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We consider a system $\displaystyle \frac{dx}{dt}=r_1(t) G_1(x) \left[\int_{h_1(t)}^t f_1(y(s))~d_s R_1 (t,s) - x(t) \right], \frac{dy}{dt}=r_2(t)G_2(y) \left[ \int_{h_2(t)}^t f_2(x(s))~d_s R_2 (t,s) - y(t)\right]$ withincreasing functions $f_1$ and $f_2$, which has at most one positiveequilibrium. Here the values of the functions $r_i,G_i,f_i$ are positive forpositive arguments, the delays in the cooperative term can be distributed andunbounded, both systems with concentrated delays and integro-differentialsystems are a particular case of the considered system. Analyzing the relationof the functions $f_1$ and $f_2$, we obtain several possible scenarios of theglobal behaviour. They include the cases when all nontrivial positive solutionstend to the same attractor which can be the positive equilibrium, the origin orinfinity. Another possibility is the dependency of asymptotics on the initialconditions: either solutions with large enough initial values tend to theequilibrium, while others tend to zero, or solutions with small enough initialvalues tend to the equilibrium, while others infinitely grow. In some sensesolutions of the equation are intrinsically non-oscillatory: if both initialfunctions are less/greater than the equilibrium value, so is the solution forany positive time value. The paper continues the study of equations withmonotone production functions initiated in [Nonlinearity, 2013, 2833-2849].

On stability of cooperative and hereditary systems with a distributed delay