On stability of cooperative and hereditary systems with a distributed delay

Abstract

We consider a system $\frac{dx}{dt}=r_1(t)G_1(x)\left[{\int }_{h_1(t)}^t{f}_1(y(s))d_s{R}_1(t,s)-x(t)\right],\frac{dy}{dt}=r_2(t)G_2(y)\left[{\int }_{h_2(t)}^t{f}_2(x(s))d_s{R}_2(t,s)-y(t)\right]$ with increasing functions $f_1$ and $f_2$, 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 $f_1$ and $f_2$, 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 non-oscillatory: 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 [Nonlinearity, 2013, 2833-2849].

18 pages, 3 figures, published in 2015 in Nonlinearity

References in corpus (1)

• Amplitude and phase dynamics in oscillators with distributed-delay coupling

Cited by in corpus (2)

• Permanence for nonautonomous differential systems with delays in the linear and nonlinear terms
• On the global attractivity of non-autonomous neural networks with a distributed delay