## Multiple positive periodic solutions of a delayed discrete predator-prey system with type IV functional responses.(English)Zbl 1142.39015

The authors consider the following discrete predator-prey system with type IV functional responses and delays:
\begin{aligned} x_1(k+ 1)&= x_1(k)\exp\Biggl[b_1(k)-a_1(k)x_1(k- \tau_1(k))-{c(k)x_2(k-\sigma(k))\over(x^2_1(k- \tau_2(k))/n)+ x_1(k-\tau_2(k))+ a}\Biggr],\\ x_2(k+1)&= x_2(k)\exp\Biggl[-b_2(k)+ {a_2(k)x_1(k-\tau_2(k))\over (x^2_1(k-\tau_2(k))/n)+ x_1(k- \tau_2(k))+ a}\Biggr] \end{aligned}
(where for $$i=1,2$$, $$b_i: Z\to\mathbb R$$, $$c,a_i: Z\to\mathbb R^+$$, $$\tau_i,\sigma: Z\to Z^+$$ are all $$\omega$$ periodic, $$n$$ and $$a$$ are positive constants) for the initial condition
\begin{aligned} x_1(-m)&\geq 0,\quad m=1,2,\dots,\max\{\tau_1(k), \tau_2(k),\sigma(k)\},\quad x(0)> 0.\\ x_2(-m)&\geq 0,\quad m= 1,2,\dots,\max\{\tau_1(k), \tau_2(k), \sigma(k)\},\quad y(0)> 0. \end{aligned}
In the paper a theorem for the existence of positive periodic solutions of the system is given.

### MSC:

 39A11 Stability of difference equations (MSC2000) 92D25 Population dynamics (general) 39A20 Multiplicative and other generalized difference equations
Full Text:

### References:

 [1] Agarwal, R. P., (Difference Equations and Inequalities: Theory, Methods and Applications. Difference Equations and Inequalities: Theory, Methods and Applications, Monographs and Textbooks in Pure and Applied Mathematics, vol. 228 (2000), Marcel Dekker Inc.: Marcel Dekker Inc. New York) · Zbl 0952.39001 [2] Arditi, R.; Perrin, N.; Saiah, H., Functional response and heterogeneities: An experimental test with cladocerans, OIKOS, 60, 69-75 (1991) [3] Berryman, A. A., The origins and evolution of predator-prey theory, Ecology, 75, 1530-1535 (1992) [4] Chen, Y. M., Multiple periodic solutions of delayed predator-prey systems with type IV functional responses, Nonlinear Anal., 5, 45-53 (2004) · Zbl 1066.92050 [5] Fan, M.; Wang, K., Periodic solutions of a discrete time non-autonomous ratio-dependent predator-prey system, Math. Comput. Modelling, 35, 951-961 (2002) · Zbl 1050.39022 [6] Freedman, H. I., Deterministic Mathematical Models in Population Ecology (1980), Marcel Dekker: Marcel Dekker New York · Zbl 0448.92023 [7] Gaines, R. E.; Mawhin, J. L., Coincidence Degree and Nonlinear Differential Equations (1977), Springer-Verlag: Springer-Verlag Berlin · Zbl 0326.34021 [8] Gopalsamy, K., Stability and Oscillations in Delay Differential Equations of Population Dynamics (1992), Kluwer Academic: Kluwer Academic Boston · Zbl 0752.34039 [9] Hanski, I., The functional response of predator: Worries about scale, TREE, 6, 141-142 (1991) [10] Holling, C. S., The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Ent. Soc. Can., 45, 1-60 (1965) [11] Maynard, S. J., Models in Ecology (1974), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0312.92001 [12] May, R. M., Stability and Complexity in Model Ecosystems (1974), Princeton University Press [13] Murry, J. D., Mathematical Biology (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0682.92001 [14] Rosenzweig, M. L.; MacArthur, R. H., Graphical representation and stability conditions of predator-prey interactions, Amer. Nat., 47, 209-223 (1963) [15] Rosenzweig, M. L., Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time, Science, 171, 385-387 (1969) [16] Xu, R.; Chaplain, M. A.J.; Davidson, F. A., Periodic solutions for a predator-prey model with Holling-type functional response and time delays, Appl. Math. Comput., 161, 637-654 (2005) · Zbl 1064.34053 [17] 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 (2004) · Zbl 1029.92025 [18] 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
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.