Large scavenging birds of prey are a crucial component of the natural environment, providing one of the most important ecosystem services of avian taxa (Šekercioğlu, 2006; Moleón and Sánchez-Zapata, 2015). A relatively recent review has emphasized the significance of carrion as a huge reservoir of nutrients and energy and the crucial role of scavenging in food web theory and community ecology (DeVault et al., 2016). Obligate scavengers, like vultures, deliver a wide range of biological, ecological, economic and cultural services, but primarily they recycle dead biomass and affect the rate of energy flow within the ecosystem (DeVault et al., 2003; Moleón et al., 2014). In fact, scavenging is more critical in energy transfer than actual predation (Şekercioğlu et al., 2004); for example, a minimum of 124-fold more energy was transferred per scavenging link than was transferred per predation link of 10 food webs examined by Wilson and Wolkovich (2011). In the meantime, by consuming wildlife and livestock carcasses scavengers contribute significantly to the reduction in related health risks to humans from infectious diseases, which spread by population outbreaks of opportunistic scavengers hosting dangerous microbes (Pain et al., 2003; Markandya et al., 2008; Ogada et al., 2012). However, the ecosystem services delivered by healthy vulture populations have only recently been recognized following their large-scale decline or total collapse and the observed environmental and economic costs triggered by their absence (Prakash et al., 2003; Oaks et al., 2004; Markandya et al., 2008; Morales-Reyes et al., 2015; Panagiotopoulou et al., 2018).

At present, avian scavengers are among the most threatened species in the world (McClure et al., 2018; Safford et al., 2019). Their populations are endangered by various anthropogenic factors including food poisoning or contamination, reduction in available food supplies due to changes in livestock management practices and carcass disposal driven by new sanitary legislation, and collision with wind turbines or power lines (Šekercioğlu, 2006; Ogada et al., 2011; Margalida, 2012). Old-world vultures are long-lived species with low reproductive rates and delayed maturity; thus they have been particularly affected by anthropogenic mortality such as poisoning and persecution in the past (Whitfield et al., 2004; Hernández and Margalida, 2009). In Europe, all vulture species have continued to suffer severe population declines in recent decades due to the consumption of poison baits, which is still considered the most important threat to this avian group (Hernández and Margalida, 2009; Margalida, 2012; McClure et al., 2018; Margalida and Mateo, 2019).

The risk posed by baiting and the number of vultures affected depend on a number of factors, namely the type and amount of toxin used, the bait type and density (e.g., the number per unit area), and the time interval between bait placing and consumption. The most important factor appears to be the group size of foraging vultures that is related to the targeted poisoning animal, which could be a single small carnivore, a flock of large ungulates or even an elephant (Ogada et al., 2011; Parvanov et al., 2018). The infrequent large catastrophes are caused mainly after large-carnivore predation on big livestock (bovine, equine), when their kill remains are laced with agrochemicals, so the entire carcass can be considered a bait and subsequently cause the secondary death of numerous vultures (Stoynov et al., 2018). Poisoning, usually unintentional (sometimes also intentional), has proved to be the greatest threat to vulture populations globally, acting synergistically with other mortality factors such as collision with power lines or wind turbines (Ogada et al., 2015; Velevski et al., 2015; Vasilakis et al., 2016). Though poison baits are the most serious threat, international media attention has focused on impressive mass-poisoning events (Ogada et al., 2016; Henriquez et al., 2020), while more frequent events caused by baits aimed at small carnivores such as foxes and stray dogs are mostly ignored (Ntemiri et al., 2018). Recent field data using advanced technology (e.g., miniature GPS transmitters mounted on vultures) reveal that most of these poisoning events are not discovered (Stoynov et al., 2019). Though killing only a few individuals at a time, these baits may be the main constraint on the recovery of small and isolated vulture populations.

Current supplementary feeding schemes are developed following several different strategies (e.g., pure artificial, predator simulation, large “vulture restaurants”, small feeding sites near farms) and play a key role in vulture conservation especially in small and highly threatened population nuclei (Piper, 2006; Mateo-Tomás and Olea, 2010; Duriez et al., 2012; Moreno-Opo et al., 2015). However, vultures have an extremely complex foraging behavior involving social facilitation and information exchange mechanisms and perform long-distance movements, sometimes traveling hundreds of kilometers in search of food (Monsarrat et al., 2013; Kane et al., 2014; van Overveld et al., 2018, 2020; Peshev et al., 2021). This fact makes them especially vulnerable to a wide range of environmental hazards that are difficult to control, including poisoning (Ruxton and Houston, 2004; Houston, 2006).

Another approach is through reintroduction projects that attempt to reestablish sustainable populations within the historical range of each species by the release of wild or captive-bred individuals in cases where local populations are extinct (IUCN/SSC, 2013). Vulture reintroduction or restocking projects (the latter applies when local populations are not extinct but are not yet able to recover alone) have been applied with exceptionally positive results due to certain aspects of vulture biology, such as passive behavior, easy reproduction in captivity, and simple post-release food management through controlled feeding and easy adaptation in natural habitat through collective foraging with wild conspecifics (Houston, 2006). In the last few decades, reintroductions and restocking projects have been popular as a management tool for the restoration of vulture populations in Europe, Asia and America (Griffith et al., 1989; Wilson and Stanley Price, 1994; Bustamante, 1998; Terrasse et al., 2004; Sarrazin, 2007; Stoev et al., 2016; Stoynov et al., 2016; Colomer et al., 2020).

The aim of this study was to assess the risk of extinction of a typical vulture population due to poisoning. Emphasis was placed on limitations associated with small population size such as demographic stochasticity and other factors such as Allee effects that could affect the probability of species recovery (Deredec and Courchamp, 2007; Van Houtan et al., 2010). More specifically, we explored the effect of poisoning events in relation to their intensity (i.e., temporal frequency and mortality), focusing on the effect of large versus small poisoning events since the latter are most frequent in anthropogenic landscapes aimed at small carnivores.

In this context, the relevant research questions were as follows: (1) does one major poisoning event constitute a more serious threat than many small ones to the recovery of the species? (2) Do poisoning events accentuate the stochastic Allee effect of an already-small population? Our ultimate objective was to investigate the cumulative effect of demographic stochasticity and poisoning on the population dynamics and extinction probability of a species which has a wide range and is genetically close to most other old-world vulture species that numbered a million individuals until recently (e.g., the genus Gyps has 8 species from the 16 found in vulture guilds in Europe, Africa and Asia). For this we use two models: firstly, with a mathematical model we derive a simple measure of how the growth rate is affected by frequent poisoning events to test if frequent small events are more of a problem than large infrequent events of the same cumulative mortality. Then we use a second, more detailed, model of the population including demographic stochasticity, uncertainty in timing of poisoning events, age structure and density dependence to carry out a more realistic series of tests of the same question.

Our simple mathematical model of population shows the effect of frequent poisoning events on population dynamics and yields quantitative estimates of the population growth rate. When a population with a positive growth rate (r>0) is subjected to poisoning events, the effect is to depress the growth rate. In the presence of poison baits, the growth rate decreases when the population size decreases, as it does with an Allee effect. For a sufficiently small population, the growth rate may become zero; this is the minimum viable population (MVP; Fagan et al., 2001; Flather et al., 2011). Thus, for viability a population N must exceed this value (N>M). If the population exceeds this, the resulting growth rate is positive, but if the population is smaller than this critical size, then the growth rate is negative. The structure of Eq. (2) reveals that the gain of individuals through regular population growth between episodes needs to be greater, on average, than the number of individuals lost in an episode (N>M). Equation (3) also shows (see Appendix A) that a sequence of small, frequent poison baits can be worse for a colony than a sequence of large catastrophes with the same total mortality, as small, frequent events have greater impact on growth than large but very rare ones that have a negligible effect on an otherwise healthy population. Of course, if such an event does happen in the initial stage, it can extirpate the whole population in one poisoning episode. The more realistic simulation model, where the timing of poisonings is random with an average frequency 1/T, supports these findings. Simulation runs with the same parameter set show that a population may survive or become extinct when the population is small (Fig. 1).

In the early stages of a newly formed colony, when the population numbers tens of individuals, the extinction probability remains high, but once the population becomes large, the extinction probability becomes negligible. Furthermore, the growth of the population suggests that an Allee effect is present depending on the initial group size (Fig. 2). A delay in the population growth rate is thus prominent among populations where the initial colony size is small (Fig. 2a). In contrast, this pattern is rather muted when we consider only the populations that survived (Fig. 2b). The discernable leveling off of population growth above 1000 individuals is density dependent because the population has a ceiling at 10 000 (Fig. 2). The leveling off (Fig. 2a) below 10 000, associated with smaller initial populations, is because there are more extinct populations leading to a lower average. Given the same amount of deaths per unit time, the impact of small and frequent poisoning incidents was more significant than the impact of large but rare ones.

More specifically by investigating the probability of population survival as a function of initial colony size (N0) for four scenarios with different poisoning intensity, namely (a) small and frequent (D=1), (b) large and rare (D=30), (c) intermediate (D=10), and (d) no poisoning, showed that for any starting colony size, the survival rate is lowest for small, frequent events (Fig. 3).

We also found through simulations that for models with different ceiling values (100, 200 and 10 000), the results were rather insensitive to the value of the ceiling (Fig. 3). The proportion of populations surviving in the three cases is more or less the same, with the coefficient of variation between them being less than 5 %. Thus, the basic persistence probability will not be so sensitive to changes in the ceiling value (or carrying capacity) if this is greater than 100 individuals.

Poisoning events range from small but frequent to large but rare. Media headlines tend to concentrate on large-scale catastrophic events with the power to shock (Smith, 2021; Murn and Botha, 2018), neglecting the importance of small ones. In this study, the values of the respective model simulations were chosen with a constant ratio D/T, meaning that the number of poisoned vultures per unit time remained the same on average. We have shown that a sequence of small, frequent episodes of poisoning is expected (on average) to have a higher impact on the population than a few large, infrequent ones for the same cumulative mortality (Fig. 3). This can be understood by the fact that a growing population recovers exponentially in long time intervals and this geometric growth advantage is lost if intervals are small.

Poisoning events of this kind also induce Allee effects which are associated with minimum viable population sizes. Therefore, in a conservation context this can make it much more difficult for a colony to become established if it is initially small. Similarly, the initial population size has also been found to be critical, through applying different PVA models, for the two remaining griffon vulture colonies in Sardinia (Aresu et al., 2020). Specifically, there is no point in establishing a colony that is smaller than the MVP: M=D/rT (where r is the population growth rate of the vultures and where poisoning occurs on average every T years killing each time D individuals). The minimum viable population represents an important threshold, but, in the presence of environmental and demographic uncertainty, there is no certain outcome. In this context the extinction of the founder colony does not necessarily imply poor management, since even populations with N0<MVP might still survive or those with N0>MVP might still go extinct. In the present case this type of Allee effect is manifested in small initial populations that grow more slowly than larger ones when they are close to the critical number of vultures needed to be released for the recovery and the safeguarding of the species in the long term. This underlines the great need for the creation of large-sized sustainable vulture colonies. This in turn demands a large reservoir of birds available for restocking. The Allee mechanism emerging from this model, based on poison baits, is distinct from Allee effects associated with communal foraging (where large colonies share effort and hence improve survival rates, especially if food is scarce and widely scattered) or with genetic effects (Berec et al., 2007). If either of these other effects were to be present, the result would be to increase the strength of the Allee effect and raise the size of the MVP. Our model is based on demographic parameters without explicitly considering food as a limiting factor in contrast to other studies where there is more explicit treatment of food energetic requirements (Colomer et al., 2011; Margalida et al., 2018).

These results were supported using a combination of arguments from mathematical models and from simulations. The purpose of this paper was primarily to demonstrate the possibility of these things happening. In this case our approach of parameterizing the simulation model on the basis of average values based on the population in Crete is appropriate. Should specific, targeted management be required for the single population in Cyprus, a more systematic approach to parameter uncertainty might be considered (McGowan et al., 2011).

So, consistent with intuition and experience, a substantial initial group size is needed to offset mortalities due to poison baiting, consistent with the predictions of many demographic and genetic studies demonstrating that the persistence time of a colony increases with initial size (Shaffer, 1987; Griffith et al., 1989; Allen et al., 1992). Specifically, if poisoning events of impact D (i.e., the number of dead vultures) happen every T years and the species has a yearly growth rate r, the colony size should always be well in excess of D/(rT) birds in order to have a sustainable population. In this context, the restocking success would strongly increase by using already-paired breeding adults among the released founders, as this would lead to a larger effective population size which is generally a small fraction of the total population (Frankham, 1995; Sarrazin and Legrende, 2000). Our results were obtained using a model developed specifically for the parameters of the population of griffon vultures in Cyprus but are expected to be relevant to most vulture species in any place worldwide whenever restocking is used as a conservation tool.

In the absence of a stable breeding nucleus of griffon vultures on Cyprus (i.e., four to six individuals) and with the lack of other nearby mainland colonies, the restocking scheme could be seen as a colonization process. In this case, its success would be largely determined by habitat suitability, the species ranging behavior and its reproductive performance (Le Gouar et al., 2008; Knowlton and Graham, 2010). Given that the foraging and breeding habitat remains intact and within the dispersal abilities of griffon vultures, the promotion of breeding activity might be a good short-term conservation target, which would lead to a higher growth rate (see Eq. 1) and lower MVP (Eq. 2). This could be achieved by taking advantage of the species' high gregariousness and conspecific attraction (Mihoub et al., 2011), which has shown to be the main driver for the creation of new colonies (Sarrazin et al., 1996; Le Gouar et al., 2008). By supplementary feeding close to old traditional breeding or roosting cliffs, these sites would facilitate potential breeders to locate mates as well as providing safe, non-contaminated food (Bijleveld et al., 2010; Mateo-Tomás and Olea, 2010). Consequently, effort has to be focused on certain areas and time periods, as poison bait occurrence does not happen at random as in our simulations but exhibits a characteristic spatial pattern (i.e., in pastoral and hunting areas) and a distinctive period of manifestation (i.e., in the lambing or pre-shooting season).

Therefore, as small canids (e.g., feral dogs, foxes) continue to be poisoned very frequently globally but are frequently overlooked due to species-dependent detection bias that underrepresents such animals (Gil-Sánchez et al., 2021), we expect that small numbers of dead, poisoned vultures will similarly be unnoticed especially in remote places, while only major poisoning events are expected to be found easily and registered officially. This unintentional (secondary) poisoning (Xirouchakis et al., 2000), although difficult to reveal in situ, may be one of the biggest threats to vultures worldwide (Botha et al., 2017) and could explain why small, isolated vulture populations where they still exist continue to decline or are unable to recover. Clearly, small poison events should be targeted by authorities to attract the attention they deserve, as they could eliminate other endangered raptors too and be a critical determining factor in the species' population dynamics especially if they affect adult age classes of species with low reproductive rates (Hernández and Margalida, 2008).

The colony of Gyps fulvus on Cyprus still remains critically endangered. Fresh action started in 2019 via a new restocking project aiming to transfer and release at least 24 individuals of Spanish origin into Cyprus over a 2-year period. The conservation actions needed for the survival of such a small group are primarily to do with protecting the colony from poisoning such as by constant supplementary feeding and an extensive anti-poisoning campaign, e.g., by using specially trained anti-poison dogs (Ntemiri et al., 2018). There should also be some constantly monitored individual vultures tagged with GPS-tracking devices, a cheap method for locating small poisoning events over extensive rough areas (Stoynov et al., 2019). This will allow the creation of (poison-free) “vulture safe zones” (VSZs; see also IUCN Bangladesh, 2016; Botha et al., 2017). Locally adapted VSZs are critical for maintaining the population nucleus relatively safely. Since the use of poison baits to kill predators is based on complex socio-economic factors in rural areas, anti-poisoning action plans should be applied worldwide according to the specific peculiarities of each area (Safford et al., 2019). Alongside this, it is important to boost populations via targeted restocking projects. Such actions may also need to overcome Allee effects by ensuring minimum viable populations.

In conclusion, while the public concern over shrinking vulture populations has focused mainly on large mass-poisoning events, we have shown, through mathematical models and simulation, that a sequence of small episodes may actually be worse than a few large but rare events that cause the same overall mortality. Future population reinforcement actions for vultures will most likely be successful if many adult individuals are released simultaneously in poison-free vulture safe zones around remaining colonies so as to maximize initial density and minimize Allee effects. In the meantime, such action could promote social facilitation and the formation of new griffon vulture colonies throughout their former ranges, which could also be important for the other seven Gyps species, as well as for other vulture species worldwide that are threatened with extinction.