Persistence and global stability in discrete models of Lotka–Volterra type

Abstract

In this paper, we establish new sufficient conditions for global asymptotic stability of the positive equilibrium in the following discrete models of Lotka–Volterra type: { N i ( p + 1 ) = N i ( p ) exp { c i − a i N i ( p ) − ∑ j = 1 n a i j N j ( p − k i j ) } , p ⩾ 0 , 1 ⩽ i ⩽ n , N i ( p ) = N i p ⩾ 0 , p ⩽ 0 , and N i 0 > 0 , 1 ⩽ i ⩽ n , where each N i p for p ⩽ 0 , each c i , a i and a i j are finite and { a i > 0 , a i + a i i > 0 , 1 ⩽ i ⩽ n , and k i j ⩾ 0 , 1 ⩽ i , j ⩽ n . Applying the former results [Y. Muroya, Persistence and global stability for discrete models of nonautonomous Lotka–Volterra type, J. Math. Anal. Appl. 273 (2002) 492–511] on sufficient conditions for the persistence of nonautonomous discrete Lotka–Volterra systems, we first obtain conditions for the persistence of the above autonomous system, and extending a similar technique to use a nonnegative Lyapunov-like function offered by Y. Saito, T. Hara and W. Ma [Y. Saito, T. Hara, W. Ma, Necessary and sufficient conditions for permanence and global stability of a Lotka–Volterra system with two delays, J. Math. Anal. Appl. 236 (1999) 534–556] for n = 2 to the above system for n ⩾ 2 , we establish new conditions for global asymptotic stability of the positive equilibrium. In some special cases that k i j = k j j , 1 ⩽ i , j ⩽ n , and ∑ j = 1 n a j i a j k = 0 , i ≠ k , these conditions become a i > ∑ j = 1 n a j i 2 , 1 ⩽ i ⩽ n , and improve the well-known stability conditions a i > ∑ j = 1 n | a j i | , 1 ⩽ i ⩽ n , obtained by K. Gopalsamy [K. Gopalsamy, Global asymptotic stability in Volterra's population systems, J. Math. Biol. 19 (1984) 157–168].