Absolute and delay-dependent stability of equations with a distributed delay

Abstract

We study delay-independent stability in nonlinear models with a distributed delay which have a positive equilibrium. Such models frequently occur in population dynamics and other applications. In particular, we construct a relevant difference equation such that its stability implies stability of the equation with a distributed delay and a finite memory. This result is, generally speaking, incorrect for systems with infinite memory. If the relevant difference equation is unstable, we describe the general delayindependent attracting set and also demonstrate that the equation with a distributed delay is stable for small enough delays.

Global dynamics in a commodity market model

“…Let us mention that the idea of relating delay-differential equations with its formal limit given by a difference equation goes back at least to a paper of May [11]; the interested reader is referred to [12,8,[13][14][15][16]7,9,10,17,18] for more results, including equations with distributed delays, equations with infinite delay, and delayed partial differential equations.…”

Boundedness and persistence of delay differential equations with mixed nonlinearity

“…The results were applied, for example, to the Mackey-Glass equation of population dynamics with nonmonotone feedback [10]. However, they can also be applied to some other models, including the Nicholson's blowflies equation with two delayṡ x(t) = P (t)x(h(t))e −x(g(t)) − δ(t)x(t) which in the case when variable delays are equal h(t) = g(t) was studied, for example, in [7,11,12].…”

“…There are also many generalizations of Eqs. (1.1)-(1.5) to the case of distributed delays and integro-differential equations [6,11,12,22,26].…”

On stability of cooperative and hereditary systems with a distributed delay

“…Distributed delays describe a feasible fact that any interval for delay value has some probability, such models include equations with concentrated (either constant or variable) delays. Stability of equations and systems with distributed delays attracted recently much attention, see, for example, [2,3,4,6,10,11,12,17,18,19,23,24,25,26,28,31] for some recent results and their applications, also see references therein. The summary of the results obtained by the beginning of 1990ies can be found in [16].…”