On stability of cooperative and hereditary systems with a distributed delay

TL;DR

This paper analyzes the stability and long-term behavior of a class of cooperative systems with distributed delays, identifying conditions under which solutions converge to equilibria, zero, or infinity, and exploring the influence of initial conditions.

Contribution

It extends previous work by examining systems with unbounded distributed delays and provides a comprehensive analysis of their global dynamics and stability scenarios.

Findings

Solutions tend to equilibrium, zero, or infinity depending on system parameters.

Initial conditions influence asymptotic behavior, with thresholds for convergence.

Solutions are intrinsically non-oscillatory based on initial comparison to equilibrium.

Abstract

We consider a system $\displaystyle \frac{dx}{dt}=r_1(t) G_1(x) \left[ \int_{h_1(t)}^t f_1(y(s))~d_s R_1 (t,s) - x(t) \right], \frac{dy}{dt}=r_2(t) G_2(y) \left[ \int_{h_2(t)}^t f_2(x(s))~d_s R_2 (t,s) - y(t)\right]$ 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].