Insular species, particularly birds, experience high levels of speciation and endemism. Similarly, island birds experience extreme levels of extinction. Based on a 2012 taxonomic assessment, historically there were four reed-warbler species in the Mariana Islands, the Guam Reed-warbler Acrocephalus luscinia (Guam), the Nightingale Reed-warbler Acrocephalus hiwae (Saipan and Alamagan), the Aguijuan Reed-warbler A. nijoi (Aguiguan or Aguijuan), and the Pagan Reed-warbler A. yamashinae (Pagan). Between 2008 and 2010 we surveyed for three of these species on Alamagan, Aguiguan, and Pagan. Our results indicate that reed-warblers are extinct on Aguiguan, likely extinct on Pagan, and only the Nightingale Reed-warbler on Alamagan and Saipan remains. We estimated the global population at between 1,019 and 6,356 birds (95% CI; mean estimate 3,688), which has declined by more than 1,000 birds since the first quantitative surveys were conducted in 1982, i.e. a 24% decline in 28 years. Camp et al. (2009) describe the status of the Nightingale Reed-warbler on Saipan, which has also declined. We estimated the Alamagan population to be between 428 and 1,762 birds in 2010 (mean estimate 946). Thus, the Alamagan population is ~25 % of the global population, and it has declined slightly since 2000. This decline was not significant but is concerning, especially given a similar decline on Saipan. Restoration and protection of tall-stature native and secondary forest could benefit the Alamagan population, as would similar conservation on Saipan that includes wetland habitat. After suitable restoration of forest and wetland habitats on Aguiguan, Guam and Pagan, individuals from Alamagan and Saipan could serve as founder populations. Careful consideration of the extent and habitat preference of individuals translocated to Tinian, where an unknown reed-warbler species previously occurred, is warranted.

We consider a class of multitype Galton–Watson branching processes with a countably infinite type set $\mathcal{X}_d$ whose mean progeny matrices have a block lower Hessenberg form. For these processes, we study the probabilities $\textbf{\textit{q}}(A)$ of extinction in sets of types $A\subseteq \mathcal{X}_d$ . We compare $\textbf{\textit{q}}(A)$ with the global extinction probability $\textbf{\textit{q}} = \textbf{\textit{q}}(\mathcal{X}_d)$ , that is, the probability that the population eventually becomes empty, and with the partial extinction probability $\tilde{\textbf{\textit{q}}}$ , that is, the probability that all types eventually disappear from the population. After deriving partial and global extinction criteria, we develop conditions for $\textbf{\textit{q}} < \textbf{\textit{q}}(A) < \tilde{\textbf{\textit{q}}}$ . We then present an iterative method to compute the vector $\textbf{\textit{q}}(A)$ for any set A. Finally, we investigate the location of the vectors $\textbf{\textit{q}}(A)$ in the set of fixed points of the progeny generating vector.

Under mild nondegeneracy assumptions on branching rates in each generation, we provide a criterion for almost sure extinction of a multi-type branching process with time-dependent branching rates. We also provide a criterion for the total number of particles (conditioned on survival and divided by the expectation of the resulting random variable) to approach an exponential random variable as time goes to ∞.

Although the exact expressions for the extinction probabilities of the Interacting Branching Collision Processes (IBCP) were very recently given by Chen et al. [4], some of these expressions are very complicated; hence, useful information regarding asymptotic behaviour, for example, is harder to obtain. Also, these exact expressions take very different forms for different cases and thus seem lacking in homogeneity. In this paper, we show that the asymptotic behaviour of these extremely complicated and tangled expressions for extinction probabilities of IBCP follows an elegant and homogenous power law which takes a very simple form. In fact, we are able to show that if the extinction is not certain then the extinction probabilities {an} follow an harmonious and simple asymptotic law of an ∼ kn-αρcn as n → ∞, where k and α are two constants, ρc is the unique positive zero of the C(s), and C(s) is the generating function of the infinitesimal collision rates. Moreover, the interesting and important quantity α takes a very simple and uniform form which could be interpreted as the ‘spectrum’, ranging from -∞ to +∞, of the interaction between the two components of branching and collision of the IBCP.

We present two iterative methods for computing the global and partial extinction probability vectors for Galton-Watson processes with countably infinitely many types. The probabilistic interpretation of these methods involves truncated Galton-Watson processes with finite sets of types and modified progeny generating functions. In addition, we discuss the connection of the convergence norm of the mean progeny matrix with extinction criteria. Finally, we give a sufficient condition for a population to become extinct almost surely even though its population size explodes on the average, which is impossible in a branching process with finitely many types. We conclude with some numerical illustrations for our algorithmic methods.

Global biodiversity declines and increasing rates of extinction necessitate the assessment and prediction of the vulnerability of species to extinction. Here, we examine the relationships between conservation status and ecological traits of reef fish species of the Brazilian biogeographical province. We used binomial tests and a logistic regression to address two questions. Do biological attributes differ between threatened and non-threatened fishes? Which combination of traits and impacts exerts greater influence on species threat status? Of the 559 species, 36 are categorized as threatened (compiled from global, national and local Red Lists). Three species are categorized as Critically Endangered, seven as Endangered and 26 as Vulnerable. Our analyses revealed that Elasmobranchii, sex-changing bony fishes and endemic species are the most vulnerable reef fishes in Brazilian waters. Body size and trophic category were identified as good predictors of the vulnerability of a species to extinction. Small-bodied species that are exploited by the ornamental trade and have complex reproductive strategies are also of concern. Such combinations of attributes could be of value in predicting which reef fish species elsewhere have a high risk of extinction.

We focus on supercritical decomposable (reducible) multitype branching processes. Types are partitioned into irreducible equivalence classes. In this context, extinction of some classes is possible without the whole process becoming extinct. We derive criteria for the almost-sure extinction of the whole process, as well as of a specific class, conditionally given the class of the initial particle. We give sufficient conditions under which the extinction of a class implies the extinction of another class or of the whole process. Finally, we show that the extinction probability of a specific class is the minimal nonnegative solution of the usual extinction equation but with added constraints.

We consider the uniqueness and extinction properties of the interacting branching collision process (IBCP), which consists of two strongly interacting components: an ordinary Markov branching process and a collision branching process. We establish that there is a unique IBCP, and derive necessary and sufficient conditions for it to be nonexplosive that are easily checked. Explicit expressions are obtained for the extinction probabilities for both regular and irregular cases. The associated expected hitting times are also considered. Examples are provided to illustrate our results.

We introduce a class of discrete-time two-sex branching processes where the offspring probability distribution and the mating function are governed by an environmental process. It is assumed that the environmental process is formed by independent but not necessarily identically distributed random vectors. For such a class, we determine some relationships among the probability generating functions involved in the mathematical model and derive expressions for the main moments. Also, by considering different probabilistic approaches we establish several results concerning the extinction probability. A simulated example is presented as an illustration.

We introduce the idea of controlling branching processes by means of another branching process, using the fractional thinning operator of Steutel and van Harn. This idea is then adapted to the model of alternating branching, where two Markov branching processes act alternately at random observation and treatment times. We study the extinction probability and limit theorems for reproduction by n cycles, as n → ∞.

In this paper, we investigate the asymptotic behaviour of controlled branching processes with random control functions. In a critical case, we establish sufficient conditions for both their almost-sure extinction and for their nonextinction with a positive probability. For some suitably chosen norming constants, we also determine different kinds of limiting behaviour for this class of processes.

In this paper, we study a class of bisexual Galton-Watson branching processes in which the law of offspring distribution is dependent on the population size. Under a suitable condition on the offspring distribution, we prove that the limit of mean growth-rate per mating unit exists. Based on this limit, we give a criterion to identify whether the process admits ultimate extinction with probability one.

We consider a branching model, which we call the collision branching process (CBP), that accounts for the effect of collisions, or interactions, between particles or individuals. We establish that there is a unique CBP, and derive necessary and sufficient conditions for it to be nonexplosive. We review results on extinction probabilities, and obtain explicit expressions for the probability of explosion and the expected hitting times. The upwardly skip-free case is studied in some detail.

In this paper, the class of controlled branching processes with random control functions introduced by Yanev (1976) is considered. For this class, necessary and sufficient conditions are established for the process to become extinct with probability 1 and the limit probabilistic behaviour of the population size, suitably normed, is investigated.

In this paper, we introduce a bisexual Galton-Watson branching process with mating function dependent on the population size in each generation. Necessary and sufficient conditions for the process to become extinct with probability 1 are investigated for two possible conditions on the sequence of mating functions. Some results for the probability generating functions associated with the process are also given.

A curious connection exists between the theory of optimal stopping for independent random variables, and branching processes. In particular, for the branching process Zn with offspring distribution Y, there exists a random variable X such that the probability P(Zn = 0) of extinction of the nth generation in the branching process equals the value obtained by optimally stopping the sequence X1,…, Xn, where these variables are i.i.d. distributed as X. Generalizations to the inhomogeneous and infinite horizon cases are also considered. This correspondence furnishes a simple ‘stopping rule’ method for computing various characteristics of branching processes, including rates of convergence of the nth generation's extinction probability to the eventual extinction probability, for the supercritical, critical and subcritical Galton-Watson process. Examples, bounds, further generalizations and a connection to classical prophet inequalities are presented. Throughout, the aim is to show how this unexpected connection can be used to translate methods from one area of applied probability to another, rather than to provide the most general results.

The paper considers stability and instability properties of the Markov chain generated by the composition of an i.i.d. sequence of random transformations. The transformations are assumed to be either linear mappings or else mappings which can be well approximated near 0 by linear mappings. The main results concern the risk probabilities that the Markov chain enters or exits certain balls centered at 0. An application is given to the probability of extinction in a model from population dynamics.

Generalizing the classical Banach matchbox problem, we consider the process of removing two types of ‘items’ from a ‘pile’ with selection probabilities for the type of the next item to be removed depending on the current numbers of remaining items, and thus changing sequentially. Under various conditions on the probability pn1,n2 that the next removal will take away an item of type I, given that n1 and n2 are the current numbers of items of the two types, we derive asymptotic formulas (as the initial pile size tends to infinity) for the probability that the items of type I are completely removed first and for the number of items left. In some special cases we also obtain explicit results.

In a branching process with random environments, the probability of ultimate extinction is a function of the environment sequence, and is therefore a random variable. Explicit results about the distribution of this random variable are difficult to obtain in general. Here we assume independent and identically distributed environments and use the special properties of fractional linear generating functions to derive some explicit distributions, which may be singular or absolutely continuous, depending on the values of certain parameters. We also consider briefly tail behaviour close to 1, and provide an extension to cases where probability generating functions are not fractional linear.

We consider a dynamic multilevel population or information system. At each level individuals or information units undergo a Galton–Watson-type branching process in which they can be replicated or removed. In addition, a collection of individuals or information units at a given level constitutes an information unit at the next higher level. Each collection of units also undergoes a Galton–Watson branching process, either dying or replicating. In this paper, we represent this multilevel branching model as a measure-valued stochastic process, study its moment structure, identify the limiting continuous-state approximation and analyse the long-time behavior in both non-critical and critical cases. For example, we obtain an asymptotic expression for the extinction probability for the total population mass process and an analogue of Yaglom's conditioned limit theorem in the critical case.