and the existence of all possible equilibria are investigated. The investigation emphasizes the exploring of bifurcation. It is shown that the system exists several non-hyperbolic positive equilibria, such as a weak focus of multiplicities one and two, (degenerate) saddle-nodes and Bogdanov-Takens singularities (cusp case) of codimensions 2 and 3. At these equilibria, it is proved that the system undergoes various kinds of bifurcations, such as saddle-node bifurcation, Hopf bifurcation, degenerate Hopf bifurcation and Bogdanov-Takens bifurcation of codimensions 2 and 3. With the parameters selected properly, there exhibits a limit cycle, a homoclinic loop, two limit cycles, a semistable limit cycle, or the simultaneous occurrence of a homoclinic loop and a limit cycle in the system. Moreover, it is also proved that the system has a cusp of codimension at least 4. Hence, there may exist three limit cycles generated from Hopf bifurcation of codimension 3. Numerical simulations are done to support the theoretical results.

MSC:

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
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References:

[1] Lotka, A. J., Elements of Physical Biology (1925), Williams & Wilkins: Williams & Wilkins Baltimore · JFM 51.0416.06
[2] Volterra, V., Fluctuations in the abundance of a species considered mathematically, Nature, 118, 2972, 558-560 (1926) · JFM 52.0453.03
[3] Arancibia-Ibarra, C.; Flores, J. D.; Pettet, G., A Holling-Tanner predator-prey model with strong Allee effect, Int. J. Bifurc. Chaos, 29, 11, Article 1930032 pp. (2019) · Zbl 1439.34049
[4] Anh, T. T.; Du, N. H.; Trung, T. T., On the permanence of predator-prey model with the Beddington-Deangelis functional response in periodic environment, Acta Math. Vietnam., 37, 2, 267-280 (2012) · Zbl 1253.34049
[5] Liang, Z.; Pan, H., Qualitative analysis of a ratio-dependent Holling-Tanner model, J. Math. Anal. Appl., 334, 2, 954-964 (2007) · Zbl 1124.34030
[6] Huang, H., Global stability for a class of predator-prey systems, Siam J. Appl. Math., 55, 3, 763-783 (1995) · Zbl 0832.34035
[7] Chen, S.; Shi, J.; Wei, J., Global stability and hopf bifurcation in a delayed diffusive Leslie-Gower predator-prey system, Int. J. Bifurc. Chaos, 22, 3, Article 1250061 pp. (2012) · Zbl 1270.35376
[8] Wang, L.; Feng, G., Global stability and Hopf bifurcation of a predator-prey model with time delay and stage structure, J. Appl. Math., 2014 (2014) · Zbl 1406.92528
[9] Shang, Z.; Qiao, Y.; Duan, L., Stability and bifurcation analysis in a nonlinear harvested predatorCprey model with simplified Holling type IV functional response, Int. J. Bifurc. Chaos, 30, 14 (2020) · Zbl 1457.34081
[10] Li, B.; Kuang, Y., Heteroclinic bifurcation in the Michaelis-Menten-type ratio-dependent predator-prey system, SIAM J. Appl. Math., 67, 5, 1453-1464 (2007) · Zbl 1132.34320
[11] Wei, C.; Chen, L., Periodic solution and heteroclinic bifurcation in a predator-prey system with Allee effect and impulsive harvesting, Nonlinear Dynam., 76, 2, 1109-1117 (2014) · Zbl 1306.92052
[12] Xiao, Q.; Dai, B.; Xu, B., Homoclinic bifurcation for a general state-dependent Kolmogorov type predator-prey model with harvesting, Nonlinear Anal. Real., 26, 263-273 (2015) · Zbl 1331.34102
[13] Zhao, Y.; Feng, Z.; Zheng, Y., Existence of limit cycles and homoclinic bifurcation in a plant-herbivore model with toxin-determined functional response, J. Differential Equations, 258, 8, 2847-2872 (2015) · Zbl 1316.34053
[14] Ruan, S.; Xiao, D., Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61, 4, 1445-1472 (2001) · Zbl 0986.34045
[15] Xiao, D.; Zhu, H., Multiple focus and Hopf bifurcation in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 66, 3, 802-819 (2006) · Zbl 1109.34034
[16] Huang, J.; Gong, Y.; Ruan, S., Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Discrete Contin. Dyn. Syst. Ser. B, 18, 8, 2101-2121 (2013) · Zbl 1417.34092
[17] Li, Y.; Xiao, D., Bifurcations of a predator-prey system of Holling and Leslie types, Chaos Solitons Fractals, 34, 606-620 (2007) · Zbl 1156.34029
[18] Jiao, J.; Song, Y., Delay-induced Bogdanov-Takens bifurcation in a Leslie-Gower predator-prey model with nonmonotonic functional response, Commun. Nonlinear Sci. Numer. Simul., 19, 7, 2454-2465 (2014) · Zbl 1457.92140
[19] Lamontagne, Y.; Coutu, C.; Rousseau, C., Bifurcation analysis of a predator-prey system with generalised Holling type III functional response, J. Dynam. Differential Equations, 20, 3, 535-571 (2008) · Zbl 1160.34047
[20] Dumortier, F.; Roussarie, R.; Sotomayor, J., Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3, Ergodic Theory Dynam. Syst., 7, 3, 375-413 (1987) · Zbl 0608.58034
[21] Dumortier, F.; Roussarie, R.; Sotomayor, J., (Bifurcation of Planar Vector Fields, Nilpotent Singularities and Abelian Integrals. Bifurcation of Planar Vector Fields, Nilpotent Singularities and Abelian Integrals, Lecture Notes in Mathematics, vol. 1480 (1991), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0755.58002
[22] Chow, S. N.; Li, C.; Wang, D., Normal Forms and Bifurcation of Planar Vector Fields (1994), Cambridge University Press: Cambridge University Press NY · Zbl 0804.34041
[23] Huang, J.; Gong, Y.; Chen, J., Multiple bifurcations in a predator-prey system of Holling and Leslie type with constant-yield prey harvesting, Int. J. Bifurc. Chaos, 23, 10, 50164 (2013) · Zbl 1277.34059
[24] Huang, J.; Xia, X.; Zhang, X., Bifurcation of codimension 3 in a predator-prey system of Leslie type with simplified Holling type IV functional response, Int. J. Bifurc. Chaos, 26, 2, Article 1650034 pp. (2016) · Zbl 1334.34099
[25] Huang, J.; Ruan, S.; Song, J., Bifurcations in a predator-prey system of Leslie type with generalized Holling type III functional response, J. Differential Equations, 257, 6, 1721-1752 (2014) · Zbl 1326.34082
[26] Xiao, D.; Zhang, K., Multiple bifurcations of a predator-prey system, Discrete Contin. Dyn. Syst. Ser. B, 8, 417-437 (2007) · Zbl 1142.34032
[27] Etoua, R. M.; Rousseau, C., Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function of type III, J. Differential Equations, 249, 9, 2316-2356 (2010) · Zbl 1217.34080
[28] Xiao, D., Bifurcations of saddle singularity of codimension three of a planar vector field with nilpotent linear part, Sci. Sin. A, 23, 252-260 (1993)
[29] Cai, L.; Chen, G.; Xiao, D., Multiparametric bifurcations of an epidemiological model with strong Allee effect, J. Math. Biol., 67, 2, 185-215 (2013) · Zbl 1283.34049
[30] Leslie, P. H., Some further notes on the use of matrices in population mathematics, Biometrika, 35, 3-4, 213-245 (1948) · Zbl 0034.23303
[31] Hsu, S. B.; Huang, T. W., Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55, 763-783 (1995) · Zbl 0832.34035
[32] Freedman, H. I.; Mathsen, R. M., Persistence in predator-prey systems with ratio-dependent predator influence, Bull. Math. Biol., 55, 4, 817-827 (1993) · Zbl 0771.92017
[33] Leslie, P. H.; Gower, J. C., The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47, 3-4, 219-234 (1960) · Zbl 0103.12502
[34] Holling, C. S., The functional response of invertebrate predators to prey density, Mem. Entomol. Soc. Can., 98, 48, 1-86 (1966)
[35] Tanner, J. T., The stability and the intrinsic growth rates of prey and predator populations, Ecology, 56, 4, 855-867 (1975)
[36] Collings, J. B., The effects of the functional response on the bifurcation behavior of a mite predator-prey interaction model, J. Math. Biol., 36, 2, 149-168 (1997) · Zbl 0890.92021
[37] Freedman, H. I.; Wolkowicz, G. S.K., Predator-prey systems with group defence: The paradox of enrichment revisted, Bull. Math. Biol., 48, 5-6, 493-508 (1986) · Zbl 0612.92017
[38] B. González-Yaez, E. González-Olivares, J. Mena-Lorca, Multistability on a Leslie-Gower type predator-prey model with nonmonotonic funcional response, BIOMAT 2006, in: Int. Symp. Math. Comput. Biol. 2007.
[39] Zhu, H.; Campbell, S. A.; Wolkowicz, G. S.K., Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 63, 2, 636-682 (2002) · Zbl 1036.34049
[40] Dai, Y.; Zhao, Y., Hopf cyclicity and global dynamics for a predator-prey system of Leslie type with simplified Holling type IV functional response, Int. J. Bifurc. Chaos, 28, 13 (2018) · Zbl 1406.34071
[41] W.C. Allee, Animal aggregation: a study in general sociology, University of Chicago Press, Chicago.
[42] Stephens, P. A.; Freckleton, W., What is the Allee effect?, Oikos, 87, 1, 185-190 (1999)
[43] Courchamp, F.; Clutton-Brock, T.; Grenfell, B., Inverse dependence and the Allee effect, Trends Ecol. Evol., 14, 10, 405-410 (1999)
[44] González-Olivares, E.; Cabrera-Villegas, J.; Córdova-Lepe, F., Competition among predators and Allee effect on prey, their influence on a Gause-type predation model, Math. Probl. Eng., 2019, 1-19 (2019) · Zbl 1435.92053
[45] González-Olivares, E.; Rojas-Palma, A., Multiple limit cycles in a Gause type predator-prey model with Holling type III functional response and Allee effect on prey, Bull. Math. Biol., 73, 6, 1378-1397 (2011) · Zbl 1215.92061
[46] Maiti, A.; Sen, P.; Manna, D.; Samanta, G. P., A predator-prey system with herd behaviour and strong Allee effect, Nonlinear Dyn. Sys. Theory, 16, 1, 86-101 (2016) · Zbl 1343.34120
[47] Saha, S.; Maiti, A.; Samanta, G. P., A Michaelis-Menten predator-prey model with strong Allee effect and disease in prey incorporating prey refuge, Int. J. Bifur. Chaos, 28, 6, Article 1850073 pp. (2018) · Zbl 1394.34095
[48] C. Arancibia-Ibarra, E. González-Olivares, A modified Lesie-Gower predator-prey model with hyperbolic functional response and Allee effect on prey, BIOMAT 2010, in: Int. Symp. Math. Comput. Biol. 2011, pp. 146-162.
[49] Sen, M.; Banerjee, M., Rich global dynamics in a prey-predator model with Allee effect and density dependent death rate of predator, Int. J. Bifur. Chaos, 25, 3, Article 1530007 pp. (2015) · Zbl 1314.34105
[50] Ávila Vales, E.; Estrella-González, Á.; Rivero-Esquivel, E., Bifurcations of a Leslie Gower predator prey model with Holling type III functional response and Michaelis-Menten prey harvesting (2017), arXiv:1711.08081v1
[51] Georgescu, R. M., Saddle-node bifurcation in a competing species model, ROMAI J., 3, 1, 103-110 (2007) · Zbl 1265.34170
[52] Arrowsmith, D. K.; Place, C. M., Ordinary Differential Equations (1982), Chapman and Hall: Chapman and Hall London · Zbl 0481.34005
[53] Arrowsmith, D. K.; Place, C. M., An introduction to dynamical systems (1990), Cambridge University Press · Zbl 0702.58002
[54] Zhang, Z.; Ding, T.; Huang, W.; Dong, Z., Qualitative Theory of Differential Equation (1992), Science Press: Science Press Beijing, (in Chinese): English edition: Transl. Math. Monogr. Vol. 101 (Amer. Math. Soc. Providence, RI) · Zbl 0779.34001
[55] Perko, L., Differential equations and dynamical systems, (Texts in Applied Mathematics, Vol. 7 (2001), Springer-Verlag: Springer-Verlag New York) · Zbl 0973.34001
[56] Li, C.; Li, J.; Ma, Z., Codimension 3 B-T bifurcation in an epidemic model with a nonlinear incidence, Discrete Contin. Dyn. Syst. Ser. B, 20, 4, 1107-1116 (2015) · Zbl 1317.34168
[57] Gaiko, V. A.; Vuik, C., Global dynamics in the Leslie-Gower model with the Allee effect, Int. J. Bifurc. Chaos, 28, 12, Article 1850151 pp. (2018) · Zbl 1404.34058
[58] Aguirre, P.; González-Olivares, E.; Sáez, E., Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, SIAM J. Appl. Math., 69, 5, 1244-1269 (2009) · Zbl 1184.92046
[59] Dai, Y.; Zhao, Y.; Sang, B., Four limit cycles in a predator-prey system of Leslie type with generalized Holling type III functional response, Nonlinear Anal. Real., 50, 218-239 (2019) · Zbl 1432.34064
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