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Hopf bifurcations in 3D competitive system with mixing exponential and rational growth rates. (English) Zbl 07197743
Summary: This paper investigates a three-dimensional mixing competitive system with one exponential growth rate and two rational growth rates, whose nullclines are linearly determined. In total, 33 stable nullcline classes exist. Hopf bifurcations are studied in classes 26–31. We provide examples to prove the existence of at least two limit cycles in each of the classes 27–31.
MSC:
37H20 Bifurcation theory for random and stochastic dynamical systems
37G10 Bifurcations of singular points in dynamical systems
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
70K05 Phase plane analysis, limit cycles for nonlinear problems in mechanics
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[1] Franke, J. E.; Yakubu, A. A., Mutual exclusion versus coexistence for discrete competitive systems, J. Math. Biol., 30, 2, 161-168 (1991) · Zbl 0735.92023
[2] Hirsch, M. W., Systems of differential equations which are competitive or cooperative: III. competing species, Nonlinearity, 1, 51-71 (1988) · Zbl 0658.34024
[3] Ruiz-Herrera, A., Exclusion and dominance in discrete population models via the carrying simplex, J. Diff. Equ. Appl., 19, 1, 96-113 (2013) · Zbl 1303.92107
[4] Khajanchi, S., Dynamic behavior of a beddington-deangelis type stage structured predator-prey model, Appl. Math. Comput., 244, 344-360 (2014) · Zbl 1335.92080
[5] Khajanchi, S., Bifurcation analysis of a delayed mathematical model for tumor growth, Chaos Soliton. Fract., 77, 264-276 (2015) · Zbl 1353.92054
[6] Zeeman, M. L., Hopf bifurcations in competitive three-dimensional lotka-volterra systems, Dyn. Stab. Syst., 8, 189-216 (1993) · Zbl 0797.92025
[7] Jiang, J.; Niu, L., On the validity of zeeman’s classification for three dimensional competitive differential equations with linearly determined nullclines, J. Diff. Eqn., 263, 7753-7781 (2017) · Zbl 1381.34068
[8] Jiang, J.; Niu, L.; Zhu, D., On the complete classification of nullcline stable competitive three-dimensional gompertz models, Nonlinear Anal. Real World Appl., 20, 21-35 (2014) · Zbl 1300.34109
[9] Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems and Chaos (2003), Springer-Verlag: Springer-Verlag New York · Zbl 1027.37002
[10] Gantmacher, F. R., The Theory of Matrices, volume 2 (1959), AMS Chelsea Publishing Company: AMS Chelsea Publishing Company New York · Zbl 0085.01001
[11] Meyer, C. D., Matrix Analysis and Applied Linear Algebra Book and Solutions Manual, Society for Industrial and Applied Mathematics (2000)
[12] Guckenheimer, J.; Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (1983), Springer-Verlag · Zbl 0515.34001
[13] Palis, J.; Melo, W., Geometric Theory of Dynamical Systems (1982), Springer-Verlag: Springer-Verlag New York
[14] Providence, RI · Zbl 0821.34003
[15] Hofbauer, J.; Sigmund, K., Evolutionary Games and Population Dynamics (1998), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0914.90287
[16] Jiang, J.; Niu, L., On the equivalent classification of three-dimensional competitive atkinson/allen models relative to the boundary fixed points, Discrete Contin. Dyn. Syst., 36, 217-244 (2016) · Zbl 1366.37049
[17] Jiang, J.; Niu, L., On the equivalent classification of three-dimensional competitive leslie/gower models via the boundary dynamics on the carrying simplex, J. Math. Biol., 74, 1223-1261 (2017) · Zbl 1365.37063
[18] Jiang, J.; Niu, L.; Wang, Y., On heteroclinic cycles of competitive maps via carrying simplices, J. Math. Biol., 72, 939-972 (2016) · Zbl 1355.37042
[19] Gyllenberg, M.; Jiang, J.; Niu, L.; Yan, P., On the classification of generalized competitive atkinson-Allen models via the dynamics on the boundary of the carrying simplex, Discrete Contin. Dyn. Syst., 38, 615-650 (2018) · Zbl 1381.37030
[20] Gyllenberg, M.; Jiang, J.; Niu, L., A note on global stability of three-dimensional ricker models, J. Differ. Equ. Appl., 25, 142-150 (2019) · Zbl 1406.37063
[21] Gyllenberg, M.; Jiang, J.; Niu, L.; Yan, P., On the dynamics of multi-species ricker models admitting a carrying simplex, J. Differ. Equ. Appl., 25, 1489-1530 (2019) · Zbl 1429.37010
[22] Khajanchi, S.; Banerjee, S., Role of constant prey refuge on stage structure predator-prey model with ratio dependent functional response, Appl. Math. Comput., 314, 193-198 (2017) · Zbl 1426.34098
[23] Khajanchi, S., Uniform persistence and global stability for a brain tumor and immune system interaction, Biophys. Rev. Lett., 1-22 (2017)
[24] Gyllenberg, M.; Jiang, J.; Niu, L.; Yan, P., Permanence and universal classification of discrete-time competitive via the carrying simplex, Discrete Contin. Dyn. Syst. Ser. A, 40, 1621-1663 (2020) · Zbl 1432.37116
[25] Hofbauer, J.; So, J. W.H., Uniform persistence and repellors for maps, Proc. Am. Math. Soc., 4, 1137-1142 (1989) · Zbl 0678.58024
[26] Khajanchi, S., Modeling the dynamics of stage-structure predator-prey system with monod-haldane type response function, Appl. Math. Comput., 302, 122-143 (2017) · Zbl 1411.34101
[27] Chen, X.; Jiang, J.; Niu, L., On lotka-volterra equations with identical minimal intrinsic growth rate, SIAM J. Appl. Dynam. Syst., 14, 1558-1599 (2015) · Zbl 1329.34087
[28] Chen, L.; Dong, Z.; Jiang, J.; Niu, L.; Zhai, J., Decomposition formula and stationary measures for stochastic lotka-volterra system with applications to turbulent convection, J. Math.Pures Appl., 125, 43-93 (2019) · Zbl 07047961
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