Hybrid matrix models and their population dynamic consequences
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 3, p. 433-450

In this paper, the main purpose is to reveal what kind of qualitative dynamical changes a continuous age-structured model may undergo as continuous reproduction is replaced with an annual birth pulse. Using the discrete dynamical system determined by the stroboscopic map we obtain an exact periodic solution of system with density-dependent fertility and obtain the threshold conditions for its stability. We also present formal proofs of the supercritical flip bifurcation at the bifurcation as well as extensive analysis of dynamics in unstable parameter regions. Above this threshold, there is a characteristic sequence of bifurcations, leading to chaotic dynamics, which implies that the dynamical behavior of the single species model with birth pulses are very complex, including small-amplitude annual oscillations, large-amplitude multi-annual cycles, and chaos. This suggests that birth pulse, in effect, provides a natural period or cyclicity that allows for a period-doubling route to chaos. Finally, we discuss the effects of generation delay on stability of positive equilibrium (or positive periodic solution), and show that generation delay is found to act both as a destabilizing and a stabilizing effect.

DOI : https://doi.org/10.1051/m2an:2003036
Classification:  58J90,  92D40
Keywords: hybrid matrix model, birth pulse, supercritical flip bifurcation, stroboscopic map, generation delay
     author = {Tang, Sanyi},
     title = {Hybrid matrix models and their population dynamic consequences},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {3},
     year = {2003},
     pages = {433-450},
     doi = {10.1051/m2an:2003036},
     zbl = {1027.37052},
     mrnumber = {1994311},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2003__37_3_433_0}
Hybrid matrix models and their population dynamic consequences. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 3, pp. 433-450. doi : 10.1051/m2an:2003036. http://www.numdam.org/item/M2AN_2003__37_3_433_0/

[1] Z. Agur, L. Cojocaru, R. Anderson and Y. Danon, Pulse mass measles vaccination across age cohorts. Proc. Natl. Acad. Sci. USA 90 (1993) 11698-11702.

[2] W.G. Aiello and H.I. Freedman, A time delay model of single-species growth with stage structure. Math. Biosci. 101 (1990) 139-153. | Zbl 0719.92017

[3] W.G. Aiello, H.I. Freedman and J. Wu, Analysis of a model representing stage structured population growth with state-dependent time delay. SIAM J. Appl. Math. 52 (1990) 855-869. | Zbl 0760.92018

[4] D.D. Bainov and P.S. Simeonov, System with impulsive effect: stability, theory and applications. John Wiley & Sons, New York (1989). | MR 1010418 | Zbl 0683.34032

[5] J.R. Bence and R.M. Nisbet, Space limited recruitment in open systems: The importance of time delays. Ecology 70 (1989) 1434-1441.

[6] O. Bernard and J.L. Gouzé, Transient behavior of biological loop models, with application to the droop model. Math. Biosci. 127 (1995) 19-43. | Zbl 0822.92001

[7] O. Bernard and S. Souissi, Qualitative behavior of stage-structure populations: application to structure validation. J. Math. Biol. 37 (1998) 291-308. | Zbl 0919.92035

[8] L.W. Botsford, Further analysis of Clark's delayed recruitment model. Bull. Math. Biol. 54 (1992) 275-293.

[9] J.M. Cushing, Equilibria and oscillations in age-structured population growth models, in Mathematical modelling of environmental and ecological system, J.B. Shukla, T.G. Hallam and V. Capasso Eds., Elsevier, New York (1987) 153-175.

[10] J.M. Cushing, An introduction to structured population dynamics. CBMS-NSF Regional Conf. Ser. in Appl. Math. 71 (1998) 1-10. | Zbl 0939.92026

[11] I.R. Epstein, Oscillations and chaos in chemical systems. Phys. D 7 (1983) 47-56.

[12] J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer Verlag, Berlin, Heidelberg, New York, Tokyo (1990). | MR 1139515 | Zbl 0515.34001

[13] J. Guckenheimer, G. Oster and A. Ipaktchi, The dynamics of density dependent population models. J. Math. Biol. 4 (1977) 101-147. | Zbl 0379.92016

[14] W.S.C. Gurney, R.M. Nisbet and J.L. Lawton, The systematic formulation of tractable single-species population models incorporating age-structure. J. Anim. Ecol. 52 (1983) 479-495.

[15] W.S.C. Gurney, R.M. Nisbet and S.P. Blythe, The systematic formulation of model of predator prey populations. Springer, J.A.J. Metz and O. Dekmann Eds., Berlin, Heidelberg, New York, Lecture Notes Biomath. 68 (1986). | MR 860976

[16] A. Hastings, Age-dependent predation is not a simple process. I. continuous time models. Theor. Popul. Biol. 23 (1983) 347-362. | Zbl 0507.92016

[17] S.P. Hastings, J.J. Tyson and D. Webster, Existence of periodic solutions for negative feedback cellular control systems. J. Differential Equations 25 (1977) 39-64. | Zbl 0361.34038

[18] M.J.B. Hauser, L.F. Olsen, T.V. Bronnikova and W.M. Schaffer, Routes to chaos in the peroxidase-oxidase reaction: period-doubling and period-adding. J. Phys. Chem. B 101 (1997) 5075-5083.

[19] S.M. Henson, Leslie matrix models as “stroboscopic snapshots” of McKendrick PDE models. J. Math. Biol. 37 (1998) 309-328. | Zbl 0936.92027

[20] Y.F. Hung, T.C. Yen and J.L. Chern, Observation of period-adding in an optogalvanic circuit. Phys. Lett. A 199 (1995) 70-74.

[21] E.I. Jury, Inners and stability of dynamic systems. Wiley, New York (1974). | MR 366472 | Zbl 0307.93025

[22] K. Kaneko, On the period-adding phenomena at the frequency locking in a one-dimensional mapping. Progr. Theoret. Phys. 69 (1982) 403-414. | Zbl pre01662702

[23] K. Kaneko, Similarity structure and scaling property of the period-adding phenomena. Progr. Theoret. Phys. 69 (1983) 403-414. | Zbl pre01662702

[24] M.J. Kishi, S. Kimura, H. Nakata and Y. Yamashita, A biomass-based model for the sand lance in Seto Znland Sea. Japan. Ecol. Model. 54 (1991) 247-263.

[25] A. Lakmeche and O. Arino, Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment. Dynam. Contin. Discrete Impuls. Systems 7 (2000) 165-287. | Zbl 1011.34031

[26] V. Laksmikantham, D.D. Bainov and P.S. Simeonov, Theory of impulsive differential equations. World Scientific, Singapore (1989). | MR 1082551 | Zbl 0719.34002

[27] P.H. Leslie, Some further notes on the use of matrices in certain population mathematics. Biometrika 35 (1948) 213-245. | Zbl 0034.23303

[28] S.A. Levin, Age-structure and stability in multiple-age spawning populations. Springer-Verlag, T.L. Vincent and J.M. Skowrinski Eds., Berlin, Heidelberg, New York, Lecture Notes Biomath. 40 (1981) 21-45. | Zbl 0456.92016

[29] S. A. Levin and C.P. Goodyear, Analysis of an age-structured fishery model. J. Math. Biol. 9 (1980) 245-274. | Zbl 0424.92020

[30] T. Lindstrom, Dependencies between competition and predation-and their consequences for initial value sensitivity. SIAM J. Appl. Math. 59 (1999) 1468-1486. | Zbl 0991.92036

[31] J.A.J. Metz and O. Diekmann, The dynamics of physiologically structured populations. Springer, Berlin, Heidelberg, New York, Lecture notes Biomath. 68 (1986). | MR 860959 | Zbl 0614.92014

[32] A.J. Nicholson, An outline of the dynamics of animal populations. Aust. J. Zool. 2 (1954) 9-65.

[33] A.J. Nicholson, The self adjustment of populations to change. Cold Spring Harbor Symp. Quant. Biol. 22 (1957) 153-173.

[34] J.C. Panetta, A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competition environment. Bull. Math. Biol. 58 (1996) 425-447. | Zbl 0859.92014

[35] B. Shulgin, L. Stone and Z. Agur, Pulse vaccination strategy in the SIR epidemic model. Bull. Math. Biol. 60 (1998) 1-26. | Zbl 0941.92026

[36] S.Y. Tang and L.S. Chen, Density-dependent birth rate, birth pulses and their population dynamic consequences. J. Math. Biol. 44 (2002) 185-199. | Zbl 0990.92033

[37] G. Uribe, On the relationship between continuous and discrete models for size-structured population dynamics. Ph.D. dissertation, Interdisciplinary program in applied mathematics, University of Arizona, Tucson, USA (1993).