Approximating persistence in a general class of population processes

We provide a general framework for estimating persistence in populations which may be affected by catastrophic events, and which are either unbounded or have very large ceilings. We model the population using a birth-death process modified to allow for downward jumps of arbitrary size. For such processes, it is typically necessary to truncate the process in order to make the evaluation of expected extinction times (and higher-order moments) computationally feasible. Hence, we give particular attention to the selection of a cut-off point at which to truncate the process, and we present a simple method for obtaining quantitative indicators of the suitability of a chosen cut-off. (c) 2005 Elsevier Inc. All rights reserved.

On the existence of uni-instantaneous Q-processes with a given finite mu-invariant measure

Let S be a countable set and let Q = (q(ij), i, j is an element of S) be a conservative q-matrix over S with a single instantaneous state b. Suppose that we are given a real number mu >= 0 and a strictly positive probability measure m = (m(j), j is an element of S) such that Sigma(i is an element of S) m(i)q(ij) = -mu m(j), j 0 b. We prove that there exists a Q-process P(t) = (p(ij) (t), i, j E S) for which m is a mu-invariant measure, that is Sigma(i is an element of s) m(i)p(ij)(t) = e(-mu t)m(j), j is an element of S. We illustrate our results with reference to the Kolmogorov 'K 1' chain and a birth-death process with catastrophes and instantaneous resurrection.

The UN Security Council and the question of humanitarian intervention in Darfur

This article explores the different moral and legal arguments used by protagonists in the debate about whether or not to conduct a humanitarian intervention in Darfur. The first section briefly outlines four moral and legal positions on whether there is (and should be) a right and/or duty of humanitarian intervention: communitarianism, restrictionist and counter-restrictionist legal positivism and liberal cosmopolitanism. The second section then provides an overview of the Security Council's debate about responding to Darfur's crisis, showing how its policy was influenced by both normative concerns and hard-nosed political calculations. The article concludes by asking what Darfur's case reveals about the legitimacy and likelihood of humanitarian intervention in such catastrophes and the role of the UN Security Council as the primary authorising body for the use of international force. The authors argue that this case demonstrates that for the cosmopolitan/counter-restrictionist case to prevail pivotal states need to put humanitarian emergencies on the global agenda and express a willingness to act without Council authorisation, though the question of how to proceed in cases where the Council is deadlocked remains vexed.