# zbMATH — the first resource for mathematics

Some new results on nonautonomous Lotka-Volterra competitive systems with delays. (English) Zbl 0947.34066
The authors consider the following $$N$$-species nonautonomous Lotka-Volterra type system with finite or infinite delays: $\begin{split} x_i'(t)= x_i(t)\Bigl[ b_i(t)- a_i(t)x_i (t)-\sum^n_{j=1} a_{ij}(t)x_j \bigl(t- \tau_{ij} (t)\bigr) \\ -\sum^n_{j=1}\int^0_{-\sigma_{ij}} c_{ij}(t,s) x_j (t+s)ds \Bigr], \quad i=1,2, \dots,n.\end{split}\tag{1}$ They establish a series of criteria under which a part of the $$n$$ species is driven to extinction while the remaining part of the $$n$$ species is persistent or uniformly persistent, or coexists globally asymptotically stably.
The results in the paper generalize and improve some known results on nondelayed $$N$$-species Lotka-Volterra type competitive systems.
Reviewer: V.Petrov (Plovdiv)

##### MSC:
 34K20 Stability theory of functional-differential equations 92D25 Population dynamics (general)
Full Text:
##### References:
 [1] Ahmad, S.; Mohana Rao, M.R., Asymptotically periodic solutions of n-competing species problem with time delay, J. math. anal. appl., 186, 557-571, (1994) · Zbl 0818.45004 [2] Bereketoglu, H.; Gyori, I., Global asymptotic stability in a nonautonomous lotka – volterra type system with infinite delay, J. math. anal. appl., 210, 279-291, (1997) · Zbl 0880.34072 [3] Freedman, H.I.; Ruan, S., Uniform persistence in functional differential equations, J. differential equations, 115, 173-192, (1995) · Zbl 0814.34064 [4] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Dordrecht · Zbl 0752.34039 [5] Hale, J.K., Theory of functional differential equations, (1977), Springer-Verlag Heidelberg · Zbl 0425.34048 [6] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press Boston · Zbl 0777.34002 [7] Kuang, Y., Global stability in delayed nonautonomous lotka – volterra type systems without saturated equilibria, Differential integral equations, 9, 557-567, (1996) · Zbl 0843.34077 [8] Kuang, Y.; Smith, H.L., Global stability for infinite delay lotka – volterra type systems, J. differential equations, 103, 221-246, (1993) · Zbl 0786.34077 [9] Kuang, Y.; Tang, B., Uniform persistence in nonautonomous delay differential Kolmogorov type population models, Rocky mountain J. math., 24, 165-186, (1994) · Zbl 0823.92021 [10] Li, S.; Wen, L., Theory of functional differential equations, (1987), Science and Technology Press Hunan [11] de Oca, F.Montes; Zeeman, M.L., Balancing survival and extinction in nonautonomous competitive lotka – volterra systems, J. math. anal. appl., 192, 360-370, (1995) · Zbl 0830.34039 [12] de Oca, F.Montes; Zeeman, M.L., Extinction in nonautonomous competitive lotka – volterra systems, Proc. amer. math. soc., 124, 3677-3687, (1996) · Zbl 0866.34029 [13] Tang, B.; Kuang, Y., Permanence in Kolmogorov type systems of nonautonomous functional differential equation, J. math. anal. appl., 197, 427-447, (1996) · Zbl 0951.34051 [14] Vance, R.R.; Coddington, E.A., A nonautonomous model of population growth, J. math. biol., 27, 491-506, (1989) · Zbl 0716.92016 [15] Wang, W.; Chen, L.; Lu, Z., Global stability of a competition model with periodic coefficients and time delays, Canad. appl. math. quart., 3, 365-373, (1995) [16] Wang, L.; Zhang, Y., Global stability of volterra – lotka systems with delay, Differential equations dynam. sys., 3, 205-216, (1995) · Zbl 0880.34074
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.