## Global asymptotic stability of periodic Lotka-Volterra systems with delays.(English)Zbl 0995.34071

The authors consider the periodic Lotka-Volterra-type system with finite or infinite delays $\begin{split} \frac{dx_{i}(t)}{dt}=x_{i}(t)\left[b_{i}(t)-a_{i}(t)x_{i}(t)- \sum_{j=1}^{n}a_{ij}(t)x_{j}(t-\tau_{ij}(t))\right.\\ \left. -\sum_{j=1}^{n} \int_{-\sigma_{ij}}^{0} c_{ij}(t,s) x_{j}(t+s) ds\right],\;\;i=1,2,\ldots,n, \end{split} \tag{1}$ where $$b_{i}(t)$$, $$a_{i}(t)$$, $$a_{ij}(t)$$ are $$\omega$$-periodic and continuous functions on $$\mathbb{R}$$; $$c_{ij}(t,s)$$ are $$\omega$$-periodic and continuous with respect to $$t$$ on $$\mathbb{R}$$ and integrable in $$s$$ on $$[-\sigma_{ij},0]$$; there exists a continuous positive function $$h_{0}$$ defined on $$(-\infty,0]$$ and $$0<l=\int_{-\infty}^{0}h_{0}(s) ds<\infty$$ such that $$|c_{ij}(t,s)|\leq h_{0}(s)$$ for all $$(t,s)\in\mathbb{R} \times[-\sigma_{ij},0]$$; $$\tau_{ij}(t)\geq 0$$ are $$\omega$$-periodic, continuously differentiable functions on $$\mathbb{R}$$, $$d\tau_{ij}(t)/dt<1$$ and $$\sigma_{ij}$$ are nonnegative constants or $$\sigma_{ij}=\infty$$; $$i,j=1,\ldots,n$$. There are positive constants $$c_{1},\ldots, c_{n}$$ such that the functions $\gamma_{i}(t)=c_{i}a_{i}(t)-\sum_{j=1}^{n}c_{j} \left(\frac{|a_{ji}(\psi^{-1}_{ji}(t))|} {1-\dot{\tau}_{ji}(\psi^{-1}_{ji}(t))}+ \int_{-\sigma_{ji}}^{0}|c_{ji}(t-s,s)|ds\right),$ where $$\psi^{-1}_{ji}(t)$$ is the inverse function of $$\psi_{ji}(t)=1-\tau_{ji}(t)$$, are nonnegative and $$\displaystyle\sum_{j=1}^{\infty}\int_{\alpha_{j}}^{\beta_{j}} \gamma_{i}(t) dt=\infty$$ for any interval sequence $$\{[\alpha_{i},\beta_{i}]\}$$ such that $$[\alpha_{i},\beta_{i}]\cap[\alpha_{j},\beta_{j}]=\emptyset$$ and $$\beta_{i}-\alpha_{i}=\beta_{j}-\alpha_{j}>0$$ for $$i,j=1,2,\ldots$$, $$i\neq j$$.
System (1) is said to be persistent, if for any positive solution $$x_{1}(t),\ldots,x_{n}(t)$$ there exist positive constants $$m$$, $$M$$, $$T$$ such that $$m\leq x_{i}(t)\leq M$$, $$i=1,\ldots,n$$, for $$t\geq T$$.
Under the above assumptions the main result presented in the paper asserts that if system (1) is persistent, then it has a unique positive $$\omega$$-periodic solution which is globally asymptotically stable.
Sufficient conditions for the persistence of system (1) under some additional assumptions are also given. As a sequence, the authors obtain a concrete criterion for the existence and global asymptotic stability of a positive periodic solution to the competitive system (1).

### MSC:

 34K20 Stability theory of functional-differential equations 34K13 Periodic solutions to functional-differential equations 34K60 Qualitative investigation and simulation of models involving functional-differential equations 45J05 Integro-ordinary differential equations 92D25 Population dynamics (general)
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### References:

 [1] Ahlip, R.A.; King, R.R., Global asymptotic stability of a periodic system of delay logistic equations, Bull. austral. math. soc., 53, 373-389, (1996) · Zbl 0888.34061 [2] Bereketoglu, H.; Goyri, I., Global asymptotical stability in a nonautonomous lotka – volterra type system with infinite delay, J. math. anal. appl., 210, 279-291, (1997) · Zbl 0880.34072 [3] J. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathmatical Sciences, Academic Press, New York, 1979. · Zbl 0484.15016 [4] Eilbeck, J.C.; Lopez-Gomez, J., On the periodic lotka – volterra competition model, J. math. anal. appl., 210, 58-87, (1997) · Zbl 0874.34039 [5] Freedman, H.I.; Ruan, S., Uniform persistence in functional differential equations, J. differential equations, 115, 173-192, (1995) · Zbl 0814.34064 [6] 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 [7] Gopalsamy, K., Global asymptotic stability in a periodic integrodifferential system, Tohoku math. J., 37, 323-332, (1985) · Zbl 0587.45013 [8] K. Gopalsamy, Stability and Oscillations in Delay Differential Equation of Population Dynamics, Kluwer Academic, Dordrecht, 1992. · Zbl 0752.34039 [9] Hale, J.K.; Kato, J., Phase space for retarded equations with infinite delay, Funkcial. ekvac., 21, 11-41, (1978) · Zbl 0383.34055 [10] He, X.Z.; Gopalsamy, K., Persistence, stability and level crossing in an integrodifferential system, J. math. biol., 32, 395-426, (1994) · Zbl 0807.92020 [11] Y. Kuang, Delay Differential Equation with Applications in Population Dynamics, Academic Press, Boston, 1993. · Zbl 0777.34002 [12] Kuang, Y., Global stability in delayed nonautonomous lotka – volterra type systems without saturated equilibria, Differential integral equations, 9, 557-567, (1996) · Zbl 0843.34077 [13] S. Li, L. Wen, Theory of Functional Differential Equations, Science and Technology Press, Hunan, Changsha, 1987 (in Chinese). [14] Sawano, K., Exponential asymptotic stability for functional differential equations with infinite retardations, Tohoku math. J., 31, 363-382, (1979) · Zbl 0449.34053 [15] Tang, B.; Kuang, Y., Permanence in Kolmogorov-type systems of nonautonomous functional differential equations, J. math. anal. appl., 197, 427-447, (1996) · Zbl 0951.34051 [16] Tang, B.; 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 [17] Tineo, A., An iterative scheme for the N-competing species problem, J. differential equations, 116, 1-15, (1995) · Zbl 0823.34048 [18] K. Wang, Positive periodic solutions of an n-species ecologic system, Acta Math. Appl. Sinica 17 (1) (1994) 1-8 (in Chinese). [19] Wang, W.; Chen, L.; Lu, Z., Global stability of a competition model with periodic coefficients and time delays, Canad. appl. math. quart., 3, 365-378, (1995) · Zbl 0845.92020 [20] Wang, K.; Huang, Q., Norm |·|_{h} and the periodic solutions of Volterra integro-differential equations (in Chinese), J. northeast normal univ., 3, 7-16, (1985) [21] K. Wang, Q. Huang, Ch-space and the boundedness and periodic solutions of functional differential equations with infinite delay, Science in China, Ser. A 3 (1987) 242-252 (in Chinese).
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