3.1 Specifying mortality probability

TL used an ad hoc method to approximate a range of annual mortality rates (D) for a year without Endangered Species Act protections and without a wolf hunt. Assuming a wolf year from 15 Apr in year t to 14 Apr in year t+1, they calculated D = M_t/(N_t+0.5×R_t), where parameters are as described above.

The estimates for D provided in Table 1 of TL are problematic for several reasons. First, the equation to calculate D provided in the caption to TL Table 1 is (N₂₀₂₁ − N₂₀₂₀)/(N₂₀₂₀ + 0.5 × R₂₀₂₀), which cannot be correct because it yields negative D, because N₂₀₂₁ < N₂₀₂₀ (e.g., 969–1034 = −65). We suspect this reflects a typographical error and TL meant to say (N₂₀₂₀ − N₂₀₂₁)/(N₂₀₂₀ + 0.5 × R₂₀₂₀), although this formulation would then produce negative D if N_t+1 > N_t. Additionally, the assumption in Table 1 of TL that M_t = N_t − N_t+1 is self-evidently problematic, and differs with how TL previously defined M_t. For our current analysis, rather than relying on an ad hoc estimate for D, we used an estimate derived from rigorous peer-reviewed analyses of Wisconsin wolves [10, 11]. Below, we provide further discussion and defense of these peer-reviewed analyses.

Second, the min/max estimates presented by TL for N₂₀₂₁ were implied to be estimates equivalent to the previous year’s count estimate produced by the Wisconsin DNR, but the estimates are not comparable. The Wisconsin DNR did not generate a count in 2021, and TL would not have had access to the data required to generate such an estimate. Instead, the min/max estimates presented by TL for N₂₀₂₁ appear to derive from deterministic calculations assuming certain hypothetical values for annual population growth [6]. The range 695–751 does not represent a 95% confidence interval but rather extremes representing the bounding values for population growth assumed by Treves et al. [6]. We were unable to reproduce the estimates for D in TL Table 1, even using the values specified by TL. It is unclear what values TL actually used to calculate D.

Furthermore, in an attempt to justify their specification for D, TL stated that there is little scientific consensus on annual mortality rates among Wisconsin wolves, but then they dismissed or ignored several rigorous, defensible, peer-reviewed, and remarkably similar estimates from Wisconsin and neighboring states in the upper Great Lakes region [4, 10–13]. In contrast, TL uncritically accepted implausible conclusions that a wolf population could sustain an annual population growth rate of 1.13 while suffering an annual mortality rate of 0.46 [14, 15]. In particular, TL dismissed the annual mortality estimate of 0.241 (SD = 0.019) provided by Stenglein et al. [10], because it “seems low” and because “…that study failed to account for several confounding variables and took unjustified steps in analyses” [2, p. 10]. TL then specifically criticized: 1) a variable for change of slope in the year 2004; 2) unaccounted changes in census methodology; 3) pooling non-human causes of death with unknown causes of death; 4) failure to acknowledge potential differences in mortality for collared and uncollared wolves; 5) failure to acknowledge evidence that wolf survival and population growth declined when ESA protections were lifted; and 6) failure to account for seasonal changes in incidence of wolf mortality. They concluded that the estimate from Stenglein et al. [10] was “certainly too low given the conditions between 3 November 2020 and 13 April 2021.” It is unclear whether the conditions to which they refer relate to their criticism 6 or to the Feb 2021 wolf hunt. Mortality from the wolf hunt is already accounted for in the starting population for the population projection and thus should not be considered for non-hunting annual mortality.

The claimed issues in criticisms 1–3 are not relevant to the annual survival estimate from Stenglein et al. [10], nor are they germane to either TL or this response. Criticism 4 raises an important question, but the assertion is false; Stenglein et al. [10] did address this issue explicitly by citing Stenglein et al. [11], who rigorously evaluated the extent to which mortality estimated from radio-collared wolves might underestimate population level mortality, given population growth and recruitment rates. Stenglein et al. [10] concluded that only minor adjustment was needed (i.e., annual mortality was 25% instead of 24%). Moreover, the differences in mortality rates suggested by Treves et al. [14] are implausible given the estimated annual population growth rates, estimated empirically from extensive snow-tracking data [4]. Criticism 5 relates to the claims made in Chapron and Treves [16], which have been thoroughly discredited by multiple authors [17–19], and responses by Chapron and Treves [20, 21] to those criticisms did not refute or adequately address them. Moreover, Stenglein et al. [10] used a temporal spline for year that captured potential interannual differences in mortality resulting from changes in legal protections. Criticism 6 is also false; Stenglein et al. [10] fitted a temporal model that allowed mortality hazard to vary weekly throughout the year. TL failed to demonstrate here or in any of their citations that the annual mortality estimate provided by Stenglein et al. [10] is implausible or unreliable, and we conclude that estimates for wolves in Wisconsin reported by Stenglein et al. [10], incorporating Stenglein et al. [11], are the most plausible and defensible estimates available. In contrast, the estimates of D presented in TL exaggerate annual mortality by as much as 100%.

As noted above, TL recognized that pup survival (S) was misspecified in their paper, and posted an online comment on the the journal website (7 Aug 2022) specifying a somewhat ambiguously defined correction of S = 0.29 (range = 0.14–0.58) from Table 6.3 in Wydeven et al. [4]. Regardless of it’s precise definition, this value for S is still problematic for at least two reasons. First, if the new distribution with mean 0.29 has the same shape as the old distribution, then probability mass is concentrated in the lower end of the range with mode < mean. Secondly, and more importantly, the ad hoc estimates in Table 6 of Wydeven et al. [4] represent survival estimates from birth to the end of a pup’s first winter, a period of approximately 12 months. TL acknowledge that their model requires estimates for pup survival until November, which, assuming pups are born in April, is a period of about 7 months. To use these ad hoc estimates, TL should have converted them to 7-month estimates () which would result in a mean of 0.484 with [min, max] = [0.318, 0.728]. We used a beta distribution with mean 0.484 to approximate S (Table 1), which appropriately accounts for 7 months of survival but is somewhat more conservative than Table 6 in [4] in the right tail of the distribution. Alternatively, TL could have used the estimate of 0.72 (95% C.I. = 0.51–0.94) from 84 radio-collared wolves [4]. Despite the correction, S in TL remains strongly and negatively biased.

3.2 Proportion of packs reproducing

In the calculations for the ad hoc estimate of D in Table 1 of TL, B_t = 245 represents all 256 presumed packs in the population, minus the 11 packs inhabiting tribal reservations (refugia from hunting), and the fact that not all packs reproduce each year is captured in the parameter PPN = 0.72 [range = 0.55–0.89]. But, TL twice used PPN to reduce the number of packs that produced pups. TL multiplied 245 by the upper bound of PPN (245 × 0.89 = 218), and then subtracted 51 presumed breeding females (we explain below why 51 might represent a positive bias) to derive an upper bound of 167 for B_t. However, in their parameter specifications in Table S2 of their paper, TL then derived R_t = B_t × PPN × L × S. To put it succintly, TL calculated B_t = N_packs × PPN_max − 51, and then calculated R_t = (N_packs × PPN_max − 51) × PPN × L × S. We also note that the distribution specified for PPN in Table S2 of TL produces a mean of 0.68 rather than 0.72 stated in TL.

The deduction of 51 packs by TL was based on the assumption that 23% of the harvested wolves were pregnant females, rather than from an examination of the complete harvest data. TL cited a small, non-random sample of 22 wolves, and noted that 65% of adult females and 50% of yearling females were pregnant, but they did not state exactly what proportion of the sample was females. Although not explicitly stated, it was misleadingly implied that 50–65% of the sample was pregnant females, and that by comparison, 23% was a conservative estimate. In fact, TL used this implied conclusion that 65% of the harvest (not 65% of the females in the harvest) were pregnant to assert a lower bound of only 74 packs that reproduced. In their simulations, they then assumed B_t ∼ Uniform(74, 167), and they noted that B_t = 12 and B_t = N_packs × 0.89 = 218 were “implausible extreme values”, which they did not use in their simulations. The use of the uniform distribution was justified by TL in their caption to their Fig 2 (“…the uniform, uninformative distribution allows the data to influence the result rather than our preconceived notions of what is typical in biological distributions”). This statement is incorrect and misleadingly implies that the specified uniform distribution was an uninformative prior for B_t, and that fitting data to a statistical model likelihood resulted in an updated and more informative posterior estimate for B_t. TL did not fit data to a model, and consequently, it is meaningless to speak of uninformative prior distributions, as if there were a Bayesian analysis that could in any way provide updated posterior parameter estimates. Rather, the projection in TL was a simple simulation where the specified distributions for each of the parameters were arithmetically combined to produce a result that depended directly on the specified inputs. In other words, the specified parameter inputs were themselves the data and those specified parameters directly and deterministically influenced the results. In the case of B_t, the specified uniform distribution was premised on consequential assumptions and included a calculation error, both of which absolutely influenced the results in a specific direction (fewer breeding packs).

3.3 Pup production to November

In Equation 1, TL presented their simple one-step model of population size change as N_t+1 = N_t + R_t − M_t − H. In Table S2 of TL, they presented a form of the model presumably used in the simulations. That model was N_{t+ 1} = (N_t + B_t × PPN × L × S × 0.5) × (1 − D) − H. Allowing R_t = B_t × PPN × L × S (note: R_t was not defined this way in Equation 2 of TL, but was so defined in Table S2 of TL), and allowing M_t = D × (N_t + R_t/2) as in equation 3 of TL, and then rearranging terms, yields a one-step model N_{t = 1} = N_t + R_t/2 − M_t − H where the number of pups surviving until November is only half that stated in Equation 1. To reiterate, the projection model used by TL inappropriately halved pup production compared to the model specified in the text of TL; instead of using R_t as specified in Equation 1 of TL, TL used R_t/2.

3.4 Subjecting harvested individuals to additional mortality

TL defined M_t = D × (N_t + R/2) as the number of wolves that die from t to t + 1 (excluding harvest), but this definition exaggerates mortality if there is a fall harvest, because it does not subtract out harvested wolves from N_t. TL recognized this positive bias [2, p. 11], but dismissed it by suggesting that any bias was offset by also dismissing unreported deaths and excess legal killing. There may be unreported deaths and illegal killing, but TL did not provide convincing evidence that the magnitude was comparable to the positive bias introduced by counting mortalities twice, or that such unreported deaths generate substantial negative bias in mortality estimates. Moreover, TL already presumably accounted for the possibility of additional deaths in the consideration of H. TL implied that unreported deaths necessarily result in underestimates of D, which is not typically the case with analyses that rigorously account for unobserved mortality [10].

Assuming for simplicity that there are 7 months of non-harvest mortality prior to a fall hunt, a fall hunt where harvest is accomplished relatively quickly (this is usually the case—in three previous hunts in WI in 2012–2014, 75% of the total harvest was achieved in 5 weeks, 3 weeks, and 2 weeks, respectively [22]), and then 5 more months of mortality to the end of the time-step, a better accounting of non-hunt mortality is as follows. Let M_t1 = [1 − (1 − D)^7/12] × N_t be the number of individuals that die prior to the hunt and H be the number of individuals that are harvested. The number that die after the harvest is then M_t2 = [1 − (1 − D)^5/12] × (N_t − M_t1 + R_t − H). Combining expressions and collecting terms results in M_t = M_t1 + M_t2 = N_t × D + [1 − (1 − D)^5/12]×(R_t − H). Obviously, M_t decreases as H increases, so the bias that results from failing to subtract out H is greatest when H is large. This formulation assumes completely additive harvest mortality, which is unlikely, as some degree of compensation is both theoretically expected and empirically estimated [10, 23–25]. In contrast, the formulation of TL allowed some deaths to be counted twice, if there is a harvest (which essentially imposed artificial super-additive mortality).

3.5 Other errors or misstatements

TL stated that Wisconsin’s new occupancy approach to estimated wolf population size was unpublished and not peer-reviewed. This is false because the general methodology was published before the submission of TL [5].

TL also stated that they “present the first estimates for annual mortality rate between 15 April 2020 and 14 April 2021.” The Uniform(0.38, 0.56) distribution that they presented is not a rigorous mortality estimate, but rather an ad hoc back-of-the-envelope calculation that we were unable to replicate, and that differs substantially from numerous other published and more rigorous estimates [4, 10–13].

TL stated that “The traditional census method had a reliable slope as judged by its r-squared value, twice as reliable as the new census method”. First, this is a mischaracterization of statistical reliability in the face of uncertainty. In a simulation exercise as in TL, one could make the correlation perfect by ignoring uncertainty entirely, but that does not make the slope more reliable. Second, as discussed previously, the population size estimates presented by TL do not represent the traditional census method but rather deterministically calculated bounds assuming certain hypothetical values for annual population growth [6].

The introduction and conclusion of TL stated that they used “Bayesian concepts” to account for uncertainties, but it is unclear what TL meant by that other than simply using Bayesian terminology [2, p. 4]. TL did not estimate parameters by fitting data to a model likelihood, and they provided no information or data that could update model parameters at all. Rather, TL conducted a simple simulation where uncertain parameters were used to calculate an uncertain but deterministic metric of interest. There is perhaps nothing inherently wrong with such an approach, but there also is nothing Bayesian about it.

TL engaged in a lengthy discussion about why they were justified in deducting annual mortality before the wolf hunt, and they stated that doing so treated wolf harvest as purely additive. First, it exceeded purely additive mortality, as explained above, because the calculation of deaths included a math error that allowed harvested wolves to be counted twice. Second, even the assumption of complete additivity is speculative—from a theoretical and empirical standpoint it is likely that harvest mortality is partially compensatory [10, 23–25].

In their conclusion, TL wrongly cited Treves et al. [26] as the source for the annual mortality estimate of 0.235 from Stenglein et al. [13] (also, the provided DOI for this citation was incorrect), and then went on to say that this estimate “is only plausible for 2020 if one accepts a drastic rise in population size from 2020 to 2021” [2, p. 14]. TL did not justify this assertion. In fact, an annual mortality estimate of 0.235 seems entirely reasonable in light of other published survival estimates from Wisconsin and Michigan [4, 12].

TL cited Artelle et al. [27] suggesting that many wildlife hunting plans lack clear objectives, independent review, and transparency, and then expressed regret that the Wisconsin DNR was favorably scored in [27], citing several controversial papers [6, 28] as evidence that the state of Wisconsin did not merit such a favorable review. TL further accused Wisconsin of inflating wolf quotas and “distorting sound science-informed management”, and they insinuated that the Wisconsin DNR (and other agencies) “pick and choose the evidence they wish to use based on their personal or organizational values” [2, p. 15]. This is a remarkable insinuation, given the parameter specification in TL that resulted in greatly exaggerated risk of wolf population decline in response to a wolf hunt. Although the authors may not have intended it, math errors and selected parameterizations in TL were consistently biased in the direction that maximized mortality and minimized reproduction (adult mortality, pup survival, halving pup survival to November, twice discounting the number of breeding packs, and counting harvested wolves twice among the dead). The result was erroneous conclusions that were not based on the best available or unbiased science.