The coupon collector urn model with unequal probabilities in ecology and evolution

Abstract

The sequential sampling of populations with unequal probabilities and with replacement in a closed population is a recurrent problem in ecology and evolution. Examples range from biodiversity sampling, epidemiology to the estimation of signal repertoire in animal communication. Many of these questions can be reformulated as urn problems, often as special cases of the coupon collector problem, most simply expressed as the number of coupons that must be collected to have a complete set. We aimed to apply the coupon collector model in a comprehensive manner to one example-hosts (balls) being searched (draws) and parasitized (ball colour change) by parasitic waspsto evaluate the influence of differences in sampling probabilities between items on collection speed. Based on the model of a complete multinomial process over time, we define the distribution, distribution function, expectation and variance of the number of hosts parasitized after a given time, as well as the inverse problem, estimating the sampling effort. We develop the relationship between the risk distribution on the set of hosts and the speed of parasitization and propose a more elegant proof of the weak stochastic dominance among speeds of parasitization, using the concept of Schur convexity and the 'Robin Hood transfer' numerical operation. Numerical examples are provided and a conjecture about strong dominance-an ordering characteristic of random variables-is proposed. The speed at which new items are discovered is a function of the entire shape of the sampling probability distribution. The sole comparison of values of variances is not sufficient to compare speeds associated with different distributions, as generally assumed in ecological studies.

Dynamically accelerated cover times

“…Depending on the context, the objects may represent stickers in a football album, vertices on a fully connected graph, or people in an epidemic. Close analogies to the coupon collector can be found in a toy model for the buildup of strain in a seismic fault [2], the random deposition of k-mers on a substrate [3], the infection of nodes on a network [4], or the parasitization of hosts [5]. More generally, the coupon collector belongs to the family of urn problems [6,7].…”

BioCCP.jl: collecting coupons in combinatorial biotechnology

“…Importantly, if every coupon is equiprobable, one has to draw Oðn logðnÞÞ samples. This standard formulation of the CCP and several variants thereof have been subjected to extensive research over the past decades (Doumas and Papanicolaou, 2016;Erdo ¨s and Re ´nyi, 1961;Feller, 1971;Newman, 1960) and are still subject of ongoing study in practical settings, such as computational time analysis of algorithms (Zhang et al, 2020) and ecological studies (Zoroa et al, 2017). However, the practicality of the CCP in combinatorial biotechnology has not yet been thoroughly highlighted.…”

Fluctuations in population densities inform stability mechanisms in host-parasitoid interactions

“…The Nicholson-Bailey model assumes that all hosts are identical in terms of their vulnerability to parasitism. Perhaps a more realistic scenario is individual hosts differing in their risk of parasitism due to genetic factors, spatial heterogeneities, or are exposed to parasitoids for different durations, and at different times [26]–[29]. In essence, the attack rate c in (2) can be interpreted as “parasitism risk”, and by transforming it into a random variable we obtain where p ( x ) is the distribution of risk across hosts [9].…”