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Dynamic complexities in a single-species discrete population model with stage structure and birth pulses. (English) Zbl 1066.92041

Summary: Natural populations, whose population numbers are small and generations non-overlapping, can be modelled by difference equations that describe how the populations evolve in discrete time-steps. This paper investigates a recent study on the dynamic complexities in a single-species discrete population model with stage structure and birth pulses. Using the stroboscopic map, we obtain an exact cycle of the system, and obtain a threshold conditions for its stability. Above this, there is a characteristic sequence of bifurcations, leading to chaotic dynamics, which implies that the dynamical behaviors of the single-species discrete model with birth pulses are very complex, including (a) non-unique dynamics, meaning that several attractors and chaos coexist; (b) small-amplitude annual oscillations; (c) large-amplitude multi-annual cycles; (d) chaos. Some interesting results are obtained and they showed that pulsing provides a natural period or cyclicity that allows for a period-doubling route to chaos.

MSC:

92D25 Population dynamics (general)
37N25 Dynamical systems in biology
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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[1] EBenman, B.; persson, L., Size-structured populations: ecology and evolution, (1998), Spring Berlin
[2] Metz JAJ, Diekmann O. The dynamics of physiologically structured populations. Lecture Notes in Biomathematics, vol. 68, 1986 · Zbl 0614.92014
[3] Aiello, W.G.; Freedman, H.I., A time-delay model of single-species growth with stage structure, Math. biosci., 101, 139-153, (1990) · Zbl 0719.92017
[4] 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., 3, 855-869, (1992) · Zbl 0760.92018
[5] Cai, Y.; Fan, J.; Gard, T.C., The effects of a state-structured population growth model, Nonlin. anal. th. mech. appl., 16, 20, 95-105, (1992) · Zbl 0777.92014
[6] Freedman, H.I.; Wu, J.H., Persistence and global asymptotic stability of single species dispersal models with stage structure, Quart. appl. math., 2, 351-371, (1991) · Zbl 0732.92021
[7] Hastings, A., Delay in recruitment at different trophic levels: effects on stability, J. math. biol., 21, 35-44, (1984) · Zbl 0547.92014
[8] Gurney, W.S.C.; Nisbet, R.M.; Lawton, J.H., The systematic formulation of tractable single species population models incorporating age structure, J. animal ecol., 52, 479-485, (1983)
[9] Caswell, H., Matrix population models, ()
[10] Tang, S.Y.; Chen, L.S., Density-dependent birth rate, birth pulses and their population dynamic consequences, J. math. biol., 64, 169-184, (2002)
[11] Bainov, D.D.; Simeonov, P.S., Impulsive differential equations: periodic solutions and applications, Pitman monogr. surveys pure appl. math., 66, (1993) · Zbl 0793.34011
[12] Hastings, A., Delay in recruitment at different trophic levels: effects on stability, J. math. biol., 21, 35-44, (1984) · Zbl 0547.92014
[13] Hastings, A.; Higgins, K.; Hinggins, K.; Hastings, A.; Botsford, L., Density dependence and age structure: nonlinear dynamics and population behavior, Science, Am. nat., 149, 247-269, (1997)
[14] Freedman, H., Deterministic mathematical models in population ecology, (1980), Marcel Dekker New York · Zbl 0448.92023
[15] Jury, E.I., Inners and stability of dynamic system, (1974), Wiley New York · Zbl 0307.93025
[16] May, R.M., Biological population with nonoverlapping generations: stable points, stable cycles, and chaos, Science, 186, 645-647, (1974)
[17] May, R.M.; Oster, G.F., Bifurcation and synamic complexity in simple ecological models, Am. nat., 110, 573-599, (1976)
[18] Eckmann, J.P., Routes to chaos with special emphasis on period doubling, () · Zbl 0616.58032
[19] Collet, P.; Eckmann, J.P., Iterated maps of the interval as dynamical systems, Importance time delays, 70, 1434-1441, (1989)
[20] Hauser, M.J.B.; Olsen, L.F.; Bronnikova, T.V.; Schaffer, W.M., Routs to chaos in the peroxidase – oxidase reaction: period-doubling and period-adding, J. phys. chem. B, 101, 5075-5083, (1997)
[21] Hung, Y.F.; Yen, T.C.; Chern, J.L.; Kaneko, K., On the period-adding phenomena at the frequency locking in a one-dimensional mapping, Phys. lett. A, Prog. theor. phys., 69, 403-414, (1982)
[22] Guckenheimer, J.; Oster, G.; Ipaktchi, A., The dynamics of density dependent population models, J. math. biol., 4, 101-147, (1977) · Zbl 0379.92016
[23] Gakkhar, S.; Naji, R.K., Chaos in seasonally perturbed ratio-dependent prey – predator system, Chaos, solitons and fractals, 15, 107-118, (2003) · Zbl 1033.92026
[24] Upadhyay, R.K.; Rai, V.A.; lyengar, S.R., How do ecosystems respond to external perturbations?, Chaos, solitons and fractals, 11, 1963-1982, (2000) · Zbl 0985.37105
[25] Tang, S.Y.; Chen, L.S., Quasiperiodic solutions and chaos in a periodically forced predator – prey model with age structure for predator, Int. J. bifurcat. chaos, 13, 973-980, (2003) · Zbl 1063.37586
[26] Hasting, A., Comples interactions between dispersal and dynamics: lessons from coupled logistic equations, Ecology, 74, 1362-1372, (1993)
[27] Beddington, J.R.; Free, C.A.; Lawton, J.H., Dynamic complexity in predator – prey models framed in difference equations, Nature, 255, 58-60, (1975)
[28] Kaitala, V.; Ranta, E., Red/blue chaotic power spectra, Nature, 381, 198-199, (1996)
[29] Tang, S.Y.; Chen, L.S., Chaos in functional response host-parasitiod ecosystem model, Chaos, solitons and fractals, 13, 875-884, (2002) · Zbl 1022.92042
[30] Rohani, P.; Miramontes, O., Immigration and the persistence of chaos in population models, J. theor. biol., 175, 203-206, (1995)
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