To save this undefined to your undefined account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your undefined account.
Find out more about saving content to .
To save this article to your Kindle, first ensure email@example.com is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below.
Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
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.
A nonlinear system of two delay differential equations is proposed to model
hematopoietic stem cell dynamics. Each equation describes the evolution of a
sub-population, either proliferating or nonproliferating. The nonlinearity
accounting for introduction of nonproliferating cells in the proliferating phase
is assumed to depend upon the total number of cells. Existence and stability
of steady states are investigated. A Lyapunov functional is built to obtain the
global asymptotic stability of the trivial steady state. The study of
eigenvalues of a second degree exponential polynomial characteristic equation
allows to conclude to the existence of stability switches for the unique
positive steady state. A numerical analysis of the role of each parameter on the
appearance of stability switches completes this analysis.
Many models in biology and ecology can be described by reaction-diffusion
equations wit time delay. One of important solutions for these type of equations
is the traveling wave solution that shows the phenomenon of wave propagation.
The existence of traveling wave fronts has been proved for large class of
equations, in particular, the monotone systems, such as the cooperative
systems and some competition systems. However, the problem on the uniqueness of
traveling wave (for a fixed wave speed) remains unsolved. In this paper, we show
that, for a class of monotone diffusion systems with time delayed reaction term,
the mono-stable traveling wave font is unique whenever it exists.
We propose and analyze a nonlinear mathematical model of hematopoiesis,
describing the dynamics of stem cell population subject to impulsive
perturbations. This is a system of two age-structured partial differential
equations with impulses. By integrating these equations over the
age, we obtain a system of two nonlinear impulsive differential equations with
several discrete delays. This system describes the evolution of the total
hematopoietic stem cell populations with impulses. We first examine the
asymptotic behavior of the model in the absence of impulsions.
Secondly, we add the impulsive perturbations and we investigate the qualitative
behavior of the model including the global asymptotic stability of the trivial
solution and the existence of periodic solution in the case of periodic
impulsive perturbations. Finally, numerical simulations are carried
out to illustrate the behavior of the model. This study maybe helpful to
understand the reactions observed in the hematopoietic system after different
forms of stress as direct destruction by some drugs or irradiation.
In this paper, with the assumptions that an infectious disease has a fixed
latent period in a population and the latent individuals of the population may
disperse, we reformulate an SIR model for the population living in two patches
(cities, towns, or countries etc.), which is a generalization of the classic
Kermack-McKendrick SIR model. The model is given by a system of delay
differential equations with a fixed delay accounting for the latency and
non-local terms caused by the mobility of the individuals during the latent
period. We analytically show that the model preserves some properties that the
classic Kermack-McKendrick SIR model possesses: the disease always dies out,
leaving a certain portion of the susceptible population untouched (called
final sizes). Although we can not determine the two final sizes, we are able to
show that the ratio of the final sizes in the two patches is totally determined
by the ratio of the dispersion rates of the susceptible individuals between the
two patches. We also explore numerically the patterns by which the disease dies
out, and find that the new model may have very rich patterns for the disease
to die out. In particular, it allows multiple outbreaks of the disease before it
goes to extinction, strongly contrasting to the classic Kermack-McKendrick SIR
In this paper, we consider a system of nonlinear delay-differential equations
(DDEs) which models the dynamics of the interaction between chronic myelogenous
leukemia (CML), imatinib, and the anti-leukemia immune response. Because of the
chaotic nature of the dynamics and the sparse nature of experimental data, we
look for ways to use computation to analyze the model without employing direct
numerical simulation. In particular, we develop several tools using
Lyapunov-Krasovskii analysis that allow us to test the robustness of the model
with respect to uncertainty in patient parameters. The methods developed in this
paper are applied to understanding which model parameters primarily affect the
dynamics of the anti-leukemia immune response during imatinib treatment. The
goal of this research is to aid the development of more efficient modeling
approaches and more effective treatment strategies in cancer therapy.
In this survey, we briefly review some of our recent studies on predator-prey
models with discrete delay. We first study the distribution of zeros of a second
degree transcendental polynomial. Then we apply the general results on the
distribution of zeros of the second degree transcendental polynomial to various
predator-prey models with discrete delay, including Kolmogorov-type
predator-prey models, generalized Gause-type predator-prey models with
harvesting, etc. Bogdanov-Takens bifurcations in delayed predator-prey models
with nonmonotone functional response and in delayed predator-prey model with
predator harvesting are also introduced.