Stability and bifurcation analysis for a single-species discrete model with stage structure. (English) Zbl 1445.92256
Summary: In this paper, a single-species discrete model with stage structure is investigated. By analyzing the corresponding characteristic equations, the local asymptotic stability of non-negative equilibrium points and the existence of flip bifurcation are discussed. Using the center manifold theory, the stability of the non-hyperbolic equilibrium point is obtained. Based on bifurcation theory, we obtain the direction and the stability of a flip bifurcation at the positive equilibrium with the birth rate as the bifurcation parameter. Finally, some numerical simulations, including phase portraits, chaotic bands with period windows, and Lyapunov exponent methods, are performed to validate the theoretical results, which extends the results in previous papers.
References:
| [1] | Beddington, J.R., Free, C.A., Lawton, J.H.: Dynamic complexity in predator-prey models framed in difference equations. Nature 255(1), 58-60 (1975) · doi:10.1038/255058a0 |
| [2] | Liu, X., Xiao, D.: Complex dynamic behaviors of a discrete-time predator-prey system. Chaos Solitons Fractals 32(1), 80-94 (2007) · Zbl 1130.92056 · doi:10.1016/j.chaos.2005.10.081 |
| [3] | Hu, Z., Teng, Z., Zhang, L.: Stability and bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response. Nonlinear Anal., Real World Appl. 12(4), 2356-2377 (2011) · Zbl 1215.92063 · doi:10.1016/j.nonrwa.2011.02.009 |
| [4] | Chen, B., Chen, J.: Complex dynamic behaviors of a discrete predator-prey model with stage structure and harvesting. Int. J. Biomath. 10(01), 1750013 (2017) · Zbl 1366.92101 · doi:10.1142/S1793524517500139 |
| [5] | Gámez, M., López, I., Rodríguez, C., et al.: Ecological monitoring in a discrete-time prey-predator model. J. Theor. Biol. 429, 52-60 (2017) · Zbl 1382.92219 · doi:10.1016/j.jtbi.2017.06.025 |
| [6] | Ebenman, B., Persson, L.: Size-Structured Population: Ecology and Evolution. Springer, Berlin (1988) · Zbl 0688.92014 · doi:10.1007/978-3-642-74001-5 |
| [7] | Metz, J., Diekmann, O.: The Dynamics of Physiologically Structured Populations. Lecture Notes in Biomathematics. Springer, New York (1986) · Zbl 0614.92014 · doi:10.1007/978-3-662-13159-6 |
| [8] | Bergh, O., Getz, W.M.: Stability of discrete age-structured and aggregated delay difference population models. J. Math. Biol. 26(5), 551-581 (1988) · Zbl 0714.92016 · doi:10.1007/BF00276060 |
| [9] | Aiello, W.G., Freedman, H.I.: A time-delay model of single-species growth with stage structure. Math. Biosci. 101(2), 139-153 (1990) · Zbl 0719.92017 · doi:10.1016/0025-5564(90)90019-U |
| [10] | Aiello, W.G., Freedman, H.I., Wu, J.: Analysis of a model representing stage-structured population growth with state-dependent time delay. SIAM J. Appl. Math. 52(3), 855-869 (1992) · Zbl 0760.92018 · doi:10.1137/0152048 |
| [11] | Bernard, O., Souissi, S.: Qualitative behavior of stage-structure populations: application to structure validation. J. Math. Biol. 37(4), 291-308 (1998) · Zbl 0919.92035 · doi:10.1007/s002850050130 |
| [12] | Fang, H.: Chaos in a single-species discrete population model with stage structure and birth pulses. Adv. Differ. Equ. 2014, 175 (2014) · Zbl 1344.92132 · doi:10.1186/1687-1847-2014-175 |
| [13] | Tang, S., Chen, L.: Multiple attractors in stage-structured population models with birth pulses. Bull. Math. Biol. 65(3), 479-495 (2003) · Zbl 1334.92371 · doi:10.1016/S0092-8240(03)00005-3 |
| [14] | Khajanchi, S.: Dynamic behavior of a Beddington-DeAngelis type stage structured predator-prey model. Appl. Math. Comput. 244(4), 344-360 (2014) · Zbl 1335.92080 |
| [15] | Song, Y., Tao, Y., Shu, H.: Dynamics of a ratio-dependent stage-structured predator-prey model with delay. Math. Methods Appl. Sci. 40(18), 6451-6467 (2017) · Zbl 1387.34083 · doi:10.1002/mma.4467 |
| [16] | Guo, H., Chen, L.: A study on time-limited control of single-pest with stage-structure. Appl. Comput. Math. 217(2), 677-684 (2010) · Zbl 1198.92039 |
| [17] | Tang, S., Chen, L.: Density-dependent birth rate, birth pulses and their population dynamic consequences. J. Math. Biol. 44(2), 185-199 (2002) · Zbl 0990.92033 · doi:10.1007/s002850100121 |
| [18] | Gao, S., Chen, L.: The effect of seasonal harvesting on single-species discrete population model with stage structure and birth pulses. Chaos Solitons Fractals 24(4), 1013-1023 (2005) · Zbl 1061.92059 · doi:10.1016/j.chaos.2004.09.091 |
| [19] | Murray, J.: Mathematical Biology. Springer, New York (1993) · Zbl 0779.92001 · doi:10.1007/978-3-662-08542-4 |
| [20] | Carr, J.: Application of Center Manifold Theorem. Springer, New York (1981) · Zbl 0464.58001 · doi:10.1007/978-1-4612-5929-9 |
| [21] | Dannan, F.M., Elaydi, S.N., Ponomarenko, V.: Stability of hyperbolic and nonhyperbolic fixed points of one-dimensional maps. J. Differ. Equ. Appl. 9(5), 449-457 (2003) · Zbl 1054.47043 · doi:10.1080/1023619031000078315 |
| [22] | Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 2nd edn. Springer, New York (2004) · Zbl 1082.37002 · doi:10.1007/978-1-4757-3978-7 |
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