## 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)
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### References:

 [1] Kocic, V.L.; Ladas, G., Global behavior of nonlinear difference equations of higher order with applications, (1993), 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 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 New York · Zbl 0448.92023 [10] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), 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 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 [19] May, R.M., Complexity and stability in model ecosystems, (1973), 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 New York [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 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 · 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, (in Chinese)
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