Permanence for an integrodifferential model of mutualism

Sufficient conditions are obtained for the permanence of the following integrodifferential model of mutualismdN1(t)dt=r1(t)N1(t)K1(t)+α1(t)∫0∞J2(s)N2(t-s)ds1+∫0∞J2(s)N2(t-s)ds-N1(t-σ1(t)),dN2(t)dt=r2(t)N2(t)K2(t)+α2(t)∫0∞J1(s)N1(t-s)ds1+∫0∞J1(s)N1(t-s)ds-N2(t-σ2(t)),where ri, Ki, αi and σi, i = 1, 2 are continuous functions bounded above and below by positive constants. αi > Ki, i = 1, 2. Ji ∈ C([0, + ∞), [0, + ∞)) and ∫0∞Ji(s)ds=1,i=1,2.