New conditions on the existence and stability of positive periodic solutions for n-species Lotka-Volterra systems with deviating arguments

Abstract

In this paper, we study the existence and stability of positive periodic solutions for an n-species Lotka-Volterra system with deviating arguments, x i ′ (t)= x i (t)( b i r i (t)− a i i (t) x i (t− τ i i (t))− ∑ j = 1 , j ≠ i n k i j a i j (t) x j (t− τ i j (t))), i=1,2,…,n, referred to as (E). By using Mawhin’s coincidence degree, matrix spectral theory, and some new estimation techniques for the prior bounds of unknown solutions to the equation Lx=λNx, some new and interesting sufficient conditions are obtained guaranteeing the existence and global stability of positive periodic solutions of the above system. The model studied in this paper is more general, and it includes some known Lotka-Volterra type systems, such as competitive systems, predator-prey systems, and competitor-mutualist systems. Our new results are different from the known results in the previous literature.MSC:34K13, 37B25.