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We find explicit estimates for the exponential rate of long-term convergence for the ruin probability in a level-dependent Lévy-driven risk model, as time goes to infinity. Siegmund duality allows us to reduce the problem to long-term convergence of a reflected jump-diffusion to its stationary distribution, which is handled via Lyapunov functions.
This paper is devoted to studying the long-term behaviour of a continuous-time Markov chain that can be interpreted as a pair of linear birth processes which evolve with a competitive interaction; as a special case, they include the famous Lotka–Volterra interaction. Another example of our process is related to urn models with ball removal. We show that, with probability one, the process eventually escapes to infinity by sticking to the boundary in a rather unusual way.
We investigate the stability and periodic orbits of a predator-prey model with harvesting. The model has a biologically-meaningful interior, an attractor undergoing damped oscillations, and can become destabilised to produce periodic orbits via a Hopf bifurcation. Some sufficient conditions for the existence of the Hopf bifurcation are established, and a stability analysis for the periodic solutions using a Lyapunov function is presented. Finally, some computer simulations illustrate our theoretical results.
We give a criterion for unlimited growth with positive probability for a large class of multidimensional stochastic models. As a by-product, we recover the necessary and sufficient conditions for recurrence and transience for critical multitype Galton–Watson with immigration processes and also significantly improve some results on multitype size-dependent Galton–Watson processes.
We consider the flow of gas through pipelines controlled by a compressor
station. Under a subsonic flow assumption we prove the existence
of classical solutions for a given finite time interval.
The existence result is used to construct Riemannian feedback laws and
to prove a stabilization result for a coupled system of gas pipes with a compressor
station. We introduce a Lyapunov function and prove exponential decay
with respect to the L2-norm.
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.
In this paper several models in virus dynamics with and without immune response are
discussed concerning asymptotic behaviour. The case of immobile cells but diffusing viruses and
T-cells is included. It is shown that, depending on the value of the basic reproductive number R0
of the virus, the corresponding equilibrium is globally asymptotically stable. If R0 < 1 then the
virus-free equilibrium has this property, and in case R0 > 1 there is a unique disease equilibrium
which takes over this property.
Normed ergodicity is a type of strong ergodicity for which convergence of the nth step transition operator to the stationary operator holds in the operator norm. We derive a new characterization of normed ergodicity and we clarify its relation with exponential ergodicity. The existence of a Lyapunov function together with two conditions on the uniform integrability of the increments of the Markov chain is shown to be a sufficient condition for normed ergodicity. Conversely, the sufficient conditions are also almost necessary.
One of the
simplest and natural appealing motion control strategies for robot manipulators
is the PD control with feedforward compensation. Although successful experimental
tests of this control scheme have been published since the
beginning of the eighties, the proof of global asymptotic stability
has remained unattended until now. The contribution of this paper
is to prove that global asymptotic stability can be guaranteed
provided that the proportional and derivative gains are adequately selected.
The performance of the PD control with feedforward compensation evaluated
on a two degrees-of-freedom direct-drive arm appears as fine as
the classical model-based computed torque control scheme.
This paper studies the connection between the dynamical and equilibrium behaviour of large uncontrolled loss networks. We consider the behaviour of the number of calls of each type in the network, and show that, under the limiting regime of Kelly (1986), all trajectories of the limiting dynamics converge to a single fixed point, which is necessarily that on which the limiting stationary distribution is concentrated. The approach uses Lyapunov techniques and involves the evolution of the transition rates of a stationary Markov process in such a way that it tends to reversibility.
We prove a theorem which can be used to show that the expectation of a non-negative function of the state of a time-homogeneous Markov process is uniformly bounded in time. This is reminiscent of the classical theory of non-negative supermartingales, except that our analog of the supermartingale inequality need not hold almost surely. Consequently, the theorem is suitable for establishing the stability of systems that evolve in a stabilizing mode in most states, though from certain states they may jump to a less stable state. We use this theorem to show that ‘joining the shortest queue' can bound the expected sum of the squares of the differences between all pairs among N queues, even under arbitrarily heavy traffic.
Let Xn be non-negative random variables, possessing the Markov property. We given criteria for deciding whether Pr(Xn →∞) is positive or 0. It turns out that essentially this depends on the magnitude of E(Xn+1 | Xn = x) compared to that of E(X2n+1 | Xn = x) for large x. The assumptions are chosen such that for example population-dependent branching processes can be treated by our results.
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