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An endangered lemur species that lives in Madagascar’s rainforest could vanish within 25 years if deforestation on the island isn’t reduced1.

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doi: https://doi.org/10.1038/d41586-022-03116-6

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Conservation biology More

Calculation of flight-step headings and movementTerms defining flight-step movement, precision and geophysical orientation cues are listed in Table 1. Since seasonal migration nearly ubiquitously proceeds from higher to lower latitudes, it is convenient to define headings clockwise from geographic South (counter-clockwise from geographic North for migration commencing in the Southern Hemisphere). Assuming a spherical Earth, a sequence of N migratory flight-steps with corresponding headings, αi, i = 0,…, N−1, the latitudes, ∅i+1, and longitudes, λi+1, on completion of each flight-step can be calculated using the Haversine Equation76, which we approximated by stepwise planar movement using Eqs. (1) and (2). For improved computational accuracy and to accommodate within flight-step effects, we updated simulated headings and corresponding locations hourly. A migrant’s flight-step distance ({R}_{{{mathrm {step}}}}=3.6{V}_{{mathrm {a}}}{cdot n}_{{mathrm {H}}}/{R}_{{{mathrm {Earth}}}}) (in radians), depends on its flight speed, Va (m/s) relative to the mean Earth radius REarth (km), and flight-step hours, nH. With a geomagnetic in-flight compass, expected hourly geographic headings are modulated by changes in magnetic declination, i.e., the clockwise difference between geographic and geomagnetic South10,32.Formulation of compass coursesFor simplicity, we consider the case of a single inherited or imprinted heading. This can be extended to include sequences of preferred headings. Expected geographic loxodrome headings remain unchanged en route, i.e.,$${bar{{{{{{rm{alpha }}}}}}}}_{i}={bar{{{{{{rm{alpha }}}}}}}}_{0}$$
(5)
Relative to geographic axes, expected geomagnetic loxodrome headings remain unchanged relative to proximate geomagnetic South, i.e., are offset by geomagnetic declination on departure (updated hourly in simulations)$${bar{{{{{{rm{alpha }}}}}}}}_{i}={bar{{{{{{rm{alpha }}}}}}}}_{0}+{delta }_{{mathrm {m}},i}$$
(6)
As described and illustrated in detail by Kiepenheuer13, the magnetoclinic compass was hypothesized to explain the prevalence of “curved” migratory bird routes, i.e., for which local geographic headings shift gradually but substantially en route. A migrant with a magnetoclinic compass adjusts its heading at each flight-step to maintain a constant transverse component, γ′, of the experienced inclination angle, γ, so that error-free headings are (see Fig. S5 in ref. 34)$${{bar{{{{{{rm{alpha }}}}}}}}_{i}={{sin }}}^{-1}left(frac{{{tan }}{gamma }_{i}}{{{tan }}{gamma }^{{prime} }}right){={{sin }}}^{-1}left(frac{{{tan }}{gamma }_{i}{{sin }}{bar{{{{{{rm{alpha }}}}}}}}_{0}}{{{tan }}{gamma }_{0}}right).$$
(7)
In a geomagnetic dipole field, the horizontal (Bh) and vertical (Bz) field, and therefore also inclination, each depends solely on geomagnetic latitude, ∅m:(gamma ={{{tan }}}^{-1}left({B}_{{mathrm {z}}}/{B}_{{mathrm {h}}}right)={{{tan }}}^{-1}left(2{{sin }}{phi }_{{mathrm {m}}}/{{cos }}{phi }_{{mathrm {m}}}right)={{{tan }}}^{-1}left(2{{tan }}{phi }_{{mathrm {m}}}right).) The projected transverse component, therefore, becomes$${gamma }^{{prime} }={{{tan }}}^{-1}left(frac{{{tan }}{gamma }_{0}}{{{sin }}{bar{{{alpha }}}}_{0}}right)={{{tan }}}^{-1}left(frac{2{{tan }}{{{phi }}}_{{mathrm {m}},0}}{{{sin }}{bar{{{{{{rm{alpha }}}}}}}}_{0}}right),$$which can be substituted into Eq. (7) to produce a closed formula for magnetoclinic headings in a dipole as a function of geomagnetic latitude$${bar{{{{{{rm{alpha }}}}}}}}_{i}left({{{phi }}}_{{mathrm {m}},i}right)={{{sin }}}^{-1}left(frac{{{sin }}{bar{{{{{{rm{alpha }}}}}}}}_{0}}{{{tan }}{{{phi }}}_{{mathrm {m}},0}}cdot {{tan }}{{{phi }}}_{{mathrm {m}},i}right),$$
(8)
with the expected initial heading, ({bar{{{{{{rm{alpha }}}}}}}}_{0}), and initial geomagnetic latitude, ∅m,0, being constants. Equations (7) and (8) have no solution when inclination increases en route, which could occur following substantial orientation error or in strongly non-dipolar fields. We followed previous studies in allowing magnetoclinic migrants to head towards magnetic East or West until inclination decreased sufficiently33,34,46, but also included orientation error based on the modelled compass precision.To assess sun-compass sensitivity algebraically, and also to improve computational efficiency, we used a closed-form equation for sunset azimuth, θs (derived in Supplementary Note 3 and see ref. 23),$${theta }_{{mathrm {s}}}={{{cos }}}^{-1}left(frac{-{{sin }}{delta }_{{mathrm {s}}}}{{{cos }}{{phi }}}right),$$
(9)
where δs is the solar declination, which varies between −23.4° and 23.4° with season and latitude23. Sunset azimuth is the positive and sunrise azimuth is the negative solution to Eq. (9) (relative to geographic N–S).Fixed sun-compass headings represent a uniform (clockwise) offset, ({bar{{{{{{rm{alpha }}}}}}}}_{{mathrm {s}}}) to sunrise or sunset azimuth, θs,i (calculated using Eq. (9))$${{bar{{{{{{rm{alpha }}}}}}}}_{i}={bar{{{{{{rm{alpha }}}}}}}}_{{mathrm {s}}}+theta }_{{mathrm {s}},i}$$
(10)
where the preferred heading on commencement of migration, ({bar{{{{{{rm{alpha }}}}}}}}_{{mathrm {s}}}={bar{{{{{{rm{alpha }}}}}}}}_{0}-{theta }_{{mathrm {s}},0}), is presumed to be imprinted using an inherited geographic or geomagnetic heading2,10,30.With a TCSC, preferred headings relative to sun azimuth are adjusted according to the time of day. In the context of sun-compass use during migration, Alerstam and Pettersson22 related the hourly “clock-shift” induced by crossing bands of longitude (∆h = 12 ∆λ/π), to a migrant’s time-compensated adjustment given the rate of change (i.e., angular speed) of sun azimuth close to sunset$$frac{partial {theta }_{{mathrm {s}}}}{partial h}cong frac{2pi {{sin }}{{phi }}}{24},$$
(11)
resulting in a “time-compensated” offset in heading on departure ((varDelta bar{{{{{{rm{alpha }}}}}}}cong varDelta {{{{{rm{lambda }}}}}},sin phi), which Eq.(4)). Equation (4) results in near-great-circle trajectories for small ranges in latitude, ∅, until inner clocks are reset. The feasibility of TCSC courses over longer distances (latitude ranges) relies on two critical but little-explored assumptions: (1) time-compensated orientation adjustments are presumed to follow the angular speed of sun azimuth (Eq. (11)) retained from the most recent clock-reset site, and (2) to negotiate unpredictable migratory schedules, migrants are presumed to retain their preferred geographic heading on arrival at extended stopovers22.Regarding the first assumption, time-compensated adjustments could also be influenced by proximate speeds of sun azimuth even when inner clocks are not fully reset. We, therefore, use distinct indices to keep track of “reference” flight-steps for clock-resets (cref,i) and time-compensated adjustments (sref,i). TCSC flight-step headings can then be written as$${bar{{{{{{rm{alpha }}}}}}}}_{i}=left{begin{array}{cc}{bar{{{{{{rm{alpha }}}}}}}}_{{c}_{{{mathrm {ref}}},i}}+left({theta }_{{mathrm {s}},i}-{theta }_{{mathrm {s}},{c}_{{{mathrm {ref}}},i}}right)+left({{{{{{rm{lambda }}}}}}}_{i}-{{{{{{rm{lambda }}}}}}}_{{c}_{{{mathrm {ref}}},i}}right){{sin }}{phi }_{{s}_{{{mathrm {ref}}},i}}, & {i,ne, c}_{{{mathrm {ref}}},i} ; (12a)\ {{{{{{rm{alpha }}}}}}}_{i-1}, & {i=c}_{{{mathrm {ref}}},i} ; (12b)end{array}right.,$$where θs,i represents the sunset azimuth on departures, cref,i specifies the most recent clock-reset site (during which geographic headings are also retained, i.e., ({bar{{{{{{rm{alpha }}}}}}}}_{i}={{{{{{rm{alpha }}}}}}}_{i-1})), and sref,i specifies the site defining the migrant’s temporal (hourly) rate of “time-compensated” adjustments (Eq. (11)). For TCSC courses as conceived by Alerstam and Pettersson22, reference rates of adjustment to sun azimuth are reset in tandem during stopovers, i.e., ({s}_{{{mathrm {ref}}},i}={c}_{{{mathrm {ref}}},i}), but we also considered a proximately gauged TCSC, where migrants gauge their adjustments to currently experienced speed of sun azimuth, i.e., ({s}_{{{mathrm {ref}}},i}=i).Regarding the second assumption, retaining geographic headings on arrival at stopovers is not consistent with ignoring geographic headings between consecutive nightly flight-steps, and may be difficult to achieve while landing. We, therefore, examined a more parsimonious alternative (Fig. 7d, Supplementary Fig. 3) where migrants retain their (usual) TCSC heading from the first night of stopovers, i.e., as if they would have departed on the first night. This alternative also simplifies Eq. (12) to$${bar{{{{{{rm{alpha }}}}}}}}_{i}={bar{{{{{{rm{alpha }}}}}}}}_{{c}_{{{mathrm {ref}}},i}}+left({theta }_{{mathrm {s}},({t}_{i-1}+1)}-{theta }_{{mathrm {s}},{t}_{i-1}}right)+left({{{{{{rm{lambda }}}}}}}_{i}-{{{{{{rm{lambda }}}}}}}_{{c}_{{{mathrm {ref}}},i}}right){{sin }}{{{phi }}}_{{s}_{{{mathrm {ref}}},i}}$$
(12c)
where the index ti−1 here represents the departure date from the previous flight.Sensitivity of compass-course headingsSensitivity was assessed by the marginal change in expected heading from previous (imprecise) headings, (partial {bar{alpha }}_{i}/partial {alpha }_{i-1}). When this is positive, small errors in headings will perpetuate, and therefore expected errors in migratory trajectories will grow iteratively. Conversely, negative sensitivity implies self-correction between successive flight-steps. Geographic and geomagnetic loxodromes are per definition constant relative to their respective axes so have “zero” sensitivity, as long as cue-detection errors are stochastically independent.For magnetoclinic compass courses in a dipole field, sensitivity can be calculated by differentiating Eq. (8) with respect to previous headings:$$frac{{mathrm {d}}{bar{{{{{{rm{alpha }}}}}}}}_{i}}{{mathrm {d}}{{{{{{rm{alpha }}}}}}}_{i-1}}=frac{{sin bar{{{{{{rm{alpha }}}}}}}}_{0}}{tan {phi }_{{mathrm {m}},0}}cdot frac{1}{cos {bar{alpha }}_{i}{cos }^{2}{phi }_{{mathrm {m}},i}}frac{partial {phi }_{{mathrm {m}},i}}{partial {alpha }_{i-1}}=frac{{R}_{{mathrm {step}}},sin {alpha }_{i-1}{sin bar{{{{{{rm{alpha }}}}}}}}_{0}}{cos {bar{alpha }}_{i}{cos }^{2}{phi }_{{mathrm {m}},i},tan {phi }_{{mathrm {m}},0}}$$
(13)
All three terms in the denominator indicate, as illustrated in Fig. 3b, that magnetoclinic courses become unstably sensitive at both high and low latitudes, and any heading with a significantly East–West component.Sensitivity of fixed sun compass headings is non-zero due to sun azimuth dependence on location (Eq. (9)):$$frac{{mathrm {d}}{bar{{{{{{rm{alpha }}}}}}}}_{i}}{{mathrm {d}}{{{{{{rm{alpha }}}}}}}_{i-1}} = , frac{sin {delta }_{{mathrm {s}},i}}{sin {theta }_{{mathrm {s}},i}}cdot frac{sin {phi }_{i}}{{cos }^{2}{phi }_{i}}frac{partial {phi }_{i}}{partial {alpha }_{i-1}}=frac{sin {delta }_{{mathrm {s}},i}}{sin {theta }_{{mathrm {s}},i}}cdot frac{{R}_{{mathrm {step}}},sin {phi }_{i},sin {alpha }_{i-1}}{{cos }^{2}{phi }_{i}}\ = , {R}_{{mathrm {step}}}cdot ,sin {alpha }_{i-1}frac{tan {phi }_{i}}{tan {theta }_{{mathrm {s}},i}}$$
(14)
The sine factor on the right-hand side in Eq. (14) causes the sign of (partial {bar{alpha }}_{i}/partial {alpha }_{i-1}) to be opposite for East to West or West to East headings, and tan θs also change sign at the fall equinox (due to solar declination changing sign). The azimuth term in the denominator indicates heightened sensitivity closer to the summer or winter equinox and at high latitudes, and, conversely, heightened robustness to errors closer to the spring or autumnal equinox (since ({{tan }}{theta }_{{mathrm {s}},0}to pm infty)). This seasonal and directional asymmetry is illustrated in Fig. 3c, e.TCSC courses (Eq. (12)) involve up to three sensitivity terms, due to dependencies on sun azimuth, longitude and latitude:$$ frac{{mathrm {d}}{bar{{{{{{rm{alpha }}}}}}}}_{i}}{{mathrm {d}}{{{{{{rm{alpha }}}}}}}_{i-1}} = , {R}_{{{mathrm {step}}}}cdot {{sin }}{alpha }_{i-1}frac{{{tan }}{phi }_{i}}{{{tan }}{theta }_{{mathrm {s}},i}}+frac{{mathrm {d}}{lambda }_{i}}{{mathrm {d}}{{{{{{rm{alpha }}}}}}}_{i-1}}{{sin }}{{{phi }}}_{{c}_{{{mathrm {ref}}}},i}+left({{{{{{rm{lambda }}}}}}}_{i}-{{{{{{rm{lambda }}}}}}}_{{c}_{{{mathrm {ref}}}},i}right)frac{{mathrm {d}}{{sin }}{phi }_{{s}_{{{mathrm {ref}}}},i}}{{mathrm {d}}{{{{{{rm{alpha }}}}}}}_{i-1}}\ =, left{begin{array}{cc}{R}_{{{mathrm {step}}}}cdot left[{{sin }}{alpha }_{i-1}frac{{{tan }}{phi }_{i}}{{{tan }}{theta }_{{mathrm {s}},i}}-frac{{{cos }}{{{{{{rm{alpha }}}}}}}_{i-1}{{sin }}{phi }_{{s}_{{{mathrm {ref}}}},i}}{{{cos }}{phi }_{i-1}}right],hfill & {{{{{rm{classic}}}}}} ; (15{{{{{rm{a}}}}}})\ {R}_{{{mathrm {step}}}}left[{{sin }}{alpha }_{i-1}frac{{{tan }}{phi }_{i}}{{{tan }}{theta }_{{mathrm {s}},i}}-frac{{{cos }}{{{{{{rm{alpha }}}}}}}_{i-1}{{sin }}{phi }_{{s}_{{{mathrm {ref}}}},i}}{{{cos }}{phi }_{i-1}}+left({{{{{{rm{lambda }}}}}}}_{i}-{{{{{{rm{lambda }}}}}}}_{{c}_{{{mathrm {ref}}}},i}right){{sin }}{alpha }_{i-1}{{cos }}{phi }_{i}right], & {{{{{rm{proximate}}}}}} ; left(15{{{{{rm{b}}}}}}right).end{array}right.$$The first square-bracketed terms in Eqs. (15a, b) are identical to the fixed sun compass (Eq. (14)), reflecting seasonal and latitudinal dependence in sun-azimuth. For headings with a Southward component (α0  1) and nonexistent for North–South headings (G = 1, reflecting no longitude bands being crossed). We expected this factor to affect compass courses differentially according to their error-accumulating or self-correcting nature.We further modified the effective goal-area breadth to account for a (geographically) circular goal area on the sphere, i.e., effectively modulating the longitudinal component of the goal-area breadth at the arrival latitude, ∅A:$${beta }_{{mathrm {A}}}=beta sqrt{{{{{sin }}}^{2}bar{alpha }+left({{cos }}bar{alpha }/{{cos }}{{{phi }}}_{{mathrm {A}}}right)}^{2}}.$$
(19)
To account for differential sensitivity among compass-courses, we generalized the normal many-wrongs relation between performance and number of steps, (1/{hat{N}}^{eta }), from η = 0.5 (Eqs. (3) and (16)) to$$eta left({sigma }_{{step}}|s,bright)=left(0.5+bright){e}^{-s{{sigma }_{{step}}}^{2}},$$
(20)
where b  0 self-correction, and s represents a modulating exponential damping factor, consistent with the limiting circular-uniform case (as κ → 0, i.e., ({sigma }_{{{mathrm {step}}}}to infty)), where no (timely) convergence of heading is expected with an increasing number of steps.In assessing performance, we also accounted for seasonal migration constraints via a population-specific maximum number of steps, Nmax (Table 2; this became significant for the longest-distance simulations with large expected errors, i.e., small ({{{{{{rm{kappa }}}}}}}_{{{mathrm {step}}}}=1/{sigma }_{{{mathrm {step}}}}^{2})). The probability of having arrived at the goal latitude can be estimated using the Central Limit Theorem:$${p}_{{{phi }},{N}_{{max }}}cong frac{1}{2}left[1-{erf}left(left(frac{{N}_{0}}{{N}_{{max }}}-frac{{I}_{1}left({{{{{{rm{kappa }}}}}}}_{{{mathrm {step}}}}right)}{{I}_{0}left({{{{{{rm{kappa }}}}}}}_{{{mathrm {step}}}}right)}right)cdot frac{{{cos }}bar{alpha }}{{sigma }_{{mathrm {C}}}sqrt{2}}right)right],$$
(21)
where Ij is the modified Bessel function of the first kind and order j53, and σC (the standard deviation in the latitudinal component of flight-step distance) can be calculated using Bessel functions together with known properties of sums of cosines53,77 (Supplementary Note 2).Regression-estimated performanceWe fit the parameters in the spherical-geometry factor (Eq. (18)) and many-wrongs effect (Eq. (20)) according to expected performance, estimated as the product of sufficiently timely migration (Eq. (21)) and sufficiently precise migration, now generalized from Eq. (16), i.e.$${p}_{beta ,hat{N}}cong {erf}left(frac{{beta }_{{mathrm {A}}}}{{G}^{{g}}sqrt{2left({{sigma }_{{{mathrm {ind}}}}}^{2}+{sigma }_{{{mathrm {step}}}}/{hat{N}}^{n}right)}}right),$$
(22)
This resulted in up to four fitted parameters for each compass course

i.

an exponent, g, to the spherical-geometry factor (Eq. (19)), i.e., Gg, reflecting how growth or self-correction in errors between steps further augments or reduces this factor,

ii.

a baseline offset, b0, to the “normal” exponent η = 0.5, which mediates the relation between the number of steps and performance (Eq. (20)),

iii.

an exponent s reflecting how decreasing precision among flight-steps dampens the many-wrongs convergence (Eq. (20)),

iv.

for TCSC courses, a modulation, ρ, to the offset, b0, quantifying the extent to which self-correction increases with increased flight-step distance Rstep, i.e., ({{b={b}_{0}R}_{{{mathrm {step}}}}^{{prime} }}^{rho }) in Eq. (20), where ({R}_{{{mathrm {step}}}}^{{prime} })is the flight-step distance scaled by its median value among species.

Parameters were fit using MATLAB routine fitnlm based on compass course performance among species and seven error scenarios (5°, 10°, 20°, 30°, 40°, 50°, and 60° directional precision among flight-steps), for all combinations (including or excluding the four parameters). The most parsimonious combination of parameters was selected using MATLAB routine aicbic, based on the AICc, the Akaike information criterion corrected for small sample size57. Null values for the spherical-geometry parameter were set to g = 1, and for the parameters governing convergence of route-mean headings b0 = 0, s = 0, and, for TCSC courses, ρ = 0 (for loxodrome courses, ρ = 0 by default, i.e., was not fitted).Statistics and reproducibilityOur simulation results, regression fitting and AICc-model selection are reproducible using the MATLAB scripts (see the section “Code availability”).Reporting summaryFurther information on research design is available in the Nature Research Reporting Summary linked to this article. More

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Study areaWe conducted our study in two suburbs in Wellington, New Zealand (Fig. 1). The 4.7-hectare site in the suburb of Kelburn (-41.285°S, 174.770°E) was situated on the grounds of student accommodation for Victoria University of Wellington. The site comprised bungalow houses, two accommodation halls, and access roads and paths. About half of the vegetation at the Kelburn site was a mix of tended grass lawns and gardens containing a variety of native New Zealand plant species, e.g., flax (Phormium spp.), longwood tussock (Carex comans), and cabbage tree (Cordyline australis). The other half was a mix of dense ground cover dominated by invasive weed species and native and exotic trees and shrubs, e.g., pōhutukawa (Metrosideros excelsa), common oak (Quercus robur), kawakawa (Piper excelsum), and taupata (Coprosma repens). The second suburb was Roseneath (−41.292°S, 174.801°E) on a small peninsula on the north-eastern side of Mount Victoria. The site was 8.5 hectares comprising 76 residential properties, public thoroughfares, and footpaths. We conducted fieldwork in the gardens of 25 of these properties. The vegetation varied considerably between gardens, comprising native and introduced garden plants and invasive weeds, especially blackberry (Rubus fruticosus).Figure 1(A) The study was conducted in the suburbs of Kelburn (left yellow dot) and Roseneath (right yellow dot) in the city of Wellington, New Zealand. The black polygon represents the 1475 ha area that will be targeted for ship rat (Rattus rattus) eradication in Wellington city, New Zealand. In each suburb, we radio-collared ship rats and deployed three types of devices (bait stations, chew cards, and WaxTags) to estimate home range and detection parameters. (B) In Kelburn, we radio-collared 14 rats and deployed eight devices. (C) In Roseneath, we radio-collared 16 rats and deployed 30 devices. The yellow circles indicate home range centers of individual rats, the red triangles indicate the location of bait stations and detection devices, and the small black dots indicate the telemetry locations of rats.Full size imageRat capture, radio-collaring, and field methodologyWe set 100 live-capture cage traps (custom-made, spring-loaded traps) in Kelburn from 12 July to 15 August 2020, and another 100 in Roseneath from 20 August to 20 October 2020. We baited cage traps with apple coated in chocolate spread and checked them at least once every 24 h. We set cage traps in areas with complex vegetative groundcover and understorey to maximize capture rates of ship rats (see35), and to provide shelter from inclement weather. We provided additional shelter by inserting bedding inside a tin can placed in the cage traps, along with a plastic cover over the traps to limit exposure to wind and rain. Cage traps were active for 5 days per week on average. We released all non-target species (house mice Mus musculus, European hedgehogs Erinaceus europaeus, and Eurasian blackbirds Turdus merula).We transferred any trap containing a captured rat into a sealed plastic container. Depending on the estimated size of the captured rat, we placed between one and three cotton balls soaked in isoflurane (99.9%, Attane, Piramal Critical Care Inc., Bethlehem, Pennsylvania, USA) inside the plastic container. A rat was anesthetized when it lost balance and was unable to regain balance when we gently rotated the container. We then removed the rat from the cage trap and placed it next to a heat pad with its head close to the cotton balls soaked in isoflurane to maintain anaesthesia while handling them. We fitted all rats weighing  > 110 g with a V1C 118B VHF radio-collar (Lotek, Havelock North, New Zealand). We marked each collared rat with a unique pelage code using a permanent blonde hair dye60. We also recorded biometrics, including sex, weight, and length. When processing was finished, we placed the rat into another container to recover. This container had a heating pad for warmth and an apple for food to avoid a drop in body temperature and hypoglycemia, which are common problems with anaesthesia62. When the rat appeared mobile, energetic, and behaving normally, we released it at the point of capture.We monitored radio-collared rats using a Yagi antenna (Lotek, Havelock North, New Zealand) and a Telonics R-1000 receiver (Telonics Inc., Mesa, Arizona, USA). We conducted radio-telemetry work during August–November 2020, with fixes taken during the day and night. We recorded a total of three fixes per rat per night, taken at two-hour intervals between the hours of sunset (2200 h) and sunrise (0500 h). We mostly attempted one day-time fix (1200 h); however, if a tracked rat was active (determined by a VHF signal that was moving or changing amplitude), we attempted a second fix in the afternoon. To minimize location error, we used the close approach radio-tracking method described by63. Once a successful fix was made, we used a handheld GPS unit to record the location, date, and time. Telemetry fixes were collected for each radio-collared rat for 18–97 days.After approximately one week of radiotracking an animal, we obtained an initial crude estimate of the center of each rat’s home range as the mean of all eastings and northings (based on a minimum of 15 telemetry points per rat). A bait station baited with non-toxic pellets (Protecta Sidekick bait stations, Bell Laboratories Inc., Windsor, Wisconsin, USA), a WaxTag with a peanut butter odor incorporated into the wax (PCR WaxTags, Traps.co.nz, Rolleston, New Zealand), and a chew card (a corflute card baited with peanut butter) were deployed at varying distances (max. 50 m) and cardinal directions from the estimated home range center of each individual rat. This layout maximized the likelihood of encounters with devices, compared with a regular grid-type deployment where some of the devices could fall outside a collared rat’s home range and thus never be encountered. Note that the crude estimate of the location of the home range center for each rat was only used to guide device placement, i.e., it was not used in any statistical analyses, or to describe rat home range sizes. Further, to avoid a choice-type experiment (i.e., all three devices set immediately next to each other), we randomly assigned a distance and cardinal direction to each device type within each rat’s home range but ensured all devices were deployed  > 15 m apart. The three device types were chosen because they are used by Predator Free Wellington to conduct their eradication operations.Every deployed device had a trail camera (Browning Strike Force HD Pro Micro Series, Morgan, Utah, USA) taking video of rats encountering and interacting with the device. We set cameras to take 20 s of video footage when triggered, followed by a 1 s re-trigger interval. We fixed trail cameras to trees at a height of 50 cm above ground level and placed the devices 1.5 m in front of the camera (after64). This strategy allowed accurate identification of pelage codes on marked rats. We cleared vegetation in front of and immediately behind the trail cameras to avoid accidental triggers. We used pegs to mark a 30-cm-radius circle around each device and considered a rat–device encounter when a rat entered that circle. We serviced trail camera–device pairs at least once every three days. This included adding more non-lethal bait to bait stations and peanut butter to monitoring devices, installing new WaxTags or chew cards if they had been destroyed, and replacing batteries and SD cards in trail cameras. We set up 54 trail camera–device pairs. However, due to trail camera malfunctions, we were able to retrieve footage from only 38 cameras, 8 in Kelburn and 30 in Roseneath. Trail camera–device pairs were active for 20–70 days, but we retained data from only the first 20 days for the analyses.Video processingAll video footage was viewed and interpreted by the same individual (HRM) for consistency. We extracted the following information: date and time of rat sightings, rat ID (according to the pelage code, or designated as ‘R’ for unmarked rats), the duration of the visit to a device, whether or not an encounter occurred (as defined above), and whether or not an interaction occurred. We defined an interaction as a rat either gnawing on a chew card or WaxTag or entering a bait station.Data analysisWe combined all ship rat telemetry data with the device encounter and interaction data, and developed a hierarchical Bayesian model to infer factors influencing the key parameters σ, ε0, and θ. The analytical approach builds on that described in65. For the purpose of estimating ε0 and θ, multiple encounters or interactions by the same individual with the same device on the same night were counted as a single encounter or interaction.The VHF telemetry data Zij were composed of xij (eastings) and yij (northings) locations for each individual rat i at site j (either Kelburn or Roseneath). To simplify the notation, we drop the j subscript from all subsequent equations. We modelled the probability of observing Zi as a symmetric bivariate normal variable$$P({Z}_{i})= prod_{i=1}^{{L}_{i}}Normal(Delta {x}_{i}|0,{sigma }_{i}^{2})Normal(Delta {y}_{i}|0,{sigma }_{i}^{2})$$
(1)
where σi is the standard deviation of a normal distribution with zero mean, Li is the number of location fixes for individual i, and Δxi and Δyi are the straight-line distances from the home range center of individual i to xi and yi, respectively.Home range centers can be estimated using various methods, all of which have underlying assumptions (e.g.,66,67). We calculated the home range center for each individual as the mean of all xi and yi, i.e., the centroid of all locations that we recorded for each individual ( > 30 VHF fixes in all instances). Under this formulation, the home range center is assumed to be perfectly observed, an assumption that is supported by the sample size of telemetry locations that we obtained for each individual (see Supplementary Table 266).We modelled σi as a log-normal variable with mean ln(μi), which was a function of the sex of the individual:$$lnleft({sigma }_{i}right)sim Normal(mathit{ln}left({mu }_{i}right), V)$$
(2)
$$lnleft({mu }_{i}right)= {beta }_{0}+ {beta }_{1}{sex}_{i}$$
(3)
where V is the variance of ln(σi), and ln(μi) is a linear function of a categorical variable indicating whether rat i is a male (0) or a female (1). The priors on the β coefficients and V were Normal(0, 10) and InverseGamma(0.01, 0.01), respectively.The encounter data (Eimt) across all devices m and nights t was modelled as a Bernoulli process:$${E}_{imt}sim Bernoulli({gamma }_{imt})$$
(4)
$$logitleft({gamma }_{imt}right)sim MultivariateNormal(logitleft({P}_{imt}right), varSigma )$$
(5)
where γimt is a latent variable representing the degree to which the nightly probability of rat i encountering a given device is not independent of the encounter outcomes of nearby devices, i.e., we assumed there is spatial autocorrelation in the nightly probability of encountering a device. To account for the spatial autocorrelation not explained by the covariates explicitly modelled (i.e., σ and device type, see below), we included an exponential spatial covariance error structure (Σ) as follows:$$varSigma = {nu }^{2}{e}^{-varphi r}$$
(6)
where ν2 is the variance, φ is a correlation distance parameter, and r is the distance (in m) between pairs of devices68,69. Further, because not all devices were available on all nights, Σ was calculated iteratively for each night considering only those devices that were available. We used moderately informative log-normal priors for the covariance parameters to obtain proper posteriors69: ν2 ~ logN(3,1) and φ ~ logN(1,1).The nightly probability of encounter of device m by individual i on night t (Pimt) was calculated using a half-normal detection function70:$${P}_{imt}= {{left({varepsilon }_{0, im}{e}^{left(-frac{{d}_{im}^{2}}{2{sigma }_{i}^{2}}right)}right)}^{{tau E}_{it}^{*}}}times {{left({varepsilon }_{0,im}{e}^{left(-frac{{d}_{im}^{2}}{2{sigma }_{i}^{2}}right)}right)}^{1-{E}_{it}^{*}}}$$
(7)
where ε0,im is the maximum nightly probability of encounter for device m, or the probability if device m was placed at the center of the home range of rat i. The variable σi is the standard deviation from Eq. (1) (i.e., σi is estimated jointly from the telemetry and encounter data) and dim is the distance (in m) between the home range center of rat i and device m; only devices within a distance of 3.72σi from the home range center were considered in the calculation in Eq. (7)70. Finally, τ is a strictly positive parameter (i.e., τ  > 0), measuring the degree of device-shyness, which is multiplied by an indicator variable (left({E}_{it}^{*}right)) which takes a value of 0 when individual i has not encountered a device (of any type) on nights prior to night t, or a value of 1 if it had previously encountered one, regardless of the type of device it encountered. If τ  1 then rats are ‘device-shy’ and thus more likely to avoid devices on nights following an initial encounter. ({E}_{it}^{*}) was reset to 0 after 20 days of no encounters with a device. Following65 we set the prior on τ as Gamma(0.933, 8.33) (shape and rate parameters, respectively).Values of ε0,im were predicted as a function of σi, device type, and individual effects using the following equation:$$logitleft({varepsilon }_{0, im}right)={alpha }_{0}+ {alpha }_{1}mathrm{ln}left({sigma }_{i}right)+ {alpha }_{2}{chewcard}_{m}+{alpha }_{3}{waxtag}_{m}+{delta }_{i}$$
(8)
where α2 and α3 quantify the increase or decrease in the maximal probability of encountering a chew card or a WaxTag relative to a bait station (which is the reference category). The δi parameters account for individual differences in ε0. Finally, we allowed ε0 to be a function of ln(σi) because we assumed encounter probability at home range center will decrease with increasing home range size (as suggested by71 and shown by65). The priors on the α coefficients and δ were Normal(0, 10) and Normal(0, 1), respectively.The interaction data (Iimn) across all devices m and nights n when encounters occurred was modelled as a Bernoulli process with probability θ, which was a function of device type and individual effects:$${mathrm{I}}_{imn}sim Bernoullileft({theta }_{imn}right)$$
(9)
$$logitleft({theta }_{imn}right)={lambda }_{0}+ {lambda }_{1}{chewcard}_{m}+{lambda }_{2}{waxtag}_{m}+{lambda }_{3}{I}_{in}^{*}+{rho }_{i}$$
(10)
where θimn is the probability of rat i interacting with device m given that it has encountered it on night n, and λ1 and λ2 quantify the increase or decrease in the conditional probability of interaction for a chew card or a WaxTag relative to a bait station. The λ3 parameter is analogous to τ in Eq. (7) but for the process of interaction given encounter with a device. However, by incorporating λ3 directly into a linear equation, this parameter can take negative values and thus should be interpreted differently to τ: if λ3  0 indicates that individuals become ‘device-happy’ after an initial interaction. This parameter is multiplied by an indicator variable ({(I}_{in}^{*})) which takes a value of 0 when individual i has not interacted with a device (of any type) on nights prior to night n, or a value of 1 when it has interacted with one previously, regardless of the type of device it interacted with. If a rat had not interacted with a device for 20 days, ({I}_{in}^{*}) was reset to 0. Finally, the ρi parameters account for individual differences in θ. The priors on the λ coefficients and ρ were Normal(0, 10) and Normal(0, 1), respectively. Although we explicitly modelled spatial autocorrelation in the probability of encountering a device, we did not do so for the probability of interaction given an encounter. In this instance we assumed that whether an animal chose to interact with an encountered device would depend on its previous experience (as quantified by λ3) rather than the spatial location of nearby devices.We used Markov Chain Monte Carlo (MCMC) simulation to estimate model parameters using Python programming language. The variance parameter V was sampled from the full conditional posteriors, but all other parameters were estimated using the Metropolis algorithm69. Posterior summaries were taken from four chains containing 3000 samples each (with a burn-in of 2000 and a thinning rate of 30). Convergence on posteriors was assessed by visual inspection and a scale reduction factor  More

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