×

Periodicity in a nonlinear discrete predator-prey system with state dependent delays. (English) Zbl 1152.34367

Summary: With the help of a continuation theorem based on Gaines and Mawhin’s coincidence degree, easily verifiable criteria are established for the global existence of positive periodic solutions of the following nonlinear discrete state dependent delays predator-prey system
\[ \left\{\begin{aligned} N_1(k+1) & = N_1(k) \exp \left[ b_1(k)-\sum^n_{i=1} a_i(k)(N_1(k-\tau_i(k,N_1(k),N_2(k))))^{\alpha_i}\right. \\ & \left.- \sum^m_{j=1} c_j(k)(N_2(k-\sigma_j(k,N_1(k),N_2(k))))^{\beta_j}\right],\\ N_2 (k+1) & = N_2 (k) \exp \left[ -b_2(k)+\sum^n_{i=1} d_i (k) (N_1(k-\rho_i(k,N_1(k),N_2(k))))^{\gamma_i}\right]\end{aligned}\right. \]
where \(a_i,c_j,d_i:Z\rightarrow R^{+}\) are positive \(\omega \)-periodic, \(\omega \) is a fixed positive integer. \(b_{1},b_{2}:Z\rightarrow R^{+}\) are \(\omega \)-periodic and \(\sum^{\omega-1}_{k=0} b_i(k) > 0\). \(\tau_i,\sigma_j,\rho_i:Z\times R\times R\rightarrow R\) (\(i=1,2,\dots, n,j=1,2,\dots ,m)\) are \(\omega \)-periodic with respect to their first arguments, respectively. \(\alpha_i,\beta_j,\gamma_i\) (\(i=1,2,\dots ,n,j=1,2,\dots ,m)\) are positive constants.

MSC:

34K13 Periodic solutions to functional-differential equations
34C25 Periodic solutions to ordinary differential equations
34K20 Stability theory of functional-differential equations
92D25 Population dynamics (general)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] R.P. Agarwal, Difference equations and inequalities: theory, methods and applications, Monographs and Textbooks in Pure and Applied Mathematics, vol. 228, Marcel Dekker, New York, 2000.; R.P. Agarwal, Difference equations and inequalities: theory, methods and applications, Monographs and Textbooks in Pure and Applied Mathematics, vol. 228, Marcel Dekker, New York, 2000. · Zbl 0952.39001
[2] Aiello, W.; Freedman, H. I.; Wu, J., Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52, 855-869 (1992) · Zbl 0760.92018
[3] Chen, F. D.; Chen, X.; Cao, J.; Chen, A., Positive periodic solutions of a class of non-autonomous single species population model with delays and feedback control, Acta Math. Appl. Sin. Engl. Ser., 21, 6, 1319-1336 (2005) · Zbl 1110.34049
[4] Chen, F. D.; Lin, F.; Chen, X., Sufficient conditions for the existence of positive periodic solutions of a class of neutral delay models with feedback control, Appl. Math. Comput., 158, 1, 45-68 (2004) · Zbl 1096.93017
[5] Chen, F. D.; Shi, J. L., Periodicity in a Logistic type system with several delays, Comput. Math. Appl., 48, 1-2, 35-44 (2004) · Zbl 1061.34050
[6] Chen, F. D.; Shi, J. L., Periodicity in a nonlinear predator-prey system with state dependent delays periodicity in a nonlinear predator-prey system with state dependent delays, Acta Math. Appl. Sin., 26, 1, 49-60 (2005) · Zbl 1096.34050
[7] Chen, F. D.; Sun, D.; Chen, X., Periodicity in a food-limited population model with toxicants and state dependent delays, J. Math. Anal. Appl., 288, 1, 132-142 (2003)
[8] Cushing, J. M., Periodic time-dependent predator-prey system, SIAM J. Appl. Math., 32, 82-95 (1977) · Zbl 0348.34031
[9] Fan, G. H.; Ouyang, Z. G., Existence of positive periodic solution for a single species models with state dependent delay, J. Biomath., 17, 2, 173-178 (2002)
[10] Fan, M.; Agarwal, S., Periodic solutions for a class of discrete time competition systems, Nonlinear Stud., 9, 3, 249-261 (2002) · Zbl 1032.39002
[11] Fan, M.; Agarwal, S., Periodic solutions of nonautonomous discrete predator-prey system of Lotka-Volterra type, Appl. Anal., 81, 4, 801-812 (2002) · Zbl 1022.39015
[12] Fan, M.; Wang, K., Global existence of positive periodic solution of predator-prey system with deviating arguments, Acta Math. Appl. Sin., 23, 557-561 (2000) · Zbl 0969.34065
[13] Fan, M.; Wang, K., Global periodic solutions of a generalized \(n\)-species Gilpin-Ayala competition model, Comput. Math. Appl., 40, 1141-1151 (2000) · Zbl 0954.92027
[14] Fan, M.; Wang, K., Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system, Math. Comput. Modelling, 35, 9-10, 951-961 (2002) · Zbl 1050.39022
[15] Fengde, C., On a nonlinear non-autonomous predator-prey model with diffusion and distributed delay, J. Comput. Appl. Math., 180, 1, 33-49 (2005) · Zbl 1061.92058
[16] Freedman, H. I., Deterministic Mathematical Models in Population Ecology (1980), Marcel Dekker: Marcel Dekker New York · Zbl 0448.92023
[17] Freedman, H. I.; Wu, J., Periodic solutions of single-species models with periodic delay, SIAM J. Math. Anal., 23, 689-701 (1992) · Zbl 0764.92016
[18] Gaines, R. E.; Mawhin, J. L., Coincidence Degree and Nonlinear Differential Equations (1977), Springer: Springer Berlin · Zbl 0339.47031
[19] Goh, B. S., Management and Analysis of Biological Populations (1980), Elsevier Scientific: Elsevier Scientific The Netherlands
[20] K. Gopalsamy, Stability and oscillation in delay differential equations of population dynamics, Mathematics and its Applications, vol. 74, Kluwer Academic Publishers Group, Dordrecht, 1992.; K. Gopalsamy, Stability and oscillation in delay differential equations of population dynamics, Mathematics and its Applications, vol. 74, Kluwer Academic Publishers Group, Dordrecht, 1992. · Zbl 0752.34039
[21] Hale, J. K.; Lunel, S. M., Introduction to Functional Differential Equations (1993), Springer: Springer New York · Zbl 0787.34002
[22] Y. Kuang, Delay differential equations with application in population dynamics, The Series of Mathematics in Science and Engineering, vol. 191, Academic Press, Boston, 1993.; Y. Kuang, Delay differential equations with application in population dynamics, The Series of Mathematics in Science and Engineering, vol. 191, Academic Press, Boston, 1993. · Zbl 0777.34002
[23] Li, Y. K., Periodic solutions of a periodic delay predator-prey system, Proc. Amer. Math. Soc., 127, 1331-1335 (1999) · Zbl 0917.34057
[24] Li, Y. K., Periodic solutions for delay Lotka-Volterra competition systems, J. Math. Anal. Appl., 246, 230-244 (2000) · Zbl 0972.34057
[25] Li, Y. K., Periodic solutions of periodic delay Lotka-Volterra equations and systems, J. Math. Anal. Appl., 255, 265-280 (2001) · Zbl 1024.34062
[26] Li, C. R.; Lu, S. J., The qualitative analysis of \(N\)-species periodic coefficient, nonlinear relation, prey-competition systems, Appl. Math—JCU, 12, 2, 147-156 (1997), (in Chinese) · Zbl 0880.34042
[27] Murry, J. D., Mathematical Biology (1989), Springer: Springer New York · Zbl 0682.92001
[28] Zhao, J. D.; Chen, W. C., The qualitative analysis of \(N\)-species nonlinear prey competition systems, Appl. Math. Comput., 149, 2, 567-576 (2004) · Zbl 1045.92038
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.