×

Permanence for a delayed discrete ratio-dependent predator-prey system with Holling type functional response. (English) Zbl 1063.39013

For the system \[ \begin{aligned} N_1(k+1) &= N_1(k)\exp \{b_1(k)- a_1(k)N_1(k-\tau_1)- \alpha_1(k) Z(k)N_2(k)/N_1(k)\},\\ N_2(k+1) &= N_2(k)\exp\{-b_2(k)+ \alpha_2(k)Z(k-\tau_2)\},\end{aligned} \] with the abbreviation \(Z(k)=N_1^2 (k)/(N_1^2(k)+m^2N^2_2(k))\) sufficient conditions are given such that all positive solutions are asymptotically uniformly bounded and uniformly bounded away from zero.

MSC:

39A12 Discrete version of topics in analysis
92D25 Population dynamics (general)
39A20 Multiplicative and other generalized difference equations
39A11 Stability of difference equations (MSC2000)
Full Text: DOI

References:

[1] Kocic, V. L.; Ladas, G., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications (1993), Kluwer Academic: Kluwer Academic London · Zbl 0787.39001
[2] Agarwal, R. P., Difference Equations and Inequalities: Theory, Methods and Applications, Monogr. Textbooks Pure Appl. Math., vol. 228 (2000), Dekker: Dekker New York · Zbl 0952.39001
[3] Arditi, R.; Perrin, N.; Saiah, H., Functional response and heterogeneities: an experiment test with cladocerans, OIKOS, 60, 69-75 (1991)
[4] Beretta, E.; Kuang, Y., Global analysis in some delayed ratio-dependent predator-prey systems, Nonlinear Anal. TMA, 32, 381-408 (1998) · Zbl 0946.34061
[5] Berryman, A. A., The origins and evolution of predator-prey theory, Ecology, 73, 1530-1535 (1992)
[6] Fan, M.; Wang, K., Periodicity in a delayed ratio-dependent predator-prey system, J. Math. Anal. Appl., 262, 179-190 (2001) · Zbl 0994.34058
[7] Fan, M.; Wang, K., Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system, Math. Comput. Modelling, 35, 951-961 (2002) · Zbl 1050.39022
[8] Fan, Y. H.; Li, W. T.; Wang, L. L., Periodic solutions of delayed ratio-dependent predator-prey models with monotonic or nonmonotonic functional response, Nonlinear Anal. RWA, 5, 247-263 (2004) · Zbl 1069.34098
[9] Freedman, H. I., Deterministic Mathematical Models in Population Ecology (1980), Dekker: Dekker New York · Zbl 0448.92023
[10] Gopalsamy, K., Stability and Oscillations in Delay Differential Equations of Population Dynamics (1992), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0752.34039
[11] Hanski, I., The functional response of predator: worries bout scale, TREE, 6, 141-142 (1991)
[12] Holling, C. S., The functional response of predator to prey density and its role in mimicry and population regulation, Mem. Entomol. Sec. Can., 45, 1-60 (1965)
[13] Hsu, S. B.; Hwang, T. W.; Kuang, Y., Global analysis of the Michaelis-Menten type ratio-dependent predator-prey system, J. Math. Biol., 42, 489-506 (2001) · Zbl 0984.92035
[14] Hsu, S. B.; Hwang, T. W.; Kuang, Y., Rich dynamics of a ratio-dependent one-prey two-predators model, J. Math. Biol., 43, 377-396 (2001) · Zbl 1007.34054
[15] Jost, C.; Arino, O.; Arditi, R., About deterministic extinction in ratio-dependent predator-prey models, Bull. Math. Biol., 61, 19-32 (1999) · Zbl 1323.92173
[16] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press New York · Zbl 0777.34002
[17] Kuang, Y.; Beretta, E., Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36, 389-406 (1998) · Zbl 0895.92032
[18] W.T. Li, Y.H. Fan, S.G. Ruan, Periodic solutions in a delayed predator-prey model with nonmonotonic functional response, submitted for publication; W.T. Li, Y.H. Fan, S.G. Ruan, Periodic solutions in a delayed predator-prey model with nonmonotonic functional response, submitted for publication
[19] May, R. M., Complexity and Stability in Model Ecosystems (1973), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ
[20] May, R. M., Biological populations obeying difference equations: stable points, stable cycles and chaos, J. Theory Biol., 51, 511-524 (1975)
[21] Murry, J. D., Mathematical Biology (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0682.92001
[22] Rosenzweig, M. L., Paradox of enrichment: destabilization of exploitation ecosystem in ecological time, Science, 171, 385-387 (1971)
[23] Ruan, S.; Xiao, D., Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61, 1445-1472 (2001) · Zbl 0986.34045
[24] Takeuchi, Y., Global Dynamical Properties of Lotka-Volterra Systems (1996), World Scientific: World Scientific Singapore · Zbl 0844.34006
[25] Wang, L. L.; Li, W. T., Existence of periodic solutions of a delayed predator-prey system with functional response, International J. Math. Math. Sci., 1, 55-63 (2002) · Zbl 1075.34067
[26] Wang, L. L.; Li, W. T., Existence and global stability of positive periodic solutions of a predator-prey system with delays, Appl. Math. Comput., 146, 167-185 (2003) · Zbl 1029.92025
[27] Wang, L. L.; Li, W. T., Periodic solutions and permanence for a delayed nonautonomous ratio-dependent predator-prey model with Holling type functional response, J. Comput. Appl. Math., 162, 341-357 (2004) · Zbl 1076.34085
[28] L.L. Wang, W.T. Li, Periodic solutions and stability for a delayed discrete ratio-dependent predator-prey system with Holling type functional response, Discrete Dyn. Nat. Soc., in press; L.L. Wang, W.T. Li, Periodic solutions and stability for a delayed discrete ratio-dependent predator-prey system with Holling type functional response, Discrete Dyn. Nat. Soc., in press · Zbl 1073.39008
[29] Xiao, D.; Ruan, S., Global dynamics of a ratio-dependent predator-prey system, J. Math. Biol., 43, 268-290 (2001) · Zbl 1007.34031
[30] Xiao, D.; Zhang, Z., On the uniqueness and nonexistence of limit cycles for predator-prey systems, Nonlinearity, 16, 1185-1201 (2003) · Zbl 1042.34060
[31] Zhu, H. P.; Campebell, S.; Wolkowicz, G., Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 63, 636-682 (2002) · Zbl 1036.34049
[32] Wang, L.; Wang, M. Q., Ordinary Difference Equations (1989), Xinjiang Univ. Press: Xinjiang Univ. Press Xinjiang, (in Chinese)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.