Subgeometric rates of convergence for a class of continuous-time Markov process

Let (Phi(t))(t is an element of R+) be a Harris ergodic continuous-time Markov process on a general state space, with invariant probability measure pi. We investigate the rates of convergence of the transition function P-t(x, (.)) to pi; specifically, we find conditions under which r(t) vertical bar vertical bar P-t (x, (.)) - pi vertical bar vertical bar -> 0 as t -> infinity, for suitable subgeometric rate functions r(t), where vertical bar vertical bar - vertical bar vertical bar denotes the usual total variation norm for a signed measure. We derive sufficient conditions for the convergence to hold, in terms of the existence of suitable points on which the first hitting time moments are bounded. In particular, for stochastically ordered Markov processes, explicit bounds on subgeometric rates of convergence are obtained. These results are illustrated in several examples.

Altitudinal variation in behavioural thermoregulation: Local adaptation vs. plasticity in California grasshoppers

We investigated the adaptive significance of behavioural thermoregulation in univoltine populations of the grasshopper Melanoplus sanguinipes along an altitudinal gradient in California using laboratory tests of animals raised under different temperatures. Trials consisted of continuous body temperature measurements with semi-implanted microprobes in a test arena, and observation and simultaneous recording of behavioural responses. These responses included mobility, basking and orientation of the body axes (aspect angle) towards a radiation source. Mobility and basking are determined by the altitudinal origin of the parental generation and not by the temperature treatments. With increasing altitude, individuals tend increasingly to raise body temperatures via mobility and increased basking. In contrast, body orientation towards the radiation source is influenced by the temperature treatments but not by the altitude of origin. Individuals experiencing higher temperatures during rearing show a lower tendency to lateral flanking. We conclude that body orientation responses are not adapted locally. In contrast other components of the behavioural syndrome that increase body temperature, such as mobility and basking, are adaptive in response to local selection pressure. The thermoregulatory syndrome of these grasshoppers is an important contribution to life-history adaptations that appropriately match season lengths.

Genetic population structure and call variation in a passerine bird, the satin bowerbird, Ptilonorhynchus violaceus

Geographic variation in vocalizations is widespread in passerine birds, but its origins and maintenance remain unclear. One hypothesis to explain this variation is that it is associated with geographic isolation among populations and therefore should follow a vicariant pattern similar to that typically found in neutral genetic markers. Alternatively, if environmental selection strongly influences vocalizations, then genetic divergence and vocal divergence may be disassociated. This study compared genetic divergence derived from 11 microsatellite markers with a metric of phenotypic divergence derived from male bower advertisement calls. Data were obtained from 16 populations throughout the entire distribution of the satin bowerbird, an Australian wet-forest-restricted passerine. There was no relationship between call divergence and genetic divergence, similar to most other studies on birds with learned vocalizations. Genetic divergence followed a vicariant model of evolution, with the differentiation of isolated populations and isolation-by-distance among continuous populations. Previous work on Ptilonorhynchus violaceus has shown that advertisement call structure is strongly influenced by the acoustic environment of different habitats. Divergence in vocalizations among genetically related populations in different habitats indicates that satin bowerbirds match their vocalizations to the environment in which they live, despite the homogenizing influence of gene flow. In combination with convergence of vocalizations among genetically divergent populations occurring in the same habitat, this shows the overriding importance that habitat-related selection can have on the establishment and maintenance of variation in vocalizations.