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Permanence and global stability in a Lotka-Volterra predator-prey system with delays. (English) Zbl 1054.34127

The author considers permanence and global asymptotic stability of models governed by the following Lotka-Volterra-type system

x ˙ i (t)=x i (t){r i -α i x i (t)-a i x i-1 (t-τ i,i-1 )-b i x i (t-τ i,i )-c i x i+1 (t-τ i,i+1 )},

tt 0 ,1in, with the initial conditions

x i (t)=φ i (t)0,tt 0 ,andφ i (t 0 )>0,1in,

where x 0 (t)=x n+1 (t)0 and φ i (t),1in, are bounded continuous functions on [t 0 ,+) and r i >0, α i >0, c i >0, τ i,j 0, for all relevant i,j. The author derives sufficient conditions for permanence and global asymptotic stability of solutions.

MSC:
34K25Asymptotic theory of functional-differential equations
34K20Stability theory of functional-differential equations
92D25Population dynamics (general)