Predicting performance of naïve migratory animals, from many wrongs to self-correction
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
