A discrete predator-prey system with age-structure for predator and natural barriers for prey

Sanyi Tang; Lansun Chen

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2001)

  • Volume: 35, Issue: 4, page 675-690
  • ISSN: 0764-583X

Abstract

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We analyze a two species discrete predator-prey model in which the prey disperses between two patches of a heterogeneous environment with barriers and the mature predator disperses between the patches with no barrier. By using the discrete dynamical system generated by a monotone, concave maps for subcommunity of prey, we obtain the subcommunity of prey exists an equilibrium which attracts all positive solutions, and using the stability trichotomy results on the monotone and continuous operator, we obtain some sufficient conditions for the permanence of species. These results are applied to the models with rational growth functions and exponential growth functions. We also present numerical examples to illustrate the dynamic complexity of systems.

How to cite

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Tang, Sanyi, and Chen, Lansun. "A discrete predator-prey system with age-structure for predator and natural barriers for prey." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 35.4 (2001): 675-690. <http://eudml.org/doc/194068>.

@article{Tang2001,
abstract = {We analyze a two species discrete predator-prey model in which the prey disperses between two patches of a heterogeneous environment with barriers and the mature predator disperses between the patches with no barrier. By using the discrete dynamical system generated by a monotone, concave maps for subcommunity of prey, we obtain the subcommunity of prey exists an equilibrium which attracts all positive solutions, and using the stability trichotomy results on the monotone and continuous operator, we obtain some sufficient conditions for the permanence of species. These results are applied to the models with rational growth functions and exponential growth functions. We also present numerical examples to illustrate the dynamic complexity of systems.},
author = {Tang, Sanyi, Chen, Lansun},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {age-structure; natural barrier; subcommunity; persistence; discrete predator-prey model; discrete dynamic system; stability trichotomy},
language = {eng},
number = {4},
pages = {675-690},
publisher = {EDP-Sciences},
title = {A discrete predator-prey system with age-structure for predator and natural barriers for prey},
url = {http://eudml.org/doc/194068},
volume = {35},
year = {2001},
}

TY - JOUR
AU - Tang, Sanyi
AU - Chen, Lansun
TI - A discrete predator-prey system with age-structure for predator and natural barriers for prey
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2001
PB - EDP-Sciences
VL - 35
IS - 4
SP - 675
EP - 690
AB - We analyze a two species discrete predator-prey model in which the prey disperses between two patches of a heterogeneous environment with barriers and the mature predator disperses between the patches with no barrier. By using the discrete dynamical system generated by a monotone, concave maps for subcommunity of prey, we obtain the subcommunity of prey exists an equilibrium which attracts all positive solutions, and using the stability trichotomy results on the monotone and continuous operator, we obtain some sufficient conditions for the permanence of species. These results are applied to the models with rational growth functions and exponential growth functions. We also present numerical examples to illustrate the dynamic complexity of systems.
LA - eng
KW - age-structure; natural barrier; subcommunity; persistence; discrete predator-prey model; discrete dynamic system; stability trichotomy
UR - http://eudml.org/doc/194068
ER -

References

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  1. [1] J.S. Allen, M.P. Moulton and F.L. Rose, Persistence in an age-structure population for a patch-type environment. Nat. Resour. Model. 4 (1990) 197–214. Zbl0850.92063
  2. [2] J.R. Beddington, C.A. Free and J.H. Lawton, Dynamical complexity in predator-prey models framed in difference equations. Nature 255 (1975) 58–60. 
  3. [3] J.F. Chen, Influence of high dimension terms for qualitative structure of solutions of a second order linear difference system with ordinary coefficient in the neighborhood of a singular point. Acta Math. Appl. Sinica (China) 11 (1988) 299–311. Zbl0673.39002
  4. [4] M.E. Clark and L.J. Gross, Periodic solutions to nonautonomous difference equations. Math. Biosci. 102 (1990) 105–119. Zbl0712.39014
  5. [5] J.M. Cushing, An introduction to structured population dynamics. SIAM Soc. Indus. Appl. Math., Philadelphia (1998). Zbl0939.92026MR1636703
  6. [6] H.I. Freedman and J.W.-H. So, Persistence in discrete models of a population which may be subjected to harvesting. Nat. Resour. Model. 2 (1987) 135–145. Zbl0850.92074
  7. [7] H.I. Freedman and Y. Takeuchi, Global stability and predator dynamics in a model of prey dispersal in a patchy environment. Nonlinear Anal. TMA 13 (1989) 993–1002. Zbl0685.92018
  8. [8] H.I. Freedmen and W.H. Josephso, Persistence in discrete semidynamical systems. SIAM J. Math. Anal. 20 (1989) 930–938. Zbl0676.92011
  9. [9] I. Gumowski and C. Mira, Recurrences and discrete dynamics systems. Lect. Notes Math. 809 (1980) 61–96. Zbl0449.58003
  10. [10] A. Hastings, Complex interactions between dispersal and dynamics: Lessons from coupled logistic equations. Ecology 74 (1993) 1362–1372. 
  11. [11] V. Hutson and K. Schmitt, Permanence and the dynamics of biological systems. Math. Biosci. 111 (1992) 1–71. Zbl0783.92002
  12. [12] U. Krause and P. Ranet, A limit set trichotomy for monotone nonlinear dynamical systems. Nonlinear Anal. TMA 19 (1992) 375–392. Zbl0779.34038
  13. [13] Y. Kuang and Y. Takeuchi, Predator-prey dynamics in models of prey dispersal in two patch environments. Math. Biosci. 120 (1994) 77–98. Zbl0793.92014
  14. [14] N.R. Leblond, Porcupine caribou herd. Canadian Arctic Resources Comn., Offuwa (1979). 
  15. [15] S.A. Levin, Dispersion and population interactions. Amer. Natur. 108 (1974) 207–228. 
  16. [16] J. Li, Persistence in discrete age-structure population models. Bull. Math. Biol. 50 (1992) 351–366. Zbl0659.92019
  17. [17] A. Okubo, Diffusion and ecological problems, math. models. Springer, Berlin (1980). Zbl0422.92025MR572962
  18. [18] K. Schumacher, Regions and oscillations in second order predator-prey recurrences. J. Math. Biol. 16 (1983) 221–231. Zbl0513.92018
  19. [19] J.F. Selgrade and G. Namkoong, Stable periodic behavior in a pioneer-climax model. Nat. Resour. Model. 4 (1990) 215–227. Zbl0850.92060
  20. [20] J.G. Skellam, Random dispersal in theoretical populations. Biometrika 38 (1951) 196–218. Zbl0043.14401
  21. [21] H.L. Smith, Cooperative systems of differential equations with concave nonlinearities. Nonlinear Anal. TMA 10 (1986) 1037–1052. Zbl0612.34035
  22. [22] Y. Takeuchi, Cooperative systems theory and global stability of diffusion models. Acta Appl. Math. 14 (1989) 49–57. Zbl0665.92017
  23. [23] W.D. Wang and L.S. Chen, A predator-prey system with stage-structure for predator. Comput. Math. Appl. 33 (1997) 83–91. 
  24. [24] A.-A. Yakubu, Prey dominance in discrete predator-prey system with a prey refuge. Math. Biosci. 144 (1997) 155–178. Zbl0896.92031

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