Provisioning rules and chick competition in asynchronously hatching common terns (Sterna hirundo)

Abstract

Interactions between nestling birds and their parents are models for examining parent–offspring communication and sibling competition. Most studies have focused on species where young are restricted to a nest. However, offspring of many species are mobile and fed by parents for an extended period post-hatch. These chicks’ mobility may provide an opportunity to examine the role of signalling and physical competition on parental feeding decisions. We examined parental provisioning rules in relation to offspring behaviour and hatching order (i.e., competitive ability) in a species with mobile young, the common tern. We determined that about 95% of feedings were directed to the first chick to reach the parent when it landed with food. We developed a probabilistic model to predict the likelihood of a chick reaching the parent first, and thus receiving food. Our model showed that begging intensity, feeding history, and the interaction between begging intensity and relative proximity to the parent best predicted which chick would arrive first. Increased begging was associated with arriving first significantly more when a chick was relatively further from the parent than when it was closer than its siblings. Independently of these factors, larger, earlier-hatched chicks were more likely to be fed than smaller, later-hatched chicks. Additional analyses showed that parents landed closer to more intensively begging chicks, however, increased begging did not explain the advantage of earlier-hatched chicks because begging intensity did not vary with hatching order. Instead, earlier-hatched chicks were more likely to outrun later-hatched siblings and reach the parent first.

Mathematical detail for the provisioning model

Explanatory factors:

We scored binomial provisioning outcomes as fi,r=1 when chick i arrived first, else f _i,r =0; with i indexing chicks A, B, ... up to m chicks in brood r of s broods. In our study m was either 2 or 3.

The first of two model assumptions is that at any given feeding event the likelihood that a chick reaches the parent first depends on circumstances specific to that event, such as the chick’s behaviour and proximity to the parent. Thus, each brood r can be described by a suite of k=1–n factors C _{i , j , k , r} , that predict the ‘objective probabilities’, ${}^{\rm o}p_{i,r}$ (Walpole et al. 1998) of chick i reaching the parent first. The factors chosen (with first-order coefficients β _k and interactions coefficients $\beta _{k_1 ,k_2 }$) include the following.

• 1. Begging intensity (B _{i , r} ) of chick i (ordinal variable ranging from 0 to 4).

• 2. Distance (D _{i , r} ) from chick i to parent when it lands with food (cm).

• 3. Time (T _{i , r} ) since chick i was last fed (min).

• 4. Chick i was nearest to the parent, or tied in distance, when the parent lands with food (N _{i , r} ; 1 if true, else 0).

• 5. Length (H _{i , r} ) of the last prey item eaten by chick i (cm).

• 6. Chick i was the last chick fed (F _{i , r} ; 1 if true, else 0).

• 7. Number of feeding visits between successful (i.e. chick is fed) feeding events for chick i (M _{i , r} ).

To retain the symmetry of competition among all chicks A to m, each of the seven potential explanatory factors (B _{i , r} , D _{i , r} , T _{i , r} , N _{i , r} , H _{i , r} , F _{i , r} , M _{i , r} ) is defined relative to competing chicks j=1 to m; j≠i. For example, if factor k=2 is the distance between chick A and the parent when it lands (D _{A , r} ), then $C_{A,B,k = 2,r} = D_{B,r} - D_{A,r}$, and symmetrically, $C_{B,A,k = 2,r} = D_{A,r} - D_{B,r}$, describes the relationship with respect to that distance between chicks A and B. Likewise, for first-order interactions, say between D _{A , r} and begging intensity (B _{A , r} , i.e., k=1), we have $C_{A,B,k_1 = 2,k_2 = l,r} = D_{B,r} B_{B,r} - D_{A,r} B_{A,r}$, and symmetrically $C_{B,A,k_1 = 2,k_2 = l,r} = D_{A,r} B_{A,r} - D_{B,r} B_{B,r}$. These operations apply identically to continuous factors, such as distance (D _{i , r} ), ordinal variables such as begging intensity (B _{i , r} ), or categorical variables such as whether the chick was the last one fed (F _{i , r} ).

The second assumption of the model was that each chick has an intrinsic and immeasurable (or measurable but unmeasured) competitive advantage over its siblings that is independent of the factors listed above. We define this intrinsic ability in the language of Bayesian statistics as a ‘prior probability’, ^Bp _i . Both the objective probabilities calculated from the above factors and these prior probabilities are estimated thereby yielding ‘posterior probabilities’, ^Ap _{i , r} of our observed provisioning outcomes.

Statistical model:

Our model for calculating objective probabilities is described by:

$$^{\rm o} p_{i,r} = \frac{{^{\rm o} p'_{i,r} }}{{\sum\limits_{j = 1}^m {^{\rm o} p'_{j,r} } }}$$

(1a)

where

$$^{\rm o} p'_{i,r} = \frac{1}{{1 + \left( {m - 1} \right)e^{ - \sum\limits_{k = 1}^n {\beta _k } \sum\limits_{j = 1}^m {C_{i,j,k,r} \left( {1 - f_{j,r} } \right) - \sum\limits_{k_1 = 1}^n {\sum\limits_{k_2 = 1}^n {\beta _{k_1 } ,k_{_2 } } \sum\limits_{j = 1}^m {C_{i,j,k_1 ,k_2 ,r} \left( {1 - f_{j,r} } \right)} } } } }}$$

(1b)

for i=A, B, ..., m (or equivalently i=1, 2, ..., m) when f_i,r=1 and where β _k is the coefficient for main factor k, and $\beta _{k_1 ,k_2 }$ the coefficient for the interaction of main factors k₁ and k₂, with k₁≠k₂. Note, that we formulate Eq. 1b as a logistic model to retain estimates of ${}^{\rm o}p\prime _{i,r}$ between 0 and 1, though the values of ${}^{\rm o}p\prime _{i,r}$ must be scaled to assure that total probability sums to unity (Eq. 1a ). Posterior probabilities are calculated (Walpole et al. 1998) as:

$${}^{\rm A}p_{i,r} = \frac{{{}^{\rm o}p_{i,r} \times {}^Bp_i }}{{\sum\limits_{j = 1}^m {{}^{\rm o}p_{j,r} \times {}^Bp_j } }}$$

(2)

Parameter estimation:

We obtain maximum likelihood estimates for the values for the prior probabilities ${}^Bp_i$ and factor coefficients β _k and $\beta _{k_1 ,k_2 }$ by minimizing the negative ln-likelihood (L) of our observed binomial provisioning outcomes f _{i , r} with respect to our posterior provisioning probabilities ${}^Ap_{i,r}$ (Walpole et al. 1998),

$$L\left( {\hat \beta _k ;\hat \beta _{k_1 ,k_2 } ;{}^B\hat p_{i = A,B,m - 1} } \right) = - \sum\limits_{r = 1}^s {\ln \left[ {{}^Ap_{i,r} } \right]}$$

(3)

for i=A, B, ..., or m when f _{i , r} =1, noting that

$${}^B\hat p_m = 1 - \sum\limits_{i = 1}^{m - 1} {^B \hat p_i } .$$

(4)

Estimates of the model parameters were obtained by minimizing Eq. (3) simultaneously for our m=2 and 3 brood size data sets.We examined candidate models, including one that hypothesized that the values for the ^Bp _i s changed with chick age, to determine which factors and their first-order interactions best predicted the outcome f _i,r =1, (i.e., to reach the parent first), and to challenge the null hypothesis that there was no difference among hatching order in a chick’s probability of reaching the parent first, i.e., ${}^Bp_A = {}^Bp_B = \cdots = {}^Bp_m = 1/m$ for broods of size m. Note that for such null values of ${}^Bp_i$ and with all β _{k , m} =0, the posterior probabilities, ${}^Ap_i$, equal 1/m and thus each chick in a brood is equally likely to achieve f _i,r =1.