Harmless delays in a discrete ratio-dependent periodic predator-prey system

Fan, Yong-Hong; Li, Wan-Tong

doi:https://doi.org/10.1155/DDNS/2006/12176

Discrete Dynamics in Nature and Society

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Volume 2006 | Article ID 012176 | https://doi.org/10.1155/DDNS/2006/12176

Harmless delays in a discrete ratio-dependent periodic predator-prey system

Yong-Hong Fan¹and Wan-Tong Li²

Received18 Dec 2005

Accepted13 Feb 2006

Published25 Jun 2006

Abstract

Verifiable criteria are established for the existence of positive periodic solutions and permanence of a delayed discrete periodic predator-prey model with Holling-type II functional response N1(k+1)=N1(k)exp⁡{b1(k)−a1(k)N1(k−[τ1])−α1(k)N2(k)/(N1(k)+m(k)N2(k))} and N2(k+1)=N2(k)exp⁡{−b2(k)+α2(k)N1(k−[τ2])/(N1(k−[τ2])+m(k)N2(k−[τ2]))}. Our results show that the delays in the system are harmless for the existence of positive periodic solutions and permanence of the system. In particular our investigation confirms that if the death rate of the predator is rather small as well as the intrinsic growth rate of the prey is relatively large, then the species could coexist in the long run.

References

R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, vol. 228 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, 2000.
View at: Zentralblatt MATH | MathSciNet
R. Arditi, N. Perrin, and H. Saiah, “Functional response and heterogeneities: an experiment test with cladocerans,” Oikos, vol. 60, pp. 69–75, 1991.
View at: Google Scholar
E. Beretta and Y. Kuang, “Global analyses in some delayed ratio-dependent predator-prey systems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 32, no. 3, pp. 381–408, 1998.
View at: Publisher Site | Google Scholar | MathSciNet
A. A. Berryman, “The origins and evolution of predator-prey theory,” Ecology, vol. 73, pp. 1530–1535, 1992.
View at: Google Scholar
Y.-H. Fan, W.-T. Li, and L.-L. Wang, “Periodic solutions of delayed ratio-dependent predator-prey models with monotonic or nonmonotonic functional responses,” Nonlinear Analysis. Real World Applications, vol. 5, no. 2, pp. 247–263, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. Fan and K. Wang, “Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system,” Mathematical and Computer Modelling, vol. 35, no. 9-10, pp. 951–961, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H. I. Freedman, Deterministic Mathematical Models in Population Ecology, vol. 57 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, 1980.
View at: Zentralblatt MATH | MathSciNet
K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, 1992.
View at: Zentralblatt MATH | MathSciNet
I. Hanski, “The functional response of predators: worries about scale,” Trends in Ecology & Evolution, vol. 6, no. 5, pp. 141–142, 1991.
View at: Google Scholar
C. S. Holling, “The functional response of predator to prey density and its role in mimicry and population regulation,” Memoirs of Entomological Society of Canada, vol. 45, pp. 1–60, 1965.
View at: Google Scholar
S.-B. Hsu, T.-W. Hwang, and Y. Kuang, “Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system,” Journal of Mathematical Biology, vol. 42, no. 6, pp. 489–506, 2001.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S.-B. Hsu, T.-W. Hwang, and Y. Kuang, “Rich dynamics of a ratio-dependent one-prey two-predators model,” Journal of Mathematical Biology, vol. 43, no. 5, pp. 377–396, 2001.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H.-F. Huo and W.-T. Li, “Permanence and global stability of positive solutions of a nonautonomous discrete ratio-dependent predator-prey model,” Discrete Dynamics in Nature and Society, vol. 2005, no. 2, pp. 135–144, 2005.
View at: Publisher Site | Google Scholar
C. Jost, O. Arino, and R. Arditi, “About deterministic extinction in ratio-dependent predator-prey models,” Bulletin of Mathematical Biology, vol. 61, no. 1, pp. 19–32, 1999.
View at: Publisher Site | Google Scholar
V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, vol. 256 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, 1993.
View at: Zentralblatt MATH | MathSciNet
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering, Academic Press, Massachusetts, 1993.
View at: Zentralblatt MATH | MathSciNet
Y. Kuang and E. Beretta, “Global qualitative analysis of a ratio-dependent predator-prey system,” Journal of Mathematical Biology, vol. 36, no. 4, pp. 389–406, 1998.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
W.-T. Li, Y.-H. Fan, and S. Ruan, “Periodic solutions in a delayed predator-prey model with nonmonotonic functional response,” submitted.
View at: Google Scholar
R. M. May, Complexity and Stability in Model Ecosystems, Princeton University Press, New Jersey, 1973.
R. M. May, “Biological populations obeying difference equations: stable points, stable cycles, and chaos,” Journal of Theoretical Biology, vol. 51, no. 2, pp. 511–524, 1975.
View at: Publisher Site | Google Scholar
J. D. Murry, Mathematical Biology, Springer, New York, 1989.
M. L. Rosenzweig, “Paradox of enrichment: destabilization of exploitation ecosystem in ecological time,” Science, vol. 171, pp. 385–387, 1971.
View at: Google Scholar
S. Ruan and D. Xiao, “Global analysis in a predator-prey system with nonmonotonic functional response,” SIAM Journal on Applied Mathematics, vol. 61, no. 4, pp. 1445–1472, 2001.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Y. Takeuchi, Global Dynamical Properties of Lotka-Volterra Systems, World Scientific, New Jersey, 1996.
View at: Zentralblatt MATH | MathSciNet
L.-L. Wang and W.-T. Li, “Existence and global stability of positive periodic solutions of a predator-prey system with delays,” Applied Mathematics and Computation, vol. 146, no. 1, pp. 167–185, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
L.-L. Wang and W.-T. Li, “Periodic solutions and permanence for a delayed nonautonomous ratio-dependent predator-prey model with Holling type functional response,” Journal of Computational and Applied Mathematics, vol. 162, no. 2, pp. 341–357, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
D. Xiao and S. Ruan, “Global dynamics of a ratio-dependent predator-prey system,” Journal of Mathematical Biology, vol. 43, no. 3, pp. 268–290, 2001.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
D. Xiao and Z. Zhang, “On the uniqueness and nonexistence of limit cycles for predator-prey systems,” Nonlinearity, vol. 16, no. 3, pp. 1185–1201, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H. Zhu, S. A. Campbell, and G. S. K. Wolkowicz, “Bifurcation analysis of a predator-prey system with nonmonotonic functional response,” SIAM Journal on Applied Mathematics, vol. 63, no. 2, pp. 636–682, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

Copyright

Copyright © 2006 Yong-Hong Fan and Wan-Tong Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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