2,441 research outputs found
A Recipe for State Dependent Distributed Delay Differential Equations
We use the McKendrick equation with variable ageing rate and randomly
distributed maturation time to derive a state dependent distributed delay
differential equation. We show that the resulting delay differential equation
preserves non-negativity of initial conditions and we characterise local
stability of equilibria. By specifying the distribution of maturation age, we
recover state dependent discrete, uniform and gamma distributed delay
differential equations. We show how to reduce the uniform case to a system of
state dependent discrete delay equations and the gamma distributed case to a
system of ordinary differential equations. To illustrate the benefits of these
reductions, we convert previously published transit compartment models into
equivalent distributed delay differential equations.Comment: 28 page
Identification of a hereditary system with distributed delay
We study the identification problem that arises in a linear hereditary system with distributed delay. This involves estimating an infinite-dimensional parameter and we use the method of sieves, proposed by Grenander, to solve this problem
Moment Boundedness of Linear Stochastic Delay Differential Equation with Distributed Delay
This paper studies the moment boundedness of solutions of linear stochastic
delay differential equations with distributed delay. For a linear stochastic
delay differential equation, the first moment stability is known to be
identical to that of the corresponding deterministic delay differential
equation. However, boundedness of the second moment is complicated and depends
on the stochastic terms. In this paper, the characteristic function of the
equation is obtained through techniques of Laplace transform. From the
characteristic equation, sufficient conditions for the second moment to be
bounded or unbounded are proposed.Comment: 38 pages, 2 figure
On a predator prey model with nonlinear harvesting and distributed delay
A predator prey model with nonlinear harvesting (Holling type-II) with both constant and distributed delay is considered. The boundeness of solutions is proved and some sufficient conditions ensuring the persistence of the two populations are established. Also, a detailed study of the bifurcation of positive equilibria is provided. All the results are illustrated by some
numerical simulations.Ministerio de EconomÃa y CompetitividadFondo Europeo de Desarrollo RegionalConsejerÃa de Innovación, Ciencia y Empresa (Junta de AndalucÃa
Optimal linear stability condition for scalar differential equations with distributed delay
Linear scalar differential equations with distributed delays appear in the
study of the local stability of nonlinear differential equations with feedback,
which are common in biology and physics. Negative feedback loops tend to
promote oscillations around steady states, and their stability depends on the
particular shape of the delay distribution. Since in applications the mean
delay is often the only reliable information available about the distribution,
it is desirable to find conditions for stability that are independent from the
shape of the distribution. We show here that for a given mean delay, the linear
equation with distributed delay is asymptotically stable if the associated
differential equation with a discrete delay is asymptotically stable. We
illustrate this criterion on a compartment model of hematopoietic cell dynamics
to obtain sufficient conditions for stability
Spiking Activity of a LIF Neuron in Distributed Delay Framework
Evolution of membrane potential and spiking
activity for a single leaky integrate-and-fire (LIF) neuron in
distributed delay framework (DDF) is investigated. DDF provides
a mechanism to incorporate memory element in terms of delay
(kernel) function into a single neuron models. This investigation
includes LIF neuron model with two different kinds of delay kernel
functions, namely, gamma distributed delay kernel function and
hypo-exponential distributed delay kernel function. Evolution
of membrane potential for considered models is studied in terms
of stationary state probability distribution (SPD). Stationary
state probability distribution of membrane potential (SPDV)
for considered neuron models are found asymptotically similar
which is Gaussian distributed. In order to investigate the effect
of membrane potential delay, rate code scheme for neuronal
information processing is applied. Firing rate and Fano-factor
for considered neuron models are calculated and standard LIF
model is used for comparative study. It is noticed that distributed
delay increases the spiking activity of a neuron. Increase in
spiking activity of neuron in DDF is larger for hypo-exponential
distributed delay function than gamma distributed delay function.
Moreover, in case of hypo-exponential delay function, a LIF neuron
generates spikes with Fano-factor less than 1
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