The Western perspective of moral valuation encompasses a diverse set of value systems with respect to the components of nature that form the moral community. Traditionally, one can distinguish at least four archetypal value systems: anthropocentrism, sentientism, biocentrism, and ecocentrism (Rolston 2003; Palmer et al. 2014) (Table 1; Fig. 1).

Anthropocentrism values nature by the benefits it brings to people through ecosystem services, which encompasses economic, biological, and cultural benefits humans can derive from nature (Díaz et al. 2018). One justification for anthropocentrism is that humans are (arguably) the only self-reflective moral beings, and people are both the subject and object of ethics (Rolston 2003), therefore constituting the moral community. In an anthropocentric system, individuals from non-human species only have value based on their benefits or disservices for humans (instrumental or non-instrumental).

Sentientism considers that humans and all sentient animals value their life, and experience pleasure, pain, and suffering (Table 1). All sentient individuals should therefore also be part of the moral community (i.e. have an intrinsic value). In this view, it is the sentience [e.g. measured through cognitive ability, (Singer 2009)], rather than species themselves, that has intrinsic value.

Biocentrism considers that life has intrinsic value. Although different perspectives on why life has value exist (Taylor 2011), all living organisms are valued equally for being alive, and not differently based on any other trait.

Some ecocentric, or holistic, value systems consider that ecological collectives, such as species or ecosystems, have intrinsic value, independently from the individuals that comprise them. Species can have different values, i.e. speciesism (Table 1), and these values can be influenced by a multitude of factors, discussed in more detail below.

In practice, the separation between anthropocentrism, sentientism, biocentrism, and ecocentrism is blurry, and values given to different species may vary under the same general approach. For example, biocentrism can range from complete egalitarianism between organisms to a gradual valuation resembling sentientism. These four value systems can also interact with other systems that use other criteria than the intrinsic characteristics of individuals to define the moral community. For example, nativism is a system that values organisms indigenous to a spatial location or an ecosystem over those that have been introduced by humans. Nativism can therefore interact with any of the four systems presented above to alter the value attributed to a species in a given context. Finally, how someone values individuals of different species is often influenced by their personal views and experiences (Palmer et al. 2014; Waytz et al. 2019). Values and personal interests thus interact when people make and express moral judgements (Essl et al. 2017). Therefore, the archetypes of value systems presented above are rarely expressed in a clear and obvious fashion. Nonetheless, by formalising the archetypes, a framework can be created within which the consequence of conservation actions can be explored (see 'consequentialism' vs. 'deontology', Table 1).

To account for the different elements that can be combined to create the concept of value, in the following, we distinguish between ‘intrinsic’, ‘inherent’, and ‘utilitarian’, value (our definitions; Table 1). Intrinsic value is the value possessed by an individual or collective as defined by one of the systems above, and is therefore independent of context. Intrinsic value is based on objective criteria such as cognitive ability. The choice of a criteria may be subjective, but the value is independent of the assessor once the criteria has been defined. This has been termed “objective intrinsic value” by others (Sandler 2012). Inherent value is the value of an individual, species or ecosystem that results from the combination of its intrinsic value and context-specific and subjective factors (note that other scholars have used ‘inherent’ differently, e.g. (Taylor 1987; Regan 2004); here it corresponds to what has also been termed “subjective intrinsic value” (Sandler 2012)). These factors include charisma (Courchamp et al. 2018; Jarić et al. 2020), anthropomorphism (Tam et al. 2013; Table 1), organismic complexity (Proença et al. 2008), neoteny (Stokes 2007; Table 1), cultural importance (Garibaldi and Turner 2004), religion (Bhagwat et al. 2011), parochialism (Waytz et al. 2019; Table 1), and more generally the relationship between humans and elements of nature (Chan et al. 2016). For example, dogs and wolves may be considered to have similar cognitive abilities objectively, and therefore a similar intrinsic value under sentientism, but dogs may have a higher inherent value for some people because they are in close contact with individuals from this species, i.e. parochialism. Some alien species that did not have any inherent value prior to their introduction have been incorporated in local cultures, therefore providing them a novel and higher inherent value such as horses being linked to a strong local cultural identity in some parts of the USA (Rikoon 2006). Inherent value can often be considered to be fixed at the time scale of a management action, but can nonetheless vary over short time scales in some situations (see the example of the Oostvaardersplassen nature reserve below). Utilitarian value is determined only from an anthropocentric perspective. It is context-dependent and can change rapidly, for example in the case of commercial exploitation.

Conservation practices can historically be divided into three main categories, closely related to specific systems of moral valuation (Mace 2014). At one extreme, a ‘nature for itself’ or 'nature despite people' (Table 1) view mostly excludes humans from the assessment of the efficacy of conservation management actions (Fig. 2). This ecocentric perspective is the foundation of traditional conservation as defined by Soulé (1985), and relies on the four following normative postulates: “diversity of organisms is good”, “ecological complexity is good”, “evolution is good”, and “biotic diversity has intrinsic value” (Soulé 1985). It historically underlies widely-used conservation tools, like the IUCN Red List of Threatened Species (IUCN 2019), in which threat categories are defined in terms of probability of extinction (Mace and Lande 1991) (i.e. a species-level criterion aimed at preserving biodiversity). Ecocentrism is often not limited to the valuation of species, but can encompass wider collectives, i.e. assemblages of species and functions, or ecosystems. This other perspective is captured, for example, by the IUCN Red List of Ecosystems (IUCN-CEM 2016), and it is strongly reflected in international conservation agreements such as the Convention on Biological Diversity (UNEP CBD 2010). In the following we refer to traditional conservation as an ecocentric value system where species are intrinsically valuable (nature for itself; Fig. 1, 2) and humans are mostly excluded from management. We acknowledge that this is an archetypal view of traditional conservation, which is used here simply for illustrative purposes.

Figure 2.

Different value systems (or combination of) correspond to different conservation perspectives, which were introduced at different points in time (the timeline is approximate for illustrative purpose; see also Mace 2014). A nature for itself perspective can be either ecocentric, biocentric, or both. Under new conservation, an anthropocentric perspective is considered necessary to achieve a desirable outcome under a biocentric perspective (⟹). Under the people and nature approach, anthropocentric, biocentric and ecocentric perspectives are considered simultaneously (+). Underlying concepts and movements pre-dating conservation approaches are indicated in grey italic at the approximate period they originated.

By contrast the more recent, anthropocentric ‘nature for people’ perspective (Mace 2014) values species and ecosystems only to the extent that they contribute to the wellbeing of humans (Fig. 2). These values encompass ecosystem services that help sustain human life (Bolund and Hunhammar 1999) or economic assets (Fisher et al. 2008), and can rely on the assessment of species and ecosystem services in terms of their economic value (Costanza et al. 1997), which can be considered as the most general form of utilitarian value, and has also been termed economism (Norton 2000). The ‘nature for people’ perspective can nonetheless incorporate additional measures linked to human wellbeing, such as poverty alleviation or political participation. This more holistic measure of impacts on humans is exemplified by ‘new conservation’, also termed ‘social conservation’ (Miller et al. 2011; Kareiva 2014; Doak et al. 2015) (Table 1; Fig. 2). It has been argued that such an anthropocentric perspective will, by extension, help and even be necessary to maximise the conservation of nature (Kareiva and Marvier 2012). Although new conservation was introduced relatively recently (Fig. 2), it follows an older perspective termed the convergence hypothesis, which argues that if human interests depend on the elements of nature, conservation approaches motivated by anthropogenic instrumental or non-anthropogenic intrinsic values should be the same (Norton 1986; Table 1). It is important to note that the exact set of normative postulates proposed by the proponents of new conservation is not clearly defined (Miller et al. 2011), leading to differences of interpretation and heated debates in recent years (Kareiva and Marvier 2012; Kareiva 2014; Soulé 2014; Doak et al. 2015).

More recently, the necessity to account for the interdependence between the health of nature and human wellbeing [i.e. ‘people and nature’ (Mace 2014); Fig. 2] has been advocated in the United Nations Sustainable Development Goals (Weitz et al. 2018). This approach lies on the notion of weak anthropocentrism, introduced by the environmental pragmatism movement (Norton 1984; Katz and Light 2013), in which the value of elements of the environment is not only utilitarian, but defined by the relationship between humans and nature (Chan et al. 2016), and therefore is influenced by context and people’s experience (see also the notion of inherent value described above). Similarly, “nature-based solutions” is an approach endorsed by the IUCN, which aims at protecting, sustainably managing, and restoring, natural or modified ecosystems, to simultaneously provide human wellbeing and biodiversity benefits (Cohen-Shacham et al. 2016). The ‘One Health’ approach, endorsed by the Food and Agriculture Organization, the World Health Organization, and the World Organisation for Animal Health also acknowledges the interdependence between the state of ecosystems, human health, and zoonoses (Gibbs 2014). The difference between people and nature and new conservation approaches therefore lies in the fact that it merges anthropocentric and ecocentric systems, rather than considering that the latter will be addressed by focusing on the former (see Section “Nature despite/for/and people” below for details).

Finally, although the animal rights movement, based on sentientism, originated in the 19^th century (Salt 1894), it has not, to our knowledge, been formally considered in conservation approaches until recently. Two main approaches can be found in the literature. Conservation welfare (Beausoleil et al. 2018) is a consequentialist perspective that considers conservation under the prism of animal welfare maximisation (Fig. 2). Compassionate conservation (Ramp and Bekoff 2015; Wallach et al. 2018), also incorporates animal sentience, but from a virtue ethics perspective (Table 1). Although conservation welfare aims at aligning with more traditional conservation approaches presented above (Beausoleil et al. 2018), compassionate conservation appears to be set on different values and proposes, for example, to incorporate emotion to provide insight in conservation (Batavia et al. 2021).

Many of the conflicts in conservation are grounded in the failure to identify and formalise differences in world views, which contain elements of the four archetypes presented above, influenced by cultural norms, economic incentives etc. (Essl et al. 2017). Here, we propose a mathematical formulation as a method to clarify moral discourses in conservation, based on a consequentialist perspective. We therefore consider an objective-driven type of conservation. Our purpose is not to argue about the relevance of consequentialism vs. deontology, or on the place of virtue ethics in conservation. Rather, we consider that, from a management perspective, conservation necessarily includes objective-driven considerations. A better understanding of how and why objectives can differ between stakeholders as a result of their value systems is therefore useful to anticipate and manage potential conflicts. Although some participants of the discourse will be more receptive to discursive than mathematical conceptualisation, we argue that defining concepts as mathematical terms can make differences in value systems and their normative postulates more explicit and transparent, which will be beneficial when used with appropriate stakeholders, even when these terms would be hard to quantify in real life. A mathematical formulation can be seen as a logical way to express relationships between different elements. Doing so can help to identify and facilitate the discussion of shared values and incompatibilities between different environmental policies and management options (Miller et al. 2011), and contribute to manage conflicts (Redpath et al. 2013). In a similar vein, Parker et al. (1999) proposed a mathematical framework for assessing the environmental impacts of alien species. This work was highly influential in the conceptualisation of biological invasions (being cited over 2,000 times until April 2021 according to Google Scholar), rather than by its direct quantitative application. We also acknowledge that this approach has specific limitations, which are discussed below.

Our mathematical formalisation conceptualises the consequences of environmental management actions. As we develop below, these consequences will be defined differently depending on the value system, but can be understood generally as the consequences for the members of the moral community. Under anthropocentrism, these will be consequences for humans; under sentientism, these will be consequences for sentient individuals; under biocentrism and ecocentrism, these will be consequences for biodiversity. We argue that our mathematical formalisation can account for these different value systems (see Suppl. material 1: Appendix S1 for an extension to ecocentrism beyond species and considering wider collectives, i.e. ecosystems), while also accounting for cultural and personal contexts. These consequences C can be conceptualised as a combination of the impact of an action on the different species or individuals involved and the value given to said species and individuals under different value systems as follows:

$C=\sum _{\text{species}s}{\overline{I}}_s\times V_s\times N_s{}^a$ Eq.1

where Ī_s is a function (e.g. mean, maximum, etc.) of the impact (direct and indirect) resulting from the management action on all individuals of species s, V_s is the inherent value attributed to an individual of species s (as described above), N_s is the abundance of species s, and a determines the importance given to a species based on its abundance or rarity (and enables to account for the importance of a species rather than an individual, see below). The unit of C depends on how other parameters are defined, which themselves depend on the value system considered. In summary, the higher the impact on species with high values, the higher the consequences.

Inherent value V_s can have a monetary unit or be unit-less depending on how it is defined. It can be continuous or categorical (e.g. null, low, high – quantifiable as 0, 1, 2 or any other quantitative scale). Our definition of inherent value here is extremely broad, as the purpose of this work is not to define what such value should be, rather, it is to be flexible enough to encompass multiple perspectives and the subjectivity of the assessor, and be based on intrinsic, utilitarian or relational values (Chan et al. 2016; Table 1).

The parameter a can take both positive and negative values. A value of 1 means that consequences are computed over individuals. If all values V_s were the same, a = 1 implies that all individuals in the moral community (Table 1) weigh the same when computing C, irrespective of the species they belong to. This is typical of individual-centred value systems, i.e. sentientism, and biocentrism, whose characteristics (sentience and life) are defined at the individual level. As a result, impacts on larger populations would weigh more on the consequences. As a decreases towards 0, the correlation between the value of a species and its abundance decreases. For a = 0, the consequence of a management action becomes abundance-independent. For a < 0, rare species would be valued higher than common species (or the same impact would be considered to be higher for rare species), for example due to the higher risk of extinction. And for a > 1, disproportionate weight is given to abundant species, which are often important for providing ecosystem services (Gaston 2010).

The impact I_s is computed at the individual level. It can be limited to the probability of death of individuals or changes in per capita recruitment rate, thus allowing to compute a proxy for extinction risk if a ≤ 0, but can also include animal welfare, biophysical states, etc. As for V_s, continuous or categorical scales may be used. Different measures of impact can be considered under a same system of value, in which case Equation 1 should be applied to each one separately (see section “Application of the mathematical framework” below for details). I_s can only encompass the direct impact of a management action (in a narrow view that only the direct impact of humans, i.e. the moral agents, should be considered, and that the direct impacts from non-moral agents should not be considered), but also include its indirect impact resulting from biotic interactions (considering that, in the context of management and therefore human actions, these indirect impacts are ultimately the result of the actions of the moral agents). One would therefore need to define a baseline corresponding to either i) the lowest possible measurable level of impact (e.g. being alive if death is the only measure of impact, or no sign of disease and starvation for biophysical states; this would obviously be more complicated for welfare), so that I would only be positive; ii) the current state of the system, in which case impacts could be positive or negative for different species; or iii) the past state of a system, for example prior to the introduction of alien species (see (Rohwer and Marris 2021) for a discussion on the notion of ecosystem integrity). The duration over which to measure such impact should also be determined. The exact quantification of impact will be influenced by different value systems and personal subjectivity. Some impacts may be considered incommensurable (Essl et al. 2017), therefore falling out of the scope of the framework. The average impact Ī_s over all considered individuals from a species could be used as a measure at the species-level, as different individuals may experience different impacts, if the management action targets only part of a given population. Using the average impact is not without shortcomings though, since it does not account for potential discrepancies in impacts suffered by different individuals in a population. In other words, to which point do “the needs of the many outweigh the needs of the few” (Littmann 2016)? Other measures such as the maximum impact experienced by individuals, or more complex functions accounting for the variability of impacts and values across individuals of a same species may also be used, to account for potential disproportionate impacts on a subset of the considered individuals. Under anthropocentric perspectives, impacts are influenced by the utilitarian values of species.

Considering Equation 1 in an operational fashion, the consequences C computed from it can be interpreted as a constructed attribute to measure the achievement of objectives in conservation under different value systems (sensu Keeney and Gregory 2005). This may be possible for simple systems with few species and clear categories of values and impacts (Fig. 3). However, for complex systems, a quantitative evaluation of Equation 1 will be difficult or impossible. For such systems, the purpose of the framework is not to prescribe how such a constructed attribute should be computed, nor to be used directly as a decision analysis tool (i.e. not to be applied directly). To be used in such a fashion, constructed attributes need to be unambiguous, comprehensive, direct, operational, and understandable by the general public (Keeney and Gregory 2005). Because value systems can be complex, meeting all five criteria is necessarily difficult. Instead, Equation 1 should be seen as a guide to ask questions that are relevant if management shall account for different value systems. By trying to evaluate Equation 1, one will have to ask such questions in a systematic fashion (Table 2), while understanding how these questions are conceptually linked with each other.

Figure 3.

Applying the framework presented in Equation 1 to determine the likely consequence of a management action on a system with two species, highlighting possible moral dilemmas in red. In the case shown a is set to 1 for simplicity, but the two species have different inherent values V_high and V_low (i.e. how individuals are valued does not vary with abundance, but individuals of one species are valued more than the other species). The likely consequence changes with the relative abundance of the two species [top row (a) vs. bottom row (b)] and with whether the impact of the management intervention is positive (I₊) or negative (I-) on the respective species [columns (i-iv)]. a) The species with high value has higher or similar abundance to the species with low value. If the impacts I₊ and I- have similar orders of magnitudes or |I₊| > |I-|, scenario (a,ii) generates positive consequences (C₊) because V_high × N_high > V_low × N_low. Similarly, if the impacts I₊ and I- have similar orders of magnitudes or |I₊| < |I-|, scenario (a,iii) generates negative consequences (C-). If |I₊| |I-| or |I₊| > |I-| (for scenarios (a,ii) and (a,iii) respectively), the difference of impact can counter-balance V_high × N_high > V_low × N_low, making desirable consequences undesirable and vice versa. However, the difference of magnitude between I₊ and I- at which this switch occurs is difficult to determine due to the different units of V, N, and I. This uncertainty corresponds to a moral dilemma due to a conflict between the desire to have a small positive impact on the species with the larger value and abundance, and the desire to avoid a very negative impact on the species with the lower value and abundance for scenario (a,ii). For scenario (a,iii), the dilemma is due to a conflict between the desire to avoid a small negative impact on the species with the higher value and abundance, and the desire to have a very positive impact on the species with the lower value and abundance. b) The species with higher value V_high has the lower abundance N_low. If impacts are different between the two species, the opposition between V and N will most likely generate moral dilemmas (C_?). If V_high × N_low > V_low × N_high, scenario (b,ii) is equivalent to scenario (a,ii), and to scenario (a,iii) otherwise (and scenario (b,iii) is equivalent to scenario (b,iii), and to scenario (a,ii) otherwise), but because value and abundance have different units, it is difficult to determine for which value and abundance V_high × N_low = V_low × N_high. Therefore, an additional moral dilemma arises due to a conflict between the desire to avoid a negative impact on the larger population and the desire to avoid a negative impact on the species with the higher value.

If Equation 1 could be evaluated, for each measure of impact and each system of values, Equation 1 would produce relative rather than absolute values. The values of consequences C of a management action under different value systems and measure of impact cannot be directly compared with each other, because the unit and range of values of C can vary between value systems. Instead, Equation 1 can be used to rank a set of management actions for each value system or measure of impact based on their assessed consequences, to identify management actions representing consensus, compromises or conflicts amongst value systems.

Equation 1 is particularly useful to identify potential moral dilemmas, i.e. situations in which management options are conflicting under the same value system (Table 1). For example, if different types of impacts are considered simultaneously under a value system (e.g. economic vs cultural impacts, or lethal impacts vs. those causing suffering, see sections below), Equation 1 might rank management actions differently for these different impacts under the same system of moral values.

In some situations the implication of Equation 1 is clear. For example, if an impact is positive for a highly valued, highly abundant species, but slightly negative for a few individuals of another species that is not considered very important (C = I₊ × V_high × N_high + I- × V_low × N_low), the consequence will be positive (Fig. 3aii). However, if the magnitude of the negative impact is much higher than that of the positive impact (|I₊| |I-|), the consequence can become negative. Similarly, if impact is negative for the species with the highest value and abundance, and positive for the other species (C = I- × V_high × N_high + I₊ × V_low × N_low), the situation is clear if positive and negative impacts have the same magnitude, but it will shift once the magnitude of the positive impact becomes higher than the magnitude of the negative impact (|I₊| > |I-|; the difference of magnitude will likely be lower than in the first example, because of the differences in sign; Fig. 3aiii). Since impact, value and abundance have different units, the thresholds at which these shifts occur are difficult to assess, and so the consequences can be highly debatable. This can create moral dilemmas, e.g. between the desire to have a small positive impact on a larger population with higher value and the desire to avoid a very negative impact on the species with the lower value and abundance (Fig. 3aii); and between the desire to avoid a small negative impact on the larger population with the higher value and the desire to have a very positive impact on the species with the lower value and abundance (Fig. 3aiii). Moral dilemmas will be even more likely to occur if the species with the higher value has the lower abundance (C = I₊ × V_high × N_low + I- × V_low × N_high or C = I- × V_high × N_low + I₊ × V_low × N_high; Fig. 3bii,iii). If V_high × N_low > V_low × N_high, the example depicted in Fig. 3bii is equivalent to the example depicted in Fig. 3aii described above, and Fig. 3biii is equivalent to the example depicted in Fig. 3aiii. If V_high × N_low < V_low × N_high, the example depicted in Fig. 3bii is equivalent to the example depicted in Fig. 3aiii described above, and Fig. 3biii is equivalent to the example depicted in Fig. 3aii. As above, it is difficult to determine when the inequality will change direction because of the difference in the units of V and N. This reflects a moral dilemma due to a conflict between the desire to avoid a negative impact on the larger population and the desire to avoid a negative impact on the species with the higher value. In summary, uncertainty in the computation of Equation 1, and in particular the need to compare parameters with different units (i.e. impact, value, and abundance), can therefore be interpreted as a moral dilemma (Fig. 3).

In addition, some actions might not follow moral norms compared to others despite having more desirable consequences. For example, killing individuals may be considered less moral, but more efficient to preserve biodiversity or ecosystem services than using landscape management. Solving these moral dilemmas is complex, and beyond the scope of this publication, but approaches such as multi-criteria decision analyses (MCDA; Huang et al. 2011) may offer an avenue to do so (Goetghebeur and Wagner 2017).

Similarly, environmental conflicts will likely emerge when comparing the rankings generated by Equation 1 under different value systems considering different distributions of values, and different measures of impact. MCDA (Wittmer et al. 2006) and operational research (Kunsch et al. 2009), have also been proposed to resolve such conflicts. We nonetheless argue that, regardless of the capacity to resolve environmental conflicts (or moral dilemmas), Equation 1 can help decision makers to understand how conflicts (might) emerge.

In the following, we discuss the complexity of assessing the different variables and parameters of Equation 1 under different value systems using the set of primary questions defined above. By doing so, it becomes possible to identify ambiguity, difficulty of operationality, etc., to eventually move towards a good constructed attribute (although such a constructed attribute may not be reached in practice). We also discuss how, despite the difficulty to quantify the variables described above, this framework can be used as a heuristic (rather than operational) tool to capture the implications of considering different value systems for determining the appropriateness of a conservation action, and to better understand conservation disputes.

Traditional conservation is based on an ecocentric value system and seeks to maximise diversity of organisms, ecological complexity, and to enable evolution (Soulé 1985). For the sake of simplicity, we will consider an extreme perspective of traditional conservation, championed by ‘fortress conservation’ (Siurua 2006; Büscher 2016), i.e. excluding humans from the moral community. To capture these aspects, consequences C in Equation 1 can be more specifically expressed as follows:

C = ∑_{species s (excluding humans)}Ī_s × V_s × N_s^{a < 0} Eq. 2

Assigning a stronger weight to rare species (a < 0) accounts for the fact that rare species are more likely to go extinct, decreasing the diversity of organisms. Evolution and ecological complexity are not explicitly accounted for in Equation 2. To do so, one may adapt Equation 2 and consider lineages or functional groups instead of species as the unit over which impacts are aggregated.

Because traditional conservation seeks to maximise diversity, I_s can be defined as the probability of individuals dying. I_s × N_s^{a < 0} will then be proportional to the extinction risk of a species (for an operational application, a proper model for extinction probability could be used in lieu of I_s × N_s^{a < 0}). The V_s distribution could be considered uniform over all species, in the absence of biases.

New conservation considers that many stakeholders (“resource users”, Kareiva, 2014) tend to have an anthropocentric value system, and that conservation approaches that do not incorporate such a perspective will likely not succeed at maximizing diversity of organisms (Kareiva and Marvier 2012; Kareiva 2014). Under anthropocentrism, species are only conserved due to their utilitarian value, i.e. their effect on I for humans, rather than based on an inherent value V. Different groups of stakeholders are likely to be impacted differently (e.g. different monetary benefits / losses), and we propose the following extension of Equation 1 to account for this variability:

C = ∑_{stakeholders t} Ī_t × V_t × N_t Eq. 3

where Ī_t is the average impact of management on the group of stakeholders t, including indirect impacts through the effect of management of non-human species. Ī_t can correspond to economic impacts, or encompass categorical measures of wellbeing (e.g. Bacher et al. 2018). V_t is the value of the group of stakeholders t, and N_t is its abundance (i.e. the number of people that compose it). Parameter a is set to 1; as this is considered to be an individual-based value system. Note that including inherent values V_t in Equation 3 does not imply that we consider that different humans should be valued differently — though that is a view that some people hold — this needs to appear here to capture the full spectrum of perceived consequences of a management action.

New conservation holds an ambiguous perspective, stating that it should make “sure people benefit from conservation”, while at the same time it does not “want to replace biological-diversity based conservation with a humanitarian movement” (Kareiva 2014). Using our framework, we interpret this to mean that one can design management actions that minimise consequences C under both Equations 2 and 3 (i.e. a mathematical expression of the convergence hypothesis; Norton 1986). Importantly, minimising Equation 3 is thereby a prerequisite for minimising impacts I and hence consequences C in Equation 2 (Fig. 2). Under this interpretation of new conservation, Equation 2 may therefore be rewritten as follows:

C = ∑_{species s (excluding humans)}Ī_s (C_humans) × V_s × N_s^{a < 0} Eq. 4

where C_humans is computed using Equation 3, and assuming a monotonic and positive relationship between Ī_s and C_humans.

The link between biodiversity and ecosystem services is strongly supported, even if many unknowns remain (Chivian and Bernstein 2008; Cardinale et al. 2012), implying that high biodiversity can indeed support the provision of ecosystem services to humans. Such an approach will necessarily distinguish between “useful” species and others, and impacts will be perceived differently by different groups of stakeholders. Considering multiple types of impacts (economic benefits/losses, access to nature, health, etc.) while accounting for cultural differences, would increase the pool of useful species (comparing the resulting equation outputs using, for example, MCDA). The outcome of the two approaches would then potentially be more aligned with each other. This broad utilitarian perspective is captured in the most recent developments of new conservation approaches, which consider a wide range of nature contributions to people, rather than just ecosystem services (Díaz et al. 2018).

People and nature views seek to simultaneously benefit human wellbeing and biodiversity (Fig. 2). Under this perspective, Equations 2 and 3 should therefore be computed separately (instead of being linked together as in Equation 4), before being combined in a single approach, for example using MCDA (Huang et al. 2011; assuming these equations can indeed be operationally computed), to capture a more diverse set of value systems than Equations 2 and 3 alone, even if the two approaches generate divergent results.

We expressed traditional and new conservation with Equations 2, 3 and 4, which correspond to extreme interpretations of these two approaches (excluding humans or considering specific utilities of species). Doing so illustrates how our mathematical framework can capture the pitfalls of failing to explicitly define normative postulates for conservation approaches. As a result, Equations 2, 3 and 4 will likely generate conflicting results in the ranking of different management actions, especially if few types of impacts are considered. The debates over new conservation have taken place in a discursive fashion, which has not provided a clear answer to the values defended by this approach (Kareiva 2014; Soulé 2014; Doak et al. 2015). It has therefore been argued that the normative postulates of new conservation need to be more clearly defined (Miller et al. 2011). Our framework could help doing so, by being explicit about how new conservation would be defined relative to the traditional conservation and the people and nature perspective through the addition of specific terms to Equation 3 and a thorough comparison of the resulting equations. In particular, it would be interesting to explore, how inherent values are attributed to different species under a new conservation approach compared to under a traditional conservation approach (e.g. relational vs. intrinsic value; Chan et al. 2016; Table 1) and how their distributions differ.

A consequentialist, sentientist perspective aims at maximizing happiness, or conversely minimising suffering, for all sentient beings, an approach also termed ‘utilitarianism’ (Singer 1980; Varner 2008). Suffering is therefore considered as a measure of impact (or, in mathematical terms, impact is a function of suffering, which can be expressed as I (S_s) in Equation 1).

It has become widely accepted that animals experience emotions (de Waal 2011). Emotions have been shown to be linked to cognitive processes (Boissy and Lee 2014), which differ greatly among species (MacLean et al. 2012), and behavioural approaches have been used to evaluate and grade emotional responses (e.g. (Désiré et al. 2002); but see (Shriver 2006) and (Bermond et al. 2001) for different conclusions about the capacity of animals to experience suffering). We therefore postulate that the quantification of suffering is conceptually feasible in the context of the heuristic tool presented here. In a utilitarian approach, the inherent value of a species would therefore be a function of its capacity to experience emotions and suffering E_s, which can be expressed as V (E_s) instead of V_s in Equation 1.

Under these considerations for defining impact and value of species, the consequences of a conservation action can be computed as a function of suffering of individuals from species s S_s, their capacity to experience emotion and suffering E_s, and the abundance of species s:

$C=\sum _{\text{species s}}\overline{I\left(S_s\right)}\times V\left(E_s\right)\times N_s^{a=1}$ Eq. 5

Although V (E_s) should be measured in an objective fashion, many factors may influence the relationship between the inherent value and the emotional capacity of a species. For example, high empathy (Table 1) from the observer will tend to make the distribution uniform, whereas anthropomorphism and parochialism (Table 1) may lead to higher rating of the emotional capacities of species phylogenetically close to humans or with which humans are more often in contact, such as pets. Finally, we assumed that a = 1, to give equal importance to any individual regardless of the abundance of its species, as suffering and wellbeing are usually considered at the individual level (Beausoleil et al. 2018).

The short-term suffering resulting from pain and directly caused by lethal management actions, such as the use of poison to control invasive alien species (Twigg and Parker 2010) or the use of firearms and traps to cull native species threatening other native species (Proulx et al. 2016) or humans (Gibbs and Warren 2015), is the most straightforward type of suffering that can be assessed, and is usually sought to be minimised in all conservation approaches. Suffering can have many other causes, and suffering of an individual may be assessed through a wide variety of proxies, including access to food and water, death, number of dead kin for social animals, physiological measurements of stress hormones, etc. Suffering can take various forms, and commensurability can be an issue (Table 3), making the distinction between the morality of lethal actions and non-lethal suffering complex. Non-lethal suffering can result from unfavourable environmental conditions (e.g. leading to food deprivation) and occur over long periods, while lethal actions could be carried out in a quick, non-painful fashion (Shao et al. 2018), and even lead to improved animal welfare (Wilson and Edwards 2019), but may be deemed immoral.

The concept of animal welfare and how to measure it is extremely complex (Beausoleil et al. 2018), and defining it precisely is beyond the scope of this study. We nonetheless advocate a conceptual approach that takes into account indirect consequences of management actions within a certain timeframe and consider uncertainty (Table 3). Direct and indirect biotic interactions may be explicitly modelled to quantify the impact on animals and their suffering. Simulation models can also make projections on how populations may change in time, accounting for future suffering.

It has been argued that sentientism and ecocentrism are not fully incompatible (Varner 2011). The relationship between biodiversity and animal suffering can be formalised more clearly using the traditional conservation and the sentientist Equations 2 and 4, to explore if the same management action can minimise the consequences evaluated using the two equations (see also Suppl. material 1: Appendix S2 for the application of the framework to theoretical cases). The main difference with the traditional vs new conservation debate here is that Equations 2 and 4 share a number of species, whereas the new conservation Equation 3 only contains humans, which are excluded from Equation 2. Even though the variables of Equation 5 differ from those of Equation 2 (V and I are computed differently, and the value of a is different), it is possible that these equations will vary in similar ways for different management actions due to their similar structure, although this would depend on the variety of impacts on humans that are considered in Equation 3. Finally, as for the people and nature approach, the consequences of sentientist and ecocentric approaches can be evaluated in combination, as suggested by conservation welfare (Beausoleil et al. 2018), using tools such as MCDA (Wittmer et al. 2006; Huang et al. 2011).

One issue that may be irreconcilable between ecocentric approaches such as traditional conservation and approaches based on sentientism is the fate of rare and endangered species with limited or no sentience. Under utilitarian sentientism, the conservation of non-sentient species ranks lower than the conservation of sentient species, and consequently they are not included in Equation 5. For example, endangered plant species that are not a resource for the maintenance of sentient populations would receive no attention, as there would be few arguments for their conservation. Traditional conservation would focus on their conservation, as they would have a disproportionate impact in Equation 2, due to low abundance leading to a high value for N ^{a < 0}.

Finally, it is important to note that the current body of knowledge shows that the link between biodiversity and animal welfare mentioned above especially applies to the increase of native biodiversity. Local increase of biodiversity due to the introduction of alien species may only be temporary due to extinction debt (Kuussaari et al. 2009) and often results in a reduction of ecosystem functioning (Cardinale et al. 2012). Therefore, it is important to distinguish between nativism (Table 1) and the detrimental effects of invasive alien species on biodiversity and ecosystem functioning and services (Bellard et al. 2016). Nativism would result in increasing the inherent value V_s of native species (Fig. 1), whereas in the second case, insights from science on the impact of invasive alien species would modify the distribution I (S_s) rather than the distribution V_s. This can also apply to native species whose impacts on other species, such as predation, are increased through environmental changes (Carey et al. 2012).

The grey squirrel (Sciurus carolinensis) is native to North America and was introduced in various locations in Europe during the late nineteenth and the twentieth century (Bertolino 2008). It threatens native European red squirrel (Sciurus vulgaris) populations through competitive exclusion and as a vector of transmission of squirrel poxvirus in Great Britain (Schuchert et al. 2014). Furthermore, it has wider impacts on woodlands and plantations, reducing the value of tree crops, and potentially affects bird populations through nest predation (Bertolino 2008).

Based on the impacts of the grey squirrel, an eradication campaign was implemented in 1997 in Italy, with encouraging preliminary results (Genovesi and Bertolino 2001). However, this eradication campaign was halted by public pressure from animal rights movements. The strategy of the animal rights activists consisted of (i) humanising the grey squirrel and using emotive messages (referring to grey squirrels as “Cip and Ciop”, the Italian names of the Walt Disney “Chip and Dale” characters) and (ii) minimising or denying the impact of grey squirrels on native taxa, especially on the red squirrel (Genovesi and Bertolino 2001). In addition, the activists did not mention (iii) the difference in abundance between a small founding population of grey squirrels that could be eradicated by managers, and a large population of native red squirrels that would be extirpated or severely impacted by grey squirrels if control was not implemented.

Genovesi and Bertolino (2001) explain that the main reason for the failure of the species management was a different perspective on primary values. The conservation managers, favouring eradication, based their decision on species valuation, following traditional conservation. The animal rights activists, opposed to control, focussed on animal welfare. Applying the framework, and assuming an individual-based value system (a = 1 in Equation 1), three questions are apparent (Table 2):

Are red and grey squirrels valued differently?

What types of impact are considered?

Is the population of red squirrels impacted by grey squirrels larger than the population of grey squirrels to be controlled?

The arguments of animal rights activists led to the following answers to these three questions. (i) The humanisation of the grey squirrel consists of increasing the perception of its emotional capacity E_gs > E_rs (and therefore V (E_gs) > V (E_rs)). (ii) Minimising the impact of the grey squirrel is equal to restricting the time scale to a short one and to likely minimising the amount of suffering S caused by grey squirrels on other species (under a sentientist perspective), or the number of red squirrels that will die because of grey squirrels (under a biocentric perspective). In other words, S_gs = S_rs (and therefore I (S_gs) = I (S_rs)) or I_gs = I_rs without management and S_gs > S_rs (and therefore I (S_gs) > I (S_rs)) or I_gs > I_rs under management. (iii) Not mentioning differences in species abundance implies that the impacted populations of red and grey squirrels would have the same size under any management. Following these three points, the consequences under management C_m = I (S_gs) × V (E_gs) + I (S_rs) × V (E_rs) are higher than without management, due to the increase in V (E_gs) and I (S_gs). The application of our framework therefore clarifies a discourse whose perception could otherwise be altered because of techniques such as an appeal to emotion.

The framework can thus be used to provide recommendations for what the advocates for the eradication campaign would have needed to have done: i) increase the value E_rs of red squirrels in a similar way as what was done for grey squirrels, so that their relative values compared to grey squirrels would remain the same as before the communication campaign by the animal rights activists; ii) better explain the differences in animal death and suffering caused by the long-term presence of the grey squirrel compared to the short-term, carefully designed euthanasia protocol, which would avoid a subjective perception of the distribution of S; and iii) highlight the differences in the number of individuals affected. The consequences would then be computed as C = V (E_gs) × I (S_gs) × N_gs + V (E_rs) × I (S_rs) × N_rs. In that case, assuming the amount grey squirrels suffer as a result of being euthanised is the same as red squirrels suffer from the grey squirrels, and all squirrels (be they grey or red) are valued the same (i.e. avoiding nativism), the mere differences N_rs > N_gs in abundance would lead to a higher value of C without management. This would further increase by extending the impacts of grey squirrels to other species, as mentioned above.

A more fundamental issue, however, is that in some value systems it would not be acceptable to actively kill individuals, even if that meant letting grey squirrels eliminate red squirrels over long periods of time (Wallach et al. 2018). The reluctance to support indirectly positive conservation programs is a common issue (Courchamp et al. 2017). Whether an acceptable threshold on consequences over which killing individuals could be determined through discussion would depend, in part, on the willingness of the affected parties to compromise.

De-domestication, the intentional reintroduction of domesticated species to the wild, is a recent practice in conservation that raises new ethical questions related to the unique status of these species (Gamborg et al. 2010). Oostvaardersplassen is a Dutch nature reserve. Reserve managers, recognising that grazing by large herbivore was a key natural ecosystem process that had been lost, decided between 1983 and 1992 to reintroduce red deer (Cervus elaphus), and two domesticated species (Heck cattle, Bos primigenius, and konik horses, Equus ferus caballus) (ICMO2 2010). The populations of these three species increased rapidly, as natural predators were missing and, as a result of a ‘non-intervention-strategy’, no active population control measures were implemented. The project was widely criticised when a considerable number of individuals died from starvation during a harsh winter, resulting in the subsequent introduction of culls.

From a traditional conservation perspective, disregarding animal welfare and focusing on species diversity and ecological restoration, the project was a success. The introduction of the three herbivore species led to sustainable populations (despite high winter mortality events), and ensured stability of bird populations without the need for further interventions (ICMO2 2010), i.e. the conditions of many species were improved (the impact was lowered), leading to improved consequences C for biodiversity overall (Equation 2). In other words, since more individuals from all species survived (I increased in Equation 2), C improved overall, regardless of differences in value or abundance between species (a multi-species generalisation of Fig. 3i).

However, the welfare of individuals from the three charismatic large herbivorous species became a point of conflict. In terms of the framework, it appears that the conflict was driven by considering the outcome of Equation 5 in addition to that of Equation 2 to estimate the overall evaluation of the management approach, i.e. a change from only considering impacts on individual survival to also considering impacts based on suffering, with the acknowledgement that E_s should be considered (Ohl and Van der Staay 2012). Not considering Equation 5 would mean that C = 0 under sentientism, but acknowledging the existence of E_s implies that C = V (E_s)×I (S_s)×N_s¹ becomes non-null. Changes in perspective over time should therefore be taken into account when implementing conservation management actions, and adaptive management approaches should be considered. A possible explanation for this shift in attitude is the notion of responsibility (Table 3). Culling animals might be acceptable in some cases, but might not be if these individuals were purposefully introduced, which may lead to considering a sentientist perspective.

The reserve managers examined a number of sustainable measures to improve the welfare of individuals from the three species (therefore decreasing S_s to compensate the increase in V_s). These included recommendations to increase access to natural shelter in neighbouring areas of woodland or forestry, to create shelter ridges to increase survival in winter as an ethical and sustainable solution, and to use early culling to regulate populations and avoid suffering from starvation in winter (ICMO2 2010). This example shows how a combination of two complementary management actions (the rewilding of the OVP and the provision of shelter) led to minimised consequences under both the traditional conservation and the sentientist Equations 2 and 5, whereas only rewilding would increase consequences under Equation 5. Interestingly, other approaches, such as the reintroduction of large predators, were also considered but discarded due to a lack of experience and too many uncertainties in efficiency (ICMO2 2010). Our suggested framework could be adapted to explore the consequences of culling vs. increased mortality through the reintroduction of large predators, noting that some stakeholders may make moral distinctions between natural mortality and human-induced mortality. Culling may also still face opposition based on moral arguments.

Trophy hunting, the use of charismatic species for hunting activities, has been argued to be good for conservation when revenues are reinvested properly into nature protection and redistributed across local communities, but faces criticisms for moral reasons (Lindsey et al. 2007b; Di Minin et al. 2016). The action of killing some individuals to save others might be incompatible with a deontological perspective, but, assuming a consequentialist perspective, the framework can be applied to formalise the assessment of different management options. Note that here, we are not considering the ethics of how the hunt itself is carried out (e.g. canned hunting vs. a “fair chase”) nor how animals are reared (i.e. whether they can express their natural behaviours), recognising that both these factors would need to be considered when making a decision.

In traditional conservation, trophy hunting is desirable if it directly contributes to the maintenance of species diversity. That is, it should decrease impacts I evaluated as individual survival over all or the majority of species with high inherent value, leading to improved consequences for biodiversity C in Equation 2 (a multi-species generalisation of Fig. 3i, ii). The potential of trophy hunting to contribute to the maintenance of biodiversity is via creating economic revenues, i.e. an anthropocentric perspective, and it therefore falls under the umbrella of new conservation (Fig. 2; Equation 4). In theory, trophy hunting should lead to lower consequences than doing nothing for both the traditional and new conservation (Equations 2, 3 and 4), and therefore for the ‘people and nature’ approach, as they are in this case not independent from each other (Lindsey et al. 2007a). Many social and biological factors currently affect the efficacy of trophy hunting as a conservation tool. Corruption and privatisation of the benefits have sometimes prevented the revenues to be reinvested into conservation, but also to be redistributed across local communities, whereas doing so has been shown to increase their participation in conservation actions with proven benefits for local biodiversity (Di Minin et al. 2016). In other words, a decrease in the anthropocentric Equation 2 leads to a decrease in the ecocentric Equation 3, but the causal link (Equation 4) is still supposed to be valid. In addition, trophy hunting can lead to unexpected evolutionary consequences (Coltman et al. 2003), overharvesting of young males (Lindsey et al. 2007b), and disproportionate pressure on threatened species (Palazy et al. 2011, 2012, 2013) and therefore to population declines and potential detrimental effects on biodiversity. That means that I(C_humans) in Equation 4 should be carefully examined. Despite these issues, it has been argued that banning trophy hunting may create replacement activities that would be more detrimental to biodiversity (Di Minin et al. 2016).

From an animal welfare perspective, trophy hunting appears to be in direct contradiction with a decrease in animal suffering, and has been criticised by proponents of compassionate conservation (Wallach et al. 2018). However, as for the culling of invasive alien species, we suspect the story is more complex. First, there may be direct benefits for animal welfare on average, if money from trophy hunting is reinvested in protection measures against poaching (if such poaching causes, on balance, more suffering). Second, to our knowledge, only a few studies have compared the welfare of individual animals to quantify the elements of the sentientist Equation 5 (for example, assessed through access to resources) in areas where trophy hunting is practised and where it is not. Given the links between biodiversity and animal welfare described above, it seems plausible that good practice in trophy hunting may benefit the welfare of individuals from other, and from the same, species.