Boundedness and partial survival of species in nonautonomous Lotka-Volterra systems

Consider the partial survival and extinction of species in models governed by the following nonautonomous Lotka–Volterra system: dxi(t)dt=xi(t)ci(t)-∑j=1n∑l=0maijl(t)xj(t-τl),t⩾t0,1⩽i⩽n,xi(t)=φi(t),t⩽t0andφi(t0)>0,1⩽i⩽n, where each φi(t) is a continuous for t⩽t0, each ci(t) and aijl(t) are continuous functions on [t0,+∞), ci(t), aii0(t), aijl(t), 1⩽i,j⩽n, 1⩽l⩽m are bounded on [t0,+∞) and inft⩾t0aii0(t)>0,aijl(t)⩾0,1⩽i,j⩽n,1⩽l⩽m,τ0=0andτl⩾0,0⩽l⩽m. This paper generalizes results of the well-known May–Leonard system and the recent work of Redheffer (Nonlinear Anal.: Real World Appl. 4 (2003) 301) to the above nonautonomous Lotka–Volterra systems with delays. We establish conditions of the boundedness and partial survival for the solution x(t)=[xi(t)] of system in the sence that max1⩽i⩽nxi(t) is bounded away from 0 even though inft⩾t0xi(t)=0 for some i.