Permanence for an integrodifferential model of mutualism

Abstract Sufficient conditions are obtained for the permanence of the following integrodifferential model of mutualism d N 1 ( t ) d t = r 1 ( t ) N 1 ( t ) K 1 ( t ) + α 1 ( t ) ∫ 0 ∞ J 2 ( s ) N 2 ( t - s ) d s 1 + ∫ 0 ∞ J 2 ( s ) N 2 ( t - s ) d s - N 1 ( t - σ 1 ( t ) ) , d N 2 ( t ) d t = r 2 ( t ) N 2 ( t ) K 2 ( t ) + α 2 ( t ) ∫ 0 ∞ J 1 ( s ) N 1 ( t - s ) d s 1 + ∫ 0 ∞ J 1 ( s ) N 1 ( t - s ) d s - N 2 ( t - σ 2 ( t ) ) , where r i , K i , α i and σ i , i  = 1, 2 are continuous functions bounded above and below by positive constants. α i  >  K i , i  = 1, 2. J i  ∈  C ([0, + ∞), [0, + ∞)) and ∫ 0 ∞ J i ( s ) d s = 1 , i = 1 , 2 .

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