Globally asymptotic stability in two periodic delayed competitive systems. (English) Zbl 1148.34046

The authors study the following Lotka-Volterra type competition systems with multiple time delays and periodic coefficients \[ \begin{aligned} y_1^{\prime}(t)=&y_1(t)[r_1(t)-a_1(t)y_1(t)+\sum_{i=1}^{n}b_{1i}(t)y_1(t-\tau_i(t)) -\sum_{j=1}^{m}c_{1j}(t)y_2(t-\rho_j(t))],\\ y_2^{\prime}(t)=&y_2(t)[r_2(t)-a_2(t)y_2(t)+\sum_{j=1}^{m}b_{2j}(t)y_2(t-\eta_j(t)) -\sum_{i=1}^{n}c_{2i}(t)y_1(t-\sigma_i(t))], \end{aligned}\tag{1} \] where \(a_1, a_2, c_{1j}, c_{2j}\in C(R, (0,+\infty)),\) \(b_{1i}, b_{2i}\in C(R, R)\), \(\tau_i, \sigma_i,\) \( \rho_j, \eta_j\in C^1(R, [0,+\infty))\) \((i=1,2,\dots, n; j=1,2,\dots, m)\) are \(\omega\)-periodic functions, \(r_k\in C(R, R)\) are \(\omega\)-periodic functions satisfying \(\int_{0}^{\omega}r_kdt>0, k=1,2\). Sufficient conditions are derived for the existence of positive periodic solutions to system (1) by using the continuation theorem developed by R. E. Gaines and J. L. Mawhin [Coincidence degree and nonlinear differential equations. Springer, Berlin (1977; Zbl 0339.47031)].


34K13 Periodic solutions to functional-differential equations
34K20 Stability theory of functional-differential equations
92D25 Population dynamics (general)


Zbl 0339.47031
Full Text: DOI


[1] Freedman, H.I.; Wu, J., Periodic solution of single species models with periodic delay, SIAM J. math. anal., 23, 689-701, (1992) · Zbl 0764.92016
[2] Halbach, U., Life table data and population dynamics of the rotifer brachionous calyciflorus palls as influenced by periodically oscillating termperature, Effects of temperature on ectothermic organisms, (1973), Springer-Verlag Heidelberg, pp. 217-228
[3] Fan, M.; Wang, K., Periodic solutions of single population model with hereditary effect, Appl. math., 13, 58-61, (2000), (in Chinese) · Zbl 1008.92028
[4] Miler, R.K., On voterra’s population equation, SIAM J. appl. math., 14, 446-452, (1996)
[5] Seifert, G., On delay-differential equation for single species population variations, Nonlinear anal. TMA, 9, 1051-1059, (1987) · Zbl 0629.92019
[6] Freedman, H.I.; Xia, H., Periodic solution of single species models with delay, differential equations, dynamical systems and control science, Lecture note pure appl. math., 152, 55-74, (1994) · Zbl 0794.34056
[7] H. Fujimoto, Dynamical behaviours for population growth equations with delays, Nonlinear Anal. TMA 31 (1998) 549-558. · Zbl 0887.34071
[8] Chen, B.S.; Liu, Q.Y., On the stable periodic solutions of single species models with hereditary effect, Math. appl., 12, 42-46, (1999), (in Chinese)
[9] Kuang, Y.; Smith, H.L., Global stability for infinite delay lotka – volterra type systems, J. differ. eq., 103, 221-246, (1993) · Zbl 0786.34077
[10] Zhang, J.; Chen, L., Periodic solutions of single-species nonautonomous diffusion models with continuous time delays, Mothl. comput. modell., 23, 17-27, (1996) · Zbl 0864.60058
[11] Liu, Z.J.; Wang, W.D., Persistence and periodic solutions of a nonautonomous predator – prey diffusion system with Holling. III: functional response and continuous delay, Discr. cont. dynam. syst. ser. B, 4, 653-662, (2004) · Zbl 1101.92052
[12] Alvarez, C.; Lazer, A., An application of topological degree to the periodic competing species problem, J. aust. math. soc., 28, 202-219, (1986) · Zbl 0625.92018
[13] Ahmad, S., Convergence and ultimate bounds of solutions of the nonautonomous volterra – lotka competition equations, J. math. anal. appl., 127, 377-387, (1987) · Zbl 0648.34037
[14] Gopalsamy, K., Exchange of equilibria in two species lotka – volterra competition models, J. aust. math. soc. ser. B, 24, 160-170, (1982) · Zbl 0498.92016
[15] Glopalsamy, K., Global asymptotic stability in a periodic lotka – volterra system, J. aust. math. soc. ser. B, 27, 66-72, (1985) · Zbl 0588.92019
[16] Wang, W.D.; Chen, L.S.; Lu, Z.Y., Global stability of a competition model with periodic coefficients and time delays, Can. appl. math. quart., 3, 365-378, (1995) · Zbl 0845.92020
[17] Burton, T.A., Stability and periodis solutions of ordinary and functional equations, (1985), Academic Press Orlando, FL · Zbl 0635.34001
[18] Liu, Z.J.; Tan, R.H.; Chen, L.S., Global stability in a periodic delayed predator – prey system, Appl. math. comput., 186, 389-403, (2007) · Zbl 1122.34048
[19] Tang, B.R.; Kuang, Y., Existence, uniqueness and asymptotic stability of periodic solutions of periodic functional differential systems, Tohoku math. J., 49, 217-239, (1997) · Zbl 0883.34074
[20] Smith, J.M., Mathematical ideas in biology, (1968), Cambridge University Press London
[21] Gaines, R.E.; Mawhin, J.L., Coincidence degree and nonlinear differential equations, (1977), Springer-Verlag Berlin · Zbl 0326.34021
[22] Gopalsamy, K., Stability and oscillation in delay differential equations of population dynamics, Mathematics and its applications, vol. 74, (1992), Kluwer Academic Dordrecht · Zbl 0752.34039
[23] Fan, M.; Wang, K.; Jiang, D.Q., Existence and global attractivity of positive periodic solutions of periodic n-species lotka – volterra competition systems with several deviating arguments, Math. biosci., 160, 47-61, (1999) · Zbl 0964.34059
[24] Wang, W.D.; Ma, Z.E., Harmless delay for uniform persistence, J. math. anal. appl., 158, 256-268, (1991) · Zbl 0731.34085
[25] Fan, M.; Wang, K.; Ke; Wong, P.J.Y.; Agarwal, R.P., Periodicity and stability in periodic n-species lotka – volterra competition system with feedback controls and deviating arguments, Acta math. sin. (engl. ser.), 19, 801-822, (2003) · Zbl 1047.34080
[26] Tang, X.H.; Cao, D.M.; Zou, X.F., Global attractivity of positive periodic solution to periodic lotka – volterra competition systems with pure delay, J. differ. eq., 228, 580-610, (2006) · Zbl 1113.34052
[27] Teng, Z.D., Nonautonomous lotka – volterra systems with delays, J. differ. eq., 179, 538-561, (2002) · Zbl 1013.34072
[28] Li, Y.K., Periodic solutions for delay lotka – volterra competition systems, J. math. anal. appl, 246, 230-244, (2000) · Zbl 0972.34057
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