On stability of cooperative and hereditary systems with a distributed delay

We consider a system dx/dt = r₁(t)G₁(x)[∫_h1(t)^t f₁ (y(s)) d_sR₁(t, s) - x (t)] dy/dt = r₂(t)G₂(y)[∫_h2(t)^t f₂ (x(s)) d₂R₂(t, s) - y(t)] with increasing functions f₁ and f₂, which has at most one positive equilibrium. Here the values of the functions r_i,G_i, f_i are positive for positive arguments, the delays in the cooperative term can be distributed and unbounded, both systems with concentrated delays and integro-differential systems are a particular case of the considered system. Analyzing the relation of the functions f₁ and f₂, we obtain several possible scenarios of the global behaviour. They include the cases when all nontrivial positive solutions tend to the same attractor which can be the positive equilibrium, the origin or infinity. Another possibility is the dependency of asymptotics on the initial conditions: either solutions with large enough initial values tend to the equilibrium, while others tend to zero, or solutions with small enough initial values tend to the equilibrium, while others infinitely grow. In some sense solutions of the equation are intrinsically nonoscillatory: if both initial functions are less/greater than the equilibrium value, so is the solution for any positive time value. The paper continues the study of equations with monotone production functions initiated in Berezansky and Braverman (2013 Nonlinearity 26 2833-49).