Ultimate Extinction of the Promiscuous Bisexual Galton—Watson Metapopulation

Abstract

A variant of a sexual Galton-Watson process is considered. At each generation the population is partitioned among n 'hosts' (population patches) and individual members mate at random only with others within the same host. This is appropriate for many macroparasite systems, and at low parasite loads it gives rise to a depressed rate of reproduction relative to an asexual system, due to the possibility that females are unmated. It is shown that stochasticity mitigates against this effect, so that for small initial populations the probability of ultimate extinction (the complement of an 'epidemic') displays a tradeoff as a function of n between the strength of fluctuations which overcome this 'mating' probability, and the probability of the subpopulation in one host being 'rescued' by that in another. Complementary approximations are developed for the extinction probability: an asymptotically exact approximation at large n , and for small n a short-time probability that is exact in the limit where the mean number of offspring per parent is large.

Stochastic and spatial dynamics of nematode parasites in farmed ruminants

“…However, owing to the seasonal nature of the system, with oscillations in the larval-development rates and the removal of lambs for slaughter, we still find that, when rare, epidemics are more likely in smaller herds. We have also shown how this phenomenon arises within a simple branching metapopulation model (Cornell & Isham 2004), and we expect it to apply in a very wide range of circumstances.…”

Evaluating the Evidence for Lymphatic Filariasis Elimination

“…The probability of elimination, given a particular prevalence (e.g., 1%), can be calculated by considering the probability that a chain of transmission will die out (in mathematics we call this chain a branching process [ 22 ] ). These types of branching process methods have been used for soil-transmitted helminths [ 23 , 24 ], but have been adapted here to account for vector-borne transmission with an aggregated bite risk [ 25 , 26 ].…”

Female mating failures in insects

“…Female mating failure has been conceptualized in ecological theory mainly from the angle of the Allee effect. Under this scenario, a large fraction of females in low density populations remain unfertilized, which may lead to a decline in per capita population growth rate (Allee, 1932; Andrewartha & Birch, 1954), extinction of local populations (Phillip, 1957; Cornell & Isham, 2004), interspecific competitive displacement (Scott, 1977), failure of establishment of biological control agents (Hopper & Roush, 1993), and restriction of geographic distributional range (Keitt et al., 2001).…”

“…Female mating failure has been conceptualized in ecological theory mainly from the angle of the Allee effect. Under this scenario, a large fraction of females in low density populations remain unfertilized, which may lead to a decline in per capita population growth rate (Allee, 1932;Andrewartha & Birch, 1954), extinction of local populations (Phillip, 1957;Cornell & Isham, 2004), interspecific competitive displacement (Scott, 1977), failure of establishment of biological control agents (Hopper & Roush, 1993), and restriction of geographic distributional range (Keitt et al, 2001). Field studies conducted over a large spatial scale across multiple populations have revealed a positive relationship between population density and female mating success in three species, the Glanville fritillary (Kuussaari et al, 1998), the invasive gypsy moth (Contarini et al, 2009), and Tinema stick insects (Schwander et al, 2010).…”