Few growth pattern descriptions in Mallard ducks (Anas platyrhynchos) are available in the academic literature. Despite this, the intrinsic growth rate, defined as the relationship between size and age, is an important factor in life history theory 12, since it plays an important role in the evolution of age trajectories for both fertility and mortality 3. In fact, it provides an optimal life history strategy for maximizing lifetime reproduction, which is determined by maximizing age survival and fecundity 45. The size of most organisms (rather than the age) directly affects both survival probabilities 678 and individual fecundity 79. Hence, we can hypothesize that organisms grow as fast as possible to achieve mature size 1011. Regarding to that, many strategies such as enlarging the initial size 1213 or expanding the period of growth and development 14 could be implemented to increase the size. Juvenile birds grow as quickly as possible in order to reduce the period of vulnerability to predators 15, but Ricklefs didn’t prove any correlation between growth rate and predation for temperate passerine birds 101617. Food scarcity and/or poor quality food 18 as well as the ability of parents to nourish their offspring 15 are invoked as factors explaining slow growth in birds. Growth rates of different morphological features have been studied for many altricial species 19202122, but few data are available for precocial species 23. The reasons why altricial species are preferred for these studies are due to the abundance of their nests and/or the ease to get repeated measurements throughout the nestling period. The problems related to repeated catching and measuring of precocial birds offspring explain why there is little literature about the quantification of precocial species growth. However, postnatal growth rate varies widely among species of birds 24. Growth rates are likely to be negatively related to the adult mass and they are higher for altricial birds than for precocial species 2425. The study of the avian growth curve and development is also important for comparisons within and among species 2627.

Nevertheless, the study of precocial species growth has raised another important issue: since the form of the avian growth curve varies considerably 28, several mathematical functions have been used to describe growth, such as the logistic, the Gompertz and the von Bartalanffy equation, but these functions are unfitting for precocial species such as Mallard ducks. Indeed for this species, there is a scarce availability of frequent known-age birds measurements as well as there is often a lack of growth data and fitting the above mentioned function using incomplete data can bias growth rate estimates 2829. Besides, the huge number of parameters involved makes some models unfeasible (the Richards model for example; 30).

The aim of this paper is to investigate development changes in the morphology of Mallard ducks during the hatching-independence stage. It is known that Mallard ducks Offspring are able to run and swim immediately after their hatching period and they can fly at an age of 50-60 days 31. Basing on these facts, we hypothesize that some mallard ducks physical features are particularly useful in order to escape from predators in the period immediately after their hatching. Regarding these features, we remark the bill for eating or the tarsus for swimming and running which should reach the mature size as quickly as possible, even though the adult weight is not achieved. For these purposes, we employed a novel statistical approach based on capture-recapture (C-R) data of young individuals and Functional Data Analysis (FDA, 32). The C-R method allowed us to collect different repeated measures from the same Mallard ducks individuals and hence this enabled us to build the mallard ducks growth curves 11. In particular, in the next pages we’ve reported growth rates, daily weight gain curves and daily weight gain curves relative to three morphological features: bill, bill+head, tarsus. Finally, we have used Functional Principal Components (FPC, 33) to discriminate males and females .

A total of 21 immature individuals and 98 adult individuals were measured. Not all parameters were recorded for some birds in order to speed up the procedure and not to stress ducklings.

The asymptotes were 43,3 mm for tarsus, 53,1 mm for bill, 112,0 mm for BH and 1053,3 g for weight.

In Table 1, we show the basic statistics, like the mean, the standard deviation, the minimum, the maximum and the growth rate for the tarsus obtained (x_i (t)) from FDA at given time points (age =t): 10 days: 30,6 (24,3 - 39,7), 20 days: 36,8 (29,1- 43,5), 30 days: 39,8 (33,9-44,0) and so on up to 100 days: 43,7 (43,7-43,7).

In Table 2, the same statistics are shown for the bill: 1 day, 10 days: 24,7 (20,0 - 38,4), 20 days: 32,3 (25,6 - 45,0), 30 days: 37,8 (31,1 - 46,6) and so on up to 100 days: 53,8 (51.2 - 56.5).

Table 3 shows these statistics for BH at the time points 26 day, 10 days: 53,0 (44,7 - 59,0), 20 days: 67,4 (52,0 - 76,5), 30 days: 79,6 (60,7 - 92,5) and so on up to 80 days: 109,4 (107,0 - 109,4).

Thanks to these data, we have been able to draw the average growth curve for each parameter. These are given in Figure 1, Figure 2, Figure 3. As far as tarsus is concerned, most of the curves is convex. The average curve reaches the asymptote at fledging age. However, from inspection of both the curves which show higher growth speed and the maximum values of the pertaining table, we can see that some individuals reach the asymptote at a precocious age (20 days). The asymptote is reached tardily for the bill (60-70 days as shown in Table 2). Also the related curves are less steep and some of them have a sigmoid trend. The asymptote for BH is not even reached at the age of 80 days (see Table 2), most of these curves are convex and only a few of them are of sigmoid type.

Figure 4 shows the growth mean curves of the three morphological features for comparison.

Besides, for all parameters the daily growth extent has been computed.

Figure 5 provides a comparison of the daily gain curves of the three morphological features.

Finally, Figure 6, Figure 7, and Figure 8 show the results of FPC analysis for tarsus, bill and BH respectively. The first two principal components explain 97,6% of the variability for tarsus, 97,1% of the variability for bill, and 98,7 of the variability for BH.

Looking at Figure 5 we can see that the speed of tarsus growth tends to zero before the other two curves. The bill and also the BH continue to grow after fledging and reach adult values at about 80 days for bill (Table 2) and over 80 for BH (Table 3). The tarsus at fledging (50-60 days of life according to Cramp 31) is 98,1% of asymptote (at 50-60 days old), bill at fledging: 86,2% (at 50 days) and 91,8% (at 60 days), BH: 86,3% (at 50 days); 92,4% (at 60 days).

Most of the vertebrate growth curves described in literature are of determinate type. This kind of model is characterized by a reachable maximum size which animals during their growth progressively tend to attain by reducing their growth speed. For this reason this model is also called asymptotic 30. Such a model is valid for bacteria, all birds 24, 40, fish 41, bugs and most of mammals 42, 43. The way the animals approach the asymptote during their growth can be described by different curves, like the exponential, the logistic and the sigmoid curve.

The results provided in this paper supplement and complete those concerning the average growth curve of weight of mallard ducks provided by Giammarino & Quatto 11. Starting from the mentioned work, where it is proved that the weight represents a valid indicators of the age of mallard ducks, in this paper we have used functional data analysis (FDA) to determine the age of these birds at the moment of their capture/recapture so as to build growth curves for other parameters.

In our analysis the trend of the average curves for the three parameters under study is the convex type (see Figure 4). The curve relative to tarsus reaches the asymptote both faster and precociously (in some Mallard ducks already apparent at day 20 of life: see Table 1). In fact the individual growth curves relative to tarsus are steeper than the others (Figure 1).

Rapid tarsus development is not unique to mallards and may be related to an efficient early locomotion in a variety of avian taxa 44. This effect has been observed also in (Strixoccidentalis) 20, Bald eagles (Haliaeetus leucocephalus; 45), Burrowing owls (Athene cunicularia; 46), and in waders 23.

Muscle tissue is supposed to develop before skeletal tissue and it is assumed to be the factor that influence the skeleton rate development 25. In precocial species the leg muscle is functionally more important. In Laruscalifornicus, semiprecocial species, the paw skeleton is more developed than wing skeleton at the hatching 39. In Dotterel (Charadriusmorinellus), precocial species, the chicks hatch with tarsus, bill and BH well developed, but a poor wing development 23.

It can be assumed that the whole skeleton, along with the tarsus, reaches final size after fledging, with rates of growth which vary for different skeleton sectors depending on the type of juvenile locomotion 30. Indeed, it is difficult to think that, being the pelvic belt already developed, the muscles involved in flight work by clinging to bone segments not mature yet. Mallard ducks’ tarsus may grow faster than other parameters because these birds need to move as fast as possible both on the ground, to escape from predators, and in the water to search for more effectively food at that time of the life when their wings are not functional.

In mallard ducks the weight increase after fledging and this could be due mainly to the muscular component induced by flight itself. However it seems that weight starts increasing later than in other birds, probably for the same growth of those muscles engaged in flight.

In our analysis, the bill growth seems protracted. Many species exhibit this type of strategy 4748, while others present a higher growth speed 23. No hypotheses have been done forward to explain this difference.

The FPC analysis performed on tarsus, bill and BH leads to divide the group under analysis into two sub-groups: one of which seems to grow more rapidly than the other.

In fact, one part of the wild ducks is over the average of the group (which is zero), another is below. It is reasonable to suppose that the two distinguished groups are respectively males and females.

We have obtained encouraging results for 11 out of the 12 mallard ducks on which the all three FPC were carried out. In fact the result is encouraging since 9 have results shared by two FPC (while the third provides uncertain results), for 2 Mallard ducks on all the FPC. Only one duck shows as uncertain (CH 9967).

Further research using C-R and FPC should be carried out to confirm the data. The FPC analysis does seem like a good approach for determining sex of growing Mallard ducks.

Functional analysis of capture-recapture data has been shown to be an effective approach for representing the growth curves and the daily values, even if observations are relatively few and the dates of birth are not known with precision. This approach could be usefully employed for studying the growth of other animals (in addition to avian species), when standard methods are not applicable, for example in the case of precocial species or species which are hard to recapture before they reach the adult size.