# Hybrid matrix models and their population dynamic consequences

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 37, Issue: 3, page 433-450
- ISSN: 0764-583X

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topTang, Sanyi. "Hybrid matrix models and their population dynamic consequences." ESAIM: Mathematical Modelling and Numerical Analysis 37.3 (2010): 433-450. <http://eudml.org/doc/194172>.

@article{Tang2010,

abstract = {
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.
},

author = {Tang, Sanyi},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Hybrid matrix model; birth pulse; supercritical flip bifurcation;
stroboscopic map; generation delay.; hybrid matrix model; stroboscopic map; generation delay},

language = {eng},

month = {3},

number = {3},

pages = {433-450},

publisher = {EDP Sciences},

title = {Hybrid matrix models and their population dynamic consequences},

url = {http://eudml.org/doc/194172},

volume = {37},

year = {2010},

}

TY - JOUR

AU - Tang, Sanyi

TI - Hybrid matrix models and their population dynamic consequences

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 37

IS - 3

SP - 433

EP - 450

AB -
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.

LA - eng

KW - Hybrid matrix model; birth pulse; supercritical flip bifurcation;
stroboscopic map; generation delay.; hybrid matrix model; stroboscopic map; generation delay

UR - http://eudml.org/doc/194172

ER -

## References

top- Z. Agur, L. Cojocaru, R. Anderson and Y. Danon, Pulse mass measles vaccination across age cohorts. Proc. Natl. Acad. Sci. USA90 (1993) 11698–11702.
- W.G. Aiello and H.I. Freedman, A time delay model of single-species growth with stage structure. Math. Biosci.101 (1990) 139–153. Zbl0719.92017
- 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. Zbl0760.92018
- D.D. Bainov and P.S. Simeonov, System with impulsive effect: stability, theory and applications. John Wiley & Sons, New York (1989). Zbl0676.34035
- J.R. Bence and R.M. Nisbet, Space limited recruitment in open systems: The importance of time delays. Ecology70 (1989) 1434–1441.
- O. Bernard and J.L. Gouzé, Transient behavior of biological loop models, with application to the droop model. Math. Biosci.127 (1995) 19–43. Zbl0822.92001
- O. Bernard and S. Souissi, Qualitative behavior of stage-structure populations: application to structure validation. J. Math. Biol.37 (1998) 291–308. Zbl0919.92035
- L.W. Botsford, Further analysis of Clark's delayed recruitment model. Bull. Math. Biol.54 (1992) 275–293.
- 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.
- J.M. Cushing, An introduction to structured population dynamics. CBMS-NSF Regional Conf. Ser. in Appl. Math.71 (1998) 1–10.
- I.R. Epstein, Oscillations and chaos in chemical systems. Phys. D7 (1983) 47–56.
- J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer Verlag, Berlin, Heidelberg, New York, Tokyo (1990). Zbl0515.34001
- J. Guckenheimer, G. Oster and A. Ipaktchi, The dynamics of density dependent population models. J. Math. Biol.4 (1977) 101–147. Zbl0379.92016
- 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.
- 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). Zbl0542.92019
- A. Hastings, Age-dependent predation is not a simple process. I. continuous time models. Theor. Popul. Biol.23 (1983) 347–362. Zbl0507.92016
- S.P. Hastings, J.J. Tyson and D. Webster, Existence of periodic solutions for negative feedback cellular control systems. J. Differential Equations25 (1977) 39–64. Zbl0361.34038
- 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. B101 (1997) 5075–5083.
- S.M. Henson, Leslie matrix models as “stroboscopic snapshots" of McKendrick PDE models. J. Math. Biol.37 (1998) 309–328. Zbl0936.92027
- Y.F. Hung, T.C. Yen and J.L. Chern, Observation of period-adding in an optogalvanic circuit. Phys. Lett. A199 (1995) 70–74.
- E.I. Jury, Inners and stability of dynamic systems. Wiley, New York (1974). Zbl0307.93025
- K. Kaneko, On the period-adding phenomena at the frequency locking in a one-dimensional mapping. Progr. Theoret. Phys.69 (1982) 403–414. Zbl1098.37520
- K. Kaneko, Similarity structure and scaling property of the period-adding phenomena. Progr. Theoret. Phys.69 (1983) 403–414. Zbl1200.37077
- 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.
- A. Lakmeche and O. Arino, Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment. Dynam. Contin. Discrete Impuls. Systems7 (2000) 165–287. Zbl1011.34031
- V. Laksmikantham, D.D. Bainov and P.S. Simeonov, Theory of impulsive differential equations. World Scientific, Singapore (1989).
- P.H. Leslie, Some further notes on the use of matrices in certain population mathematics. Biometrika35 (1948) 213–245. Zbl0034.23303
- 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. Zbl0456.92016
- S.A. Levin and C.P. Goodyear, Analysis of an age-structured fishery model. J. Math. Biol.9 (1980) 245–274. Zbl0424.92020
- T. Lindstrom, Dependencies between competition and predation-and their consequences for initial value sensitivity. SIAM J. Appl. Math.59 (1999) 1468–1486. Zbl0991.92036
- J.A.J. Metz and O. Diekmann, The dynamics of physiologically structured populations. Springer, Berlin, Heidelberg, New York, Lecture notes Biomath. 68 (1986). Zbl0614.92014
- A.J. Nicholson, An outline of the dynamics of animal populations. Aust. J. Zool.2 (1954) 9–65.
- A.J. Nicholson, The self adjustment of populations to change. Cold Spring Harbor Symp. Quant. Biol.22 (1957) 153–173.
- 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. Zbl0859.92014
- B. Shulgin, L. Stone and Z. Agur, Pulse vaccination strategy in the SIR epidemic model. Bull. Math. Biol.60 (1998) 1–26. Zbl0941.92026
- 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. Zbl0990.92033
- 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).

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