Approximating the Stability Region for a Differential Equation with a Distributed Delay

Abstract

Abstract. We discuss how distributed delays arise in biological models and review the literature on such models. We indicate why it is important to keep the distributions in a model as general as possible. We then demonstrate, through the analysis of a particular example, what kind of information can be gained with only minimal information about the exact distribution of delays. In particular we show that a distribution independent stability region may be obtained in a similar way that delay independent results are obtained for systems with discrete delays. Further, we show how approximations to the boundary of the stability region of an equilibrium point may be obtained with knowledge of one, two or three moments of the distribution. We compare the approximations with the true boundary for the case of uniform and gamma distributions and show that the approximations improve as more moments are used.

Stability Analysis of a Reaction–Diffusion Equation with Spatiotemporal Delay and Dirichlet Boundary Condition

“…From the simulation in [9,see Figs. 4,5], we see that the bifurcated positive steady state may be stable in some conditions. Actually, from Theorem 2.3, we have that the bifurcated steady state solution is stable near λ = d, and this result supplements the work of Gourley and So [9] for Dirichlet boundary problem.…”

“…(1.4) is stable near λ = d and Hopf bifurcation cannot occur. For a delayed differential equations with a Gamma distribution delay kernel, the effect of β and m on the stability and bifurcations of the equilibrium has been investigated [5,13]. However, when f (s) is given by Eq.…”

Amplitude and phase dynamics in oscillators with distributed-delay coupling

“…At the same time, in many realistic systems, the time delays themselves are not constant [53,54] and may either vary depending on the values of system variables (state-dependent delays) or just not be explicitly known. In order to account for such situations mathematically, one can use the formalism of distributed time delays, where the time delay is represented through an integral kernel describing a particular delay distribution [55][56][57]. Distributed time delay has been successfully used to describe situations when only an approximate value of time delay is known in engineering experiments [58,59], for modelling distributions of waiting times in epidemiological models [60], maturation periods in population and ecological models [61,62], as well as in models of traffic dynamics [63], neural systems [64], predator-prey and food webs [65].…”

Approximating the stability region of a neural network with a general distribution of delays

“…The main results of this section generalize to n dimensions the theorems presented in [6] for the scalar case (i.e., (8) with n = 1). The proofs are very similar to the proofs in [6], hence we omit them. (6) is locally asymptotically stable if, for each k = 1, .…”

“…, we obtain the exact expression for C(ω) and S(ω), Substituting the exact expressions for C(ω) and S(ω) into equations (15), we obtain the true boundary of stability in the zτ -plane, as shown by the black solid curve in Figures 4(a) for p = 2 and 4(b) for p = 3 (detail on how these curves are obtained is presented in [6]). Thus the true region of stability lies between the solid gray line and the solid black curve, where the region of distribution independent stability lies between the solid and dashed gray lines.…”