Persistence and global stability in Lotka-Volterra delay differential systems. (English) Zbl 1061.34057

Muroya, Y.

Appl. Math. Lett. 17, No. 7, 795-800 (2004).

Summary: Consider the persistence and the global asymptotic stability of models governed by the following Lotka-Volterra delay differential system \[ \frac {dx_i(t)}{dt}=x_i(t)\left\{c_i- a_ix_i(t)-\sum^m_{j=1}a_{ij}x_j(t-\tau_{ij}) \right\}, \quad t\geq t_0,\;1\leq i\leq n, \]
\[ x_i(t)=\varphi_i(t)\geq 0,\quad t\leq t_0,\quad\text{and}\quad \varphi_i(t_0)>0,\quad 1\leq i\leq n, \] where each \(\varphi_i(t)\) is a continuous function for \(t\leq t_0\), each \(c_i\), \(a_i\), and \(a_{ij}\) are finite and \[ a_i>0,\;a_i+a_{ii}>0,\;1\leq i\leq n,\quad\text{and} \quad \tau_{ij}\geq 0,\;1\leq i,j\leq n. \] Here, we obtain conditions for the persistence of the system, and extending a technique offered by Saito, Hara and Ma for \(n=2\) to the above system for \(n\geq 2\). We establish new conditions for the global asymptotic stability of the positive equilibrium which improve the well-known result of Gopalsamy for some special cases.

MSC:

34K20	Stability theory of functional-differential equations
34K25	Asymptotic theory of functional-differential equations
92D25	Population dynamics (general)

Keywords:

Persistence; Global asymptotic stability; Lotka-Volterra delay differential system; Lyapunov-like function

References:

[1]	Muroya, Y., Persistence and global stability for nonautonomous Lotka-Volterra delay differential systems, Communications on Applied Nonlinear Analysis, 9, 31-45 (2002) · Zbl 1056.34075
[2]	Saito, Y.; Hara, T.; Ma, W., Necessary and sufficient conditions for permanence and global stability of a Lotka-Volterra system with two delays, J. Math. Anal. Appl., 236, 534-556 (1999) · Zbl 0944.34059
[3]	Muroya, Y., Permanence and global stability in a Lotka-Volterra predator-prey system with delays, Appl. Math. Lett., 16, 8, 1245-1250 (2003) · Zbl 1054.34127
[4]	Gopalsamy, K., Global asymptotic stability in Volterra’s population systems, J. Math. Biol., 19, 157-168 (1984) · Zbl 0535.92020
[5]	Saito, Y., The necessary and sufficient condition for global stability of Lotka-Volterra cooperative or competition system with delays, J. Math. Anal. Appl., 268, 109-124 (2002) · Zbl 1012.34072
[6]	Muroya, Y., Persistence and global stability for discrete models of nonautonomous Lotka-Volterra type, J. Math. Anal. Appl., 273, 492-511 (2002) · Zbl 1033.39013
[7]	Ahmad, S.; Lazer, A. C., Necessary and sufficient average growth in a Lotka-Volterra system, Nonlinear Analysis TMA, 34, 191-228 (1998) · Zbl 0934.34037
[8]	Ahmad, S.; Lazer, A. C., Average conditions for global asymptotic stability in a nonautonomous Lotka-Volterra system, Nonlinear Analysis TMA, 40, 37-49 (2000) · Zbl 0955.34041
[9]	Muroya, Y., Uniform persistence for Lotka-Volterra-type delay differential systems, Nonlinear Analysis RWA, 4, 689-710 (2003) · Zbl 1044.34035