Hybrid matrix models and their population dynamic consequences

Sanyi Tang

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2003)

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

Abstract

top
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.

How to cite

top

Tang, Sanyi. "Hybrid matrix models and their population dynamic consequences." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 37.3 (2003): 433-450. <http://eudml.org/doc/246062>.

@article{Tang2003,
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 - Modélisation Mathématique et Analyse Numérique},
keywords = {hybrid matrix model; birth pulse; supercritical flip bifurcation; stroboscopic map; generation delay},
language = {eng},
number = {3},
pages = {433-450},
publisher = {EDP-Sciences},
title = {Hybrid matrix models and their population dynamic consequences},
url = {http://eudml.org/doc/246062},
volume = {37},
year = {2003},
}

TY - JOUR
AU - Tang, Sanyi
TI - Hybrid matrix models and their population dynamic consequences
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2003
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
UR - http://eudml.org/doc/246062
ER -

References

top
  1. [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. [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. Zbl0719.92017
  3. [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. Zbl0760.92018
  4. [4] D.D. Bainov and P.S. Simeonov, System with impulsive effect: stability, theory and applications. John Wiley & Sons, New York (1989). Zbl0683.34032MR1010418
  5. [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. [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. Zbl0822.92001
  7. [7] O. Bernard and S. Souissi, Qualitative behavior of stage-structure populations: application to structure validation. J. Math. Biol. 37 (1998) 291–308. Zbl0919.92035
  8. [8] L.W. Botsford, Further analysis of Clark’s delayed recruitment model. Bull. Math. Biol. 54 (1992) 275–293. 
  9. [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. [10] J.M. Cushing, An introduction to structured population dynamics. CBMS-NSF Regional Conf. Ser. in Appl. Math. 71 (1998) 1–10. Zbl0939.92026
  11. [11] I.R. Epstein, Oscillations and chaos in chemical systems. Phys. D 7 (1983) 47–56. 
  12. [12] J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer Verlag, Berlin, Heidelberg, New York, Tokyo (1990). Zbl0515.34001MR1139515
  13. [13] J. Guckenheimer, G. Oster and A. Ipaktchi, The dynamics of density dependent population models. J. Math. Biol. 4 (1977) 101–147. Zbl0379.92016
  14. [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. [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). Zbl0542.92019MR860976
  16. [16] A. Hastings, Age-dependent predation is not a simple process. I. continuous time models. Theor. Popul. Biol. 23 (1983) 347–362. Zbl0507.92016
  17. [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. Zbl0361.34038
  18. [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. [19] S.M. Henson, Leslie matrix models as “stroboscopic snapshots” of McKendrick PDE models. J. Math. Biol. 37 (1998) 309–328. Zbl0936.92027
  20. [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. [21] E.I. Jury, Inners and stability of dynamic systems. Wiley, New York (1974). Zbl0307.93025MR366472
  22. [22] K. Kaneko, On the period-adding phenomena at the frequency locking in a one-dimensional mapping. Progr. Theoret. Phys. 69 (1982) 403–414. Zbl1200.37077
  23. [23] K. Kaneko, Similarity structure and scaling property of the period-adding phenomena. Progr. Theoret. Phys. 69 (1983) 403–414. Zbl1200.37077
  24. [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. [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. Zbl1011.34031
  26. [26] V. Laksmikantham, D.D. Bainov and P.S. Simeonov, Theory of impulsive differential equations. World Scientific, Singapore (1989). Zbl0719.34002MR1082551
  27. [27] P.H. Leslie, Some further notes on the use of matrices in certain population mathematics. Biometrika 35 (1948) 213–245. Zbl0034.23303
  28. [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. Zbl0456.92016
  29. [29] S. A. Levin and C.P. Goodyear, Analysis of an age-structured fishery model. J. Math. Biol. 9 (1980) 245–274. Zbl0424.92020
  30. [30] T. Lindstrom, Dependencies between competition and predation-and their consequences for initial value sensitivity. SIAM J. Appl. Math. 59 (1999) 1468–1486. Zbl0991.92036
  31. [31] J.A.J. Metz and O. Diekmann, The dynamics of physiologically structured populations. Springer, Berlin, Heidelberg, New York, Lecture notes Biomath. 68 (1986). Zbl0614.92014MR860959
  32. [32] A.J. Nicholson, An outline of the dynamics of animal populations. Aust. J. Zool. 2 (1954) 9–65. 
  33. [33] A.J. Nicholson, The self adjustment of populations to change. Cold Spring Harbor Symp. Quant. Biol. 22 (1957) 153–173. 
  34. [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. Zbl0859.92014
  35. [35] B. Shulgin, L. Stone and Z. Agur, Pulse vaccination strategy in the SIR epidemic model. Bull. Math. Biol. 60 (1998) 1–26. Zbl0941.92026
  36. [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. Zbl0990.92033
  37. [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). 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.