### Ethics statement

Permission to capture and handle proboscis monkeys was granted by the Sabah Biodiversity Centre (permit JKM/MBS.1000-2/2 JLD.3 (73)) and was carried out in accordance with the current laws of Malaysia, Sabah Wildlife Department’s Standard Operation Procedures on Animal Capture, Anaesthesia and Welfare, the Weatherall report^{26}, and the guidelines for non-human primates as described by Unwin et al.^{27}. Once an risk assessment was conducted on the target individual and its surrounding sleeping site area, darting was performed by veterinarians experienced in the capture and anesthesia of wildlife using Zoletil 100 (Tiletamine + Zolazepam; 6 ± 10 mg/kg), Anaesthesia and the vital signs were monitored throughout the procedure^{28}, and once the procedure was complete, each animal was given a prophylactic dose of Alamycine LA (20 mg/kg) and Ivermectine (0.2 mg/kg) as a preventative measure to assist in the post-anesthesia recovery.

### Data collection

Between July 2011 and December 2016, we captured 28 free-ranging adult proboscis monkeys in the Lower Kinabatangan Floodplain, Sabah, Borneo, Malaysia (5°18′N to 5°42′N and 117°54′E to 18°33′E). To reduce the impact of capturing on the animal’s social system, we captured all study subjects during the night^{28}. While animals were anaesthetized, we performed in situ measurements for their body parts using a scale (body mass) and caliper (nose size and canine length). In this study, we additionally included data for maxillary canine length to those on body mass and nose size (length × width) for the same 18 males obtained by Koda et al.^{9} as well as data for 10 females’ canine length, body mass, and nose size. Note that canine length refers to the apex to base measurement (the height of the crown), and the maxillary canine length was measured because of its larger size and greater importance in behavioral displays and weaponry^{29}. All study subjects were adults, i.e., 18 harem-holding males and 10 females including one pregnant and two lactating individuals (see Supplementary Table 1 for further detailed study subject information). The veterinarian in the sampling team attempted in situ age/class estimation for the study subjects based on the three categories of wear level of their molars, i.e., low: young adult; medium: adult; high: old adult, although the estimation was rather rough due the difficult in situ condition in the forest at night, e.g., limited tools, light, and time the animal remained anaesthetized.

### Data analysis

We used a linear regression model to establish whether body mass (response variable) was related to other physical properties such as nose and canine size (explanatory variables). To obtain a linear dimension comparable to the canine dimension, body mass and nose size were cube root- and square root-transformed, respectively, to generate response and explanatory variables. The variance inflation factors were 1.00 for both nose size and body mass, indicating that collinearity among independent factors would not affect the results^{30}. We examined a set of models with all possible combinations of the explanatory variables and ranked them according to the corrected version of the Akaike information criterion (AIC) for small sample sizes, called AICc^{31}. Following guidelines published for wildlife research, we selected models with ΔAICc ≤ 2, where ΔAICc = AICc−minimum AICc within the candidate model set^{31}. Analysis of covariance served to compare the slopes and intercepts of regression lines between the sexes. We performed these analyses using R ver. 3.3.2 (ref. ^{32}).

### Mathematical model

The aim of the simulations was to examine our hypothesis regarding the positive/negative correlations between body mass and canine size. Earlier work distinguished the subadult class from fully adult males based on nasal maturation as well as a fully developed body size^{9}. In other words, two developmental stages likely exist among sexually mature males. Therefore, before the acquisition of harem status, subadult males reach a limit in body mass, which cannot increase without a status change. By contrast, in females, the development of body mass and canines is basically similar to that of males before the acquisition of harem status. In this study, we therefore assumed a two-stage model of male development, i.e., a first stage with primary development of body and canine size for both males and females and a second stage that only applies to males and only after the acquisition of harem status.

#### Primary development of body and canine size

Body mass *m* develops under the assumption that the growth rate depends on the body mass at the time, until reaching its developmental limit *K* within a fixed time *T*, i.e.,

$$frac{{{mathrm{d}}m}}{{{mathrm{d}}t}} = amleft( {1 – frac{m}{K}} right)quad forall t in left[ {0,T} right],$$

(1)

where *a* = const is the shape factor determining the sigmoidal curve of development. This is based on a previously proposed basic model for body mass development^{20,33} but with our modification to simplify the constraint of *K*. By contrast, the canines develop in the later stage of body mass development, and thus, we supposed that larger canines would reduce feeding efficiency, resulting in a growth rate reduction for body mass as follows:

$$begin{array}{*{20}{r}} hfill {frac{{{mathrm{d}}m}}{{{mathrm{d}}t}} = left( {1 – cz} right)amleft( {1 – frac{m}{K}} right),}end{array}$$

(2)

where *cz* is the reducing factor determined by the cost factor (c in [0,1)) and the size of canines *z* and *cs* < 1 always holds. For model simplification, the canines start to develop at the time *t*_{0} and linearly develop until *t*_{1} ≤ *T* as follows:

$$z = alpha (t – t_0)quad (t_0 le t le t_1),$$

(3)

where *α* > 0 is the growth rate of canines. Consequently, the differential equation of body mass in (t in [0,T],T , > , t_1) is

$$begin{array}{*{20}{r}} hfill {frac{{{mathrm{d}}m}}{{{mathrm{d}}t}} = left{ {begin{array}{*{20}{l}} {am(1 – frac{m}{K})} hfill & {left( {0 le t le t_0} right)} hfill \ {{ 1 – calpha (t – t_0)} am(1 – frac{m}{K})} hfill & {(t_0 le t le t_1)} hfill \ {(1 – cz_1)am(1 – frac{m}{K})} hfill & {left( {t_1 le t le T} right),} hfill end{array}} right.}end{array}$$

(4)

where (cz_1: = cz_{t = t_1} = calpha (t_1 – t_0)), because proboscis monkeys likely increase their body mass until *t* = *T*, with the constant of the reducing factor, after terminating canine growth (*t* = *t*_{1}). Note that (0 le z le alpha (T – t_0)) holds; therefore,

$$begin{array}{*{20}{r}} hfill {alpha le frac{z}{{T – t_0}}}end{array},$$

(5)

also holds.

In the case of *t* < *t*_{0}, i.e., before the onset of canine eruption, the equation is a simple logistic equation; therefore, the solution is

$$begin{array}{*{20}{r}} hfill {m(t) = frac{K}{{1 + A_0{mathrm{e}}^{ – at}}},}end{array}$$

(6)

where *A*_{0} is the constant determined by the initial values of *m*(0) = *m*_{0} > 0 as follows:

$$begin{array}{*{20}{r}} hfill {A_0 = frac{K}{{m_0}} – 1.}end{array}$$

(7)

Once the canines begin to develop, i.e., in the case of *t*_{0} ≤ *t* ≤ *t*_{1}, the differential equations are solved as follows:

$$begin{array}{*{20}{r}} hfill {frac{{{mathrm{d}}m}}{{{mathrm{d}}t}} = { 1 – calpha (t – t_0)} amfrac{{K – m}}{K}}\ hfill {{int} {frac{K}{{m(K – m)}}} , {mathrm{d}}m = a{int} {{ 1 – calpha (t – t_0)} } {mathrm{d}}t}\ hfill {{int} {{ frac{1}{m} + frac{1}{{(K – m)}}} } {mathrm{d}}m = a{int} {{ 1 – calpha (t – t_0)} } {mathrm{d}}t}\ hfill { – {it{mathrm{{ln}}}}|frac{{K – m}}{m}| = – frac{1}{2}acalpha t^2 + a(1 + calpha t_0)t + C}\ hfill {mleft( t right) = frac{K}{{1 + A_1{mathrm{e}}^{frac{1}{2}acalpha t^2 – aleft( {1 + calpha t_0} right)t}}}.}end{array}$$

(8)

Then, after terminating canine development (*t* = *t*_{1}), we found that the solution is simply

$$begin{array}{*{20}{r}} hfill {m(t) = frac{K}{{1 + A_2{mathrm{e}}^{ – a(1 – cz_1)t}}}.}end{array}$$

(9)

Note that the following equations must be satisfied:

$$begin{array}{l}m(t_0) = frac{K}{{1 + A_0{mathrm{e}}^{ – at_0}}} = frac{K}{{1 + A_1{mathrm{e}}^{frac{1}{2}acalpha t_0^2 – a(1 + calpha t_0)t_0}}}\ m(t_1) = frac{K}{{1 + A_1{mathrm{e}}^{frac{1}{2}acalpha t_1^2 – a(1 + calpha t_0)t_1}}} = frac{K}{{1 + A_2{mathrm{e}}^{ – a(1 – cz_1)t_1}}}.end{array}$$

(10)

Consequently, the basic models for body mass are

$$begin{array}{*{20}{r}} hfill {m(t,t_0,t_1,K,a,c,alpha ,A_0,T) = left{ {begin{array}{*{20}{l}} {frac{K}{{1 + A_0{mathrm{e}}^{ – at}}}} hfill & {(0 le t le t_0)} hfill \ {frac{K}{{1 + A_0{mathrm{e}}^{frac{1}{2}acalpha t_0^2 + frac{1}{2}acalpha t^2 – at(1 + calpha t_0)}}}} hfill & {(t_0 le t le t_1)} hfill \ {frac{K}{{1 + A_0{mathrm{e}}^{frac{1}{2}acalpha t_0^2 – frac{1}{2}acalpha t_1^2 – at + acalpha t_1t – acalpha t_0t}}}} hfill & {(t_1 le t le T),} hfill end{array}} right.}end{array}$$

(11)

and those for canine size are

$$begin{array}{*{20}{r}} hfill {z(t,t_0,t_1,alpha ,T) = left{ {begin{array}{*{20}{l}} 0 hfill & {(0 le t le t_0)} hfill \ {alpha (t – t_0)} hfill & {(t_0 le t le t_1)} hfill \ {alpha (t_1 – t_0)} hfill & {(t_1 le t le T).} hfill end{array}} right.}end{array}$$

(12)

#### Rank-dependent secondary development of body and nose size

Our previous study suggested that males who acquire harem alpha status develop their noses as a badge of status in coordination with body mass^{9}. In this study, we supposed that alpha status males can dominate both copulation opportunities and foraging resources because harem groups with larger males better defend resources from bachelor groups generally consisting of smaller males^{34}. Therefore, such alpha status males would possibly continue their body mass growth after completing their primary developmental process, whereas males who fail to acquire harem status terminate body mass growth. Then, the differential equations of the body mass in the secondary development for harem males are equivalent with Eq. (4), i.e.,

$$frac{{{mathrm{d}}m}}{{{mathrm{d}}t}} = (1 – cz_1)am(1 – frac{m}{K})quad (t ge T),$$

(13)

whereas those for non-harem males are

$$frac{{{mathrm{d}}m}}{{{mathrm{d}}t}} = (1 – cz_1){ a – frac{{t – T}}{{t_2 – T}}a} m(1 – frac{m}{K})quad left( {t ge T} right),$$

(14)

where *t*_{2} is the termination time of the body mass growth. This equation is solved in the same manner as Eq. (8), namely

$$begin{array}{c}m(t,t_0,t_1,K,a,c,alpha ,A_0,T,t_2) = frac{K}{{1 + A_3{mathrm{e}}^{frac{1}{2}beta t^2 – beta t_2t}}}\ A_3 = A_0{mathrm{e}}^{frac{{acalpha (t_0^2 – t_1^2)}}{2} – aT + acalpha T(t_1 – t_0) – beta T(frac{1}{2}T^2 – t_2T),}end{array}$$

(15)

where (beta := frac{{a(1 – calpha t_1 + calpha t_0)}}{{t_2 – T}}). Based on our findings of allometric development of ornaments and body size in free-ranging specimens^{20,33}, we supposed that nose size is primary determined by body mass.

### Numerical simulations

#### Overview of aims

We conducted the numerical simulations with the aim to clarify the mechanism by which the termination time of canine development *t*_{1} determines the final results of body mass and canine size. We simulated the growth pattern based on several deterministic parameters, considering the developmental evidence for proboscis monkeys. Additionally, we tried to apply the model for both males and females within the same developmental frameworks, considering the sex differences of the developmental parameters as follows.

#### Parameter proposals

First, the time parameter was normalized at the male’s primary maturation time *T*, i.e., *T*:= 1. Therefore, our time parameters *t*_{0}, *t*_{1}, and *t*_{2}, are the time relative to the male maturation time *T*. By definition, (0 le t_0 le t_1 le T = 1 le t_2) holds. Additionally, the maximum canine size *z* was also normalized to 1 for the case in which the monkey maximally develops its canines during the periods between *t*_{0} and *T* = 1. Following Eq. (5), (alpha = frac{1}{{1 – t_0}}).

Next, we supposed that the primary maturation age of males (*t* = *T* = 1) at which males reach sexual maturity at approximately 8 years old^{35}, although male nose enlargement does not typically start until that time^{36}. In *Cercopithecus*^{37} and proboscis monkeys^{38}, canine development starts at approximately 4 years old and continues until 8 years old. The onset time of canine development was therefore set at 4 years old in our model, or half the subadult maturation time, i.e., (t_0 = frac{1}{2}). By contrast, females mature earlier than males. We supposed that the female maturation age was 6 years^{35}; i.e., we used ({{T}} = frac{6}{8}) in the model for female development. The time of onset of canine growth is likely similar in both sexes, but females likely terminate canine growth earlier, corresponding to their earlier physical maturation. Therefore, we supposed that female canines develop from approximately 4 years old until 6 years old, representing a shorter than observed in males (i.e., (t_0 = frac{1}{2}) for females).

Records for body mass were used for the parameters in the simulations. Proboscis monkey neonatal body mass was reported as 0.45 kg^{38}, and the maximum body mass among our specimens was 25.5 kg for males and 14.5 kg for females. Therefore, we set *m*(0) = 0.45, *K*_{male} = 25.5, and *K*_{female} = 14.5. Additionally, we used the body mass record at approximately 4 years old, or the onset time of canine eruption, which was approximately 6.5 kg (*N* = 3, records in Japan Monkey Centre). Therefore, we used (m(frac{1}{2}) = 6.5) in the simulations for both males and females. Finally, we assumed that body mass development would terminate immediately after the maturation time *T* was reached; therefore, we used *t*_{2} = *T* + 0.1 = 1.1 for males and *t*_{2} = *T* + 0.1 = 0.85 for females.

Based on the aforementioned parameters, we simulated body mass at the time at which monkeys fully develop. For this purpose, we attempted to simulate/evaluate *m*(2), or body mass at approximately 16 years old, using the various combinations of cost parameters and canine termination times, i.e., (c in (0,1)) and (t_1 in (0.5,T)).

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.