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Plant species and pollinatorsMedicago sativa L. (Fabaceae), also called alfalfa or lucerne, is a perennial legume with flowers arranged in a cluster or raceme. It is a self-compatible plant with fairly high outcrossing rate (5.3–30%)46, and it requires insect visits for seed production47. No plant material was collected for this study. Honey bees, Apis mellifera, and alfalfa leafcutting bees, Megachile rotundata, are used as managed pollinators in alfalfa seed-production fields in the USA while bumble bees are commonly used in alfalfa breeding47.Experimental design and pollinator observationsFive 11 m × 11 m patches of M. sativa plants were set up in an east–west linear arrangement at the West Madison Agricultural Research Station in Madison, Wisconsin, USA. Within each patch, we transplanted 169 young plants grown from seeds in the greenhouse, each placed 90 cm apart. These plants grew and, at flowering, a plant had an average of 30.65 ± 16.4 stems per plant, with 4.93 ± 3.41 racemes per stem, and 7.53 ± 2.44 open flowers per raceme.A honey bee hive was placed approximately 100 m from the patches and a bumble bee hive was set up at the center of the southern edge of the patches. For leafcutting bees, a 60 × 30 × 7.6 cm bee board was set up in each of two boxes placed 1/3 and 2/3 along the southern edge of the patches and a half gallon of bees was released at periodic intervals throughout the alfalfa flowering season.Over two consecutive summers, observers followed bees foraging in the alfalfa patches, marked each raceme visited in succession within a foraging bout with a numbered clip, and recorded the number of flowers visited per raceme. After a bee had left a patch, observers went back to the marked racemes and measured the distance and direction traveled between consecutive racemes. Directions were recorded as one of the cardinal directions: North (N), South (S), East (E) or West (W), or inter-cardinal directions: Northeast (NE), Southeast (SE), Northwest (NW) and Southwest (SW). The frequency distributions of distances and directions traveled between two successive racemes are presented for each bee species each year in Figs. 1 (distances) and 2 (directions). The low pollinator abundance permitted observers to follow individual bees foraging in a patch. Little interference among bee species was observed in the patches.Figure 1Frequency distributions for distances traveled between consecutive racemes (cm) for each bee species each year.Full size imageFigure 2Frequency distributions of directions traveled between consecutive racemes for each bee species each year.Full size imageModel for the distance traveled between consecutive racemesWe first determined whether a statistical model best described the distance traveled between consecutive racemes (Modeled Distance), and examined whether the model differed among bee species. We used mixed effect linear models (proc Mixed in SAS 9.3)48 to identify the model that best described the distance traveled by pollinators between consecutive racemes. The model included loge distance as a linear function of loge flower number and bee species as fixed effects. The distance traveled between consecutive racemes and the number of flowers visited per raceme were log transformed prior to analyses in order to improve the models’ residuals. In addition, we included patch and foraging bout as random effects in the model. A foraging bout includes the racemes visited in succession from the time a bee is spotted in a patch to the time it leaves that patch. We used foraging bout instead of individual bee as the random effect because bees were not individually marked in this study. Moreover, to take into consideration the potential correlation between successive observations within a foraging bout, we added clip to the model. Clip 1 represents the first and second racemes visited in the foraging bout; clip 2, the second and third, and so on. Clip was added to the model either as a random effect or as a repeated measure with an AR(1) structure. The combination of random clip and random foraging bout creates a model that is sometimes called the “compound symmetry” model. The AR(1) structure represents correlations that decline exponentially as the gap between measurements increases such that measurements closer together in time are more strongly correlated than measurements further apart. Because we expected bees to visit flowers at close proximity when resources are abundant, we chose this correlation structure as a good potential descriptor of the way distances might be correlated within foraging bouts. We started with a full model which included loge flower number, bee species, patch, foraging bout, and clip either as a random effect or as a repeated measure with an AR(1) structure. We then removed variables and compared models by inspecting AIC values and the p values for each term in the model. We considered both low AIC and statistically significant (p  More

LightGBMBefore explaining LightGBM23, it is necessary to introduce XGBoost24, which is also based on the gradient boosting decision tree (GBDT) algorithm30. XGBoost integrates multiple classification and regression trees (CART) to compensate for the lack of prediction accuracy of a single CART. It is an improved boosting algorithm based on GBDT, which is popular due to its high processing speed, high regression accuracy and ability to process large-scale data31. However, XGBoost uses a presorted algorithm to find data segmentation points, which takes up considerable memory in the calculation and seriously affects cache optimization.LightGBM is improved based on XGBoost. It uses a histogram algorithm to find the best data segmentation point, which occupies less memory and has a lower complexity of data segmentation. The flow of the histogram algorithm to find the optimal segmentation point is shown in Fig. 3.Figure 3Histogram algorithm.Full size imageMoreover, LightGBM abandons the levelwise decision tree growth strategy used by most GBDT tools and uses the leafwise algorithm with depth limitations. This leaf-by-leaf growth strategy can reduce more errors and obtain better accuracy. Decision trees in boosting algorithms may grow too deep while training, leading to model overfitting. Therefore, LightGBM adds a maximum depth limit to the leafwise growth strategy to prevent this from happening and maintains its high computational efficiency. To summarize, LightGBM can be better and faster used in industrial practice and is also very suitable as the base model in our tide level prediction task. The layer-by-layer growth strategy and leaf-by-leaf growth strategy are shown in Fig. 4.Figure 4Two GBDT growth strategies.Full size imageCNN-BiGRUConvolutional neural networkA convolutional neural network (CNN) is a deep feedforward neural network with the characteristics of local connection and weight sharing. It was first used in the field of computer vision and achieved great success32,33. In recent years, CNNs have also been widely used in time series processing. For example, Bai et al.34 proposed a temporal convolutional network (TCN) based on a convolutional neural network and residual connections, which is not worse than recurrent neural networks such as LSTM in some time series analysis tasks. At present, a convolutional neural network is generally composed of convolution layers, pooling layers and a fully connected layer. Its network structure is shown in Fig. 5. The pooling layer is usually added after the convolution layers. The maximum pooling layer can retain the strong features in the data after the convolution operation, eliminate the weak features to reduce the number of parameters in a network and avoid overfitting of the model.Figure 5Schematic diagram of a convolutional neural network.Full size imageBidirectional GRUIn previous attempts at tide level prediction by scholars, bidirectional long short-term memory networks35 have achieved good prediction results. However, in our subsequent experiments, the bidirectional gated recurrent unit achieved higher prediction accuracy than BiLSTM, so we used the BiGRU network for subsequent prediction tasks.The GRU network36 adds a gating mechanism to control information updating in a recurrent neural network. Different from the mechanism in LSTM, GRU consists of only two gates called the update gate ({z}_{t}) and the reset door ({r}_{t}).The recurrent unit structure of the GRU network is shown in Fig. 6.Figure 6Recurrent unit structure of the GRU network.Full size imageEach unit of GRU is calculated as follows:$${z}_{t}= sigma ({W}_{z}{x}_{t}+{U}_{z}{h}_{t-1}+{b}_{z})$$
(7)
$${r}_{t}= sigma ({W}_{r}{x}_{t}+{U}_{r}{h}_{t-1}+{b}_{r})$$
(8)
$${widetilde{h}}_{t}=tanh({W}_{h}{x}_{t}+{U}_{h}left({r}_{t}odot {h}_{t-1}right)+{b}_{h})$$
(9)
$${h}_{t}={z}_{t}odot {h}_{t-1}+left(1-{z}_{t}right)odot {widetilde{h}}_{t}$$
(10)
In the above formula, ({z}_{t}) represents the update gate, which controls how much information is retained from the previous state ({h}_{t-1}) (without nonlinear transformation) when calculating the current state ({h}_{t}). Meanwhile, it also controls how much information will be accepted by ({h}_{t}) from the candidate states ({widetilde{h}}_{t}). ({r}_{t}) represents the reset gate, which is used to ensure whether the calculation of the candidate state ({widetilde{h}}_{t}) depends on the previous state ({h}_{t-1}). (upsigma ) is the standard sigmoid activation function; (tanh(cdot )) is the hyperbolic tangent activation function; and (odot ) indicates the Hadamard product. The weight matrices of the update gate, reset gate, and ({widetilde{h}}_{t}) calculation layer are expressed as ({W}_{z},{W}_{r},{W}_{h}); the coefficient matrices of the update gate, reset gate, and ({widetilde{h}}_{t}) calculation layer are expressed as ({U}_{z},{U}_{r},{U}_{h}); and the offset vectors of the update gate, reset gate, and ({widetilde{h}}_{t}) calculation layer are expressed as ({b}_{z},{b}_{r},{b}_{h}).A bidirectional gated recurrent unit network37 is a combination of two GRUs whose information propagating directions are reversed, and it has independent parameters in each, which makes it able to fit both forward and backward data at first and then join up the results from two directions. BiGRU can capture sequence patterns that may be ignored by unidirectional GRU. The structure of BiGRU is shown in Fig. 7.Figure 7The structure of BiGRU.Full size imageTaking the BiGRU’s forward hidden state vector at time (t) as ({h}_{t}^{(1)}) and taking the BiGRU’s backward hidden state vector at time (t) as ({h}_{t}^{(2)}), (upsigma ) indicates the standard sigmoid activation function, and (oplus ) indicates a vector splicing operation. We can calculate the output ({y}_{t}) of a BiGRU network as follows:$${h}_{t}={h}_{t}^{(1)}oplus {h}_{t}^{(2)}$$
(11)
$${y}_{t}=sigma ({h}_{t} )$$
(12)
CNN-BiGRU prediction modelBecause CNN has significant advantages in extracting useful features from a picture or a sequence and BiGRU is good at processing time series, we combine CNN and BiGRU to build the CNN-BiGRU model. The model can be mainly divided into an input layer, a convolution layer, a BiGRU network layer, a dropout layer, a fully connected layer and an output layer. The CNN layer and BiGRU layer are the core structures of the model. The function of the dropout layer is to avoid model overfitting. The CNN layer consists of two one-dimensional convolution (Conv1D) layers and a one-dimensional maximum pooling (MaxPooling1D) layer. The input of BiGRU is the output sequence of the CNN layer, and the BiGRU network is set as a one-hidden-layer structure. The structure of the CNN-BiGRU combination model is shown in Fig. 8.Figure 8The structure of CNN-BiGRU.Full size imageVariable weight combination modelWhen we analyze and predict relatively stationary tide level time series, LightGBM can perform well. However, due to environmental factors such as air pressure, wind force and terrain in reality, most tide level observation sequences are sometimes not relatively stationary, which requires that our tide level prediction model be reasonably able to “extrapolate” based on the sample observations, that is, be capable of generating values that are not in the sample. LightGBM is a tree-based model, which leads to our prediction results being between the maximum and minimum values of sequences. Therefore, LightGBM will not be able to accurately predict the situation or tidal change trend that did not appear in previous observations. However, the CNN-BiGRU model, which is a kind of neural network, has no such problem in theory and will be able to find the trend information that may be hidden in the tide level series. Therefore, we consider providing an appropriate weight for a single base model to build a combination model to improve the accuracy of the tide level prediction task.Principle of the residual weight combination model and improved variable weight combination modelTo improve the prediction accuracy of the combination model, a simple and effective idea is to determine the base models’ weights in the combination model according to the error between the prediction value and the real value. This method is also called the residual weight method, and its calculation formulas for determining the weights are:$$gleft({x}_{t}right)= sum_{i=1}^{m}{omega }_{i}left(t-1right){f}_{i}({x}_{t})$$
(13)
$${omega }_{i}left(t-1right)=frac{frac{1}{overline{{varphi }_{i}}left(t-1right)}}{sum_{i=1}^{m}frac{1}{overline{{varphi }_{i}}left(t-1right)}}$$
(14)
$$sum_{i=1}^{m}{omega }_{i}left(t-1right)=1,{omega }_{i}left(t-1right)ge 0$$
(15)

where ({omega }_{i}left(t-1right)) denotes the weight of the (i) th model at the moment (t-1), ({f}_{i}left({x}_{t}right)) denotes the prediction value of the (i) th model at the moment (t), (gleft({x}_{t}right)) denotes the prediction value of the combination model at the moment (t), and (overline{{varphi }_{i}}left(t-1right)) is the square sum of the predictive errors of the (i) th model at the moment (t-1).Our LightGBM-CNN-BiGRU (combination model) is based on the improved residual weight method. We call it the variable weight combination model. We use the weights calculated by formula (9) and formula (11) to calculate a series of new weights. The new weights from formula (11) will take the residual weight changes in (d) time steps into consideration by averaging the old weights in (d) time steps to improve the stability of the residual weight method.$${omega }_{j}left(tright)=frac{1}{d}sum_{k=1}^{d}{omega }_{i}left(t-kright)left(d=4right)$$
(16)
After obtaining a series of weights through formula (9) and formula (11), we take the absolute value of the error between the prediction value and the true value of each combination model at the moment of (t) as ({delta }_{i,t}) and ({delta }_{j,t}), respectively:$${delta }_{i,t}=mid sum_{i=1}^{m}{omega }_{i}left(tright){f}_{i}left({x}_{t}right)-{y}_{t}mid $$
(17)
$${delta }_{j,t}=mid sum_{i=1}^{m}{omega }_{j}left(tright){f}_{i}left({x}_{t}right)-{y}_{t}mid $$
(18)
Comparing ({delta }_{i,t}) and ({delta }_{j,t}), if ({delta }_{i,t} >{delta }_{j,t}), the combination model uses the new weight ({omega }_{j}left(tright)) in place of the original weight ({omega }_{i}left(tright)). Otherwise, the weight of the combination model remains unchanged.Parameter optimization of the combination modelBecause the LightGBM-CNN-BiGRU (combination model) is a variable weight combination of the prediction results from two base models, the performance of the combination model can be directly improved by separately optimizing the super parameters of the two base models. We mainly use the grid search algorithm and K-fold cross validation method to optimize the parameters. The grid search algorithm is a method to improve the performance of a certain model by iterating over a given set of parameters. With the help of the K-fold cross validation method, we can calculate the performance score of the LightGBM model on the training set and easily optimize its superparameters. The final parameters of the LightGBM model are set to num_leaves = 26, learning_rate = 0.05, and n_estimators = 46.For the CNN-BiGRU network, we mainly improve the prediction accuracy of the model by adjusting the size and number of hidden layers in the BiGRU structure and prevent the model from overfitting by changing the dropout ratio and tracking the validation loss of the network while training.The LightGBM and CNN-BiGRU variable weight combination modelThe workflow of our tide level prediction model is shown in Fig. 9. It mainly includes data preprocessing; training, optimization and prediction of the base models; construction of a variable weight combination prediction model; and evaluation and analysis of the combination model’s performance.

(1)

Data preprocessing: The quality of the data directly determines the upper limit of the prediction and generalization ability of a certain machine learning model. Standard, clean and continuous data are conducive to model training. The data used in this study are from the Irish National Tide Gauge Network, and all of them are subject to quality control. We filled in a small number of missing values and normalized the data to speed up the model training.

(2)

Construction and optimization of base models: We divide the dataset into a training set, a validation set and a test set according to the proportion of 7:1:2 and train the LightGBM model and CNN-BiGRU model with data on the training set. We optimize the parameters and monitor whether the model has been overfitted by tracking the validation loss of the network while training. Finally, we put the data into two base models for training and then obtain the prediction results of a single base model.

(3)

Construction of the variable weight combination model. Based on the prediction results of two single base models obtained in step (2), we calculate the weight of each base model according to the principle of the improved variable weight combination method and then obtain the prediction results of the variable weight combination model.

(4)

Model evaluation and analysis: According to the indexes of the model evaluation, the variable weight combination model is compared with other basic models to analyze its prediction performance after being improved.

Figure 9Prediction flow of the LightGBM-CNN-BiGRU variable weight combination model.Full size image More

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In this section, we introduce flapping flight dynamics and describe the bird model used in our computational framework. Furthermore, we describe how such a dynamical model is used in order to identify steady and level flapping flight regimes, study their stability, and assess their energetic performance.Equations of motion modelling flapping flightFlight dynamics is restricted to the longitudinal plane and thus the bird main body is captured as a rigid-body with three degrees of freedom, i.e. two in translation and one in rotation. This model preserves symmetry with respect to this plane, without any lateral force and moment. The aerodynamic model of the wing relies on the theory of quasi-steady lifting line23. Additionally, the present work does not account for the inertial forces due to the acceleration of the wing, and thus also neglecting the so-called inertial power. This inertial power was shown to be negligible in fast forward flight conditions, in comparison to the other contributions to actuation power24, and is thus systematically neglected in similar work10,11,25 since wing inertia is neglected.The body is thus modelled with a mass (m_b) and a rotational inertia (I_{yy}) about its center of mass. The equations of motion are expressed in the body frame (G(x’, y’, z’)) with unit vectors ((hat{textbf{e}}_{x’}, hat{textbf{e}}_{y’}, hat{textbf{e}}_{z’})), and an origin located at the center of mass, as pictured in Fig. 1a. The state space vector is thus$$begin{aligned} textbf{x} = {u, w, q, theta } end{aligned}$$where u and w are the body velocities along the (x’-) and (z’-)axis and (theta) and q are the pitch angle and its time derivative about the (y’-)axis, respectively. Consequently, the equations of motion read11,13,26$$begin{aligned} begin{aligned} dot{u}&= -qw – gsin theta + frac{1}{m_b}big ( {F_{x’}(textbf{x}(t), t)} \&quad + {F_{x’, t}(textbf{x}(t), t)} + D (textbf{x}(t), t) big )\ dot{w}&= qu + gcos theta +frac{1}{m_b} big ( F_{z’}(textbf{x}(t), t) + F_{z’, t}(textbf{x}(t), t) big ) \ dot{q}&=frac{1}{I_{yy}} big ( M_{y’}(textbf{x}(t), t) + M_{y’, t}(textbf{x}(t), t) big )\ dot{theta }&= q end{aligned} end{aligned}$$
(1)
Figure 1(a) Bird model for describing the flight dynamics in the longitudinal plane. The state variables are expressed with respect to the moving body-frame located at the flier’s center of mass (G(x’,z’)). These state variables are the component of forward flight velocity, u, the velocity component of local vertical velocity, w, the orientation of this body-centered moving frame with respect to the fixed frame, (theta) and its angular velocity, q. A second frame (O(x’_{w}, z’_{w})) is used to compute the position of the wing, relative to the body. The wings (dark gray) and the tail (red) are the surfaces of application of aerodynamic forces. (b) Top view of the bird model. The left wing emphasizes a cartoon model of the skeleton. The shoulder joint s connects the wing to the body via three rotational degrees of freedom (RDoF), the elbow joint e connects the arm with the forearm via one RDoF and the wrist joint w connects the forearm to the hand via two RDoF. Each feather is attached to a bone via two additional RDoF, except the most distal one ”1” which is rigidly aligned with the hand. The right wing further emphasizes the lifting line (red) which is computed as a function of the wing morphing. The aerodynamic forces generated on the wing are computed on the discretized elements (P_{i}). The tail is modelled as a triangular shape with fixed chord (c_{t}) and maximum width (b_{t}) that can be morphed as a function of its opening angle (beta). (c) Wing element i between two wing profiles, identifying a plane (Sigma) containing the lifting line (red). (d) Cross section of the wing element, containing the chord point (mathbf {P_i}) where the velocities are computed (Color figure online).Full size imageThe forcing terms in Eq. (1) are the aerodynamic forces and moments applied to the wing (namely (F_{x’}), (F_{z’}), and (M_{y’}) ) and to the tail ((F_{x’, t}), (F_{z’, t}), and (M_{y’, t})). The whole drag is captured by an extra force D that sums contributions due to the body (D_{b}), the skin friction of the wing (wing profile) (D_{p,w}), and the skin friction of the tail (tail profile) (D_{p,t}). These terms are described in detail in the next sections. Importantly, we accounted for the drag acting purely along (x’) direction, after proving that the projection of the drag forces along (z’)-axis is between two and three orders of magnitude smaller with respect to the aerodynamic forces produced by two other main lifting surfaces. This assumptions holds for the fast forward flight regime that are subject of our study, but such components of drag along (z’) axis should be accounted for other flight situations.Wing modelThe bird has two wings. Each wing is a rigid poly-articulated body, comprising the bird arm, forearm and hand, as pictured in Fig. 1b. Each segment is actuated by a joint to induce wing morphing. We refer to13,15 for a complete description of this wing kinematic model.Each joint is kinematically driven to follow a sinusoidal trajectory specified as:$$begin{aligned} q_{i}(t) = q_{0,i}(t) + A_{i} sin (omega t + phi _{0,i}) end{aligned}$$
(2)
with (omega = 2 pi f) and f being the flapping frequency which is identical for each joint, (q_{0,i}) being the mean angle over a period (or offset), (A_i) the amplitude, and (phi _{0,i}) the relative phase of joint i. A complete wingbeat cycle is therefore described through a set of 19 kinematic parameters, including the frequency f.We assume that the wing trajectory is rigidly constrained, and therefore we do not need to explicitly solve the wing dynamics. Under this assumption, the motion generation does not require the computation of joint torques. The model further embeds seven feathers of length (l_{ki}) in each wing. The feathers in the model have to be considered a representative sample of the real wing feathers. They thus have a limited biological relevance; their number is chosen so as to interpolate the planform satisfactorily and to smoothly capture the morphing generated by the bone movements. These feathers are attached to their respective wing bones via two rotational degrees of freedom allowing them to pitch and spread in the spanwise direction. These two degrees of freedom are again kinematically driven by relationships that depend on the angle between the wing segments13. This makes the feathers spreading and folding smoothly through the wingbeat cycle. In sum, the kinematic model of the wing yields the position of its bones and feathers at every time step. This provides a certain wing morphing from which the wing envelope (leading edge and trailing edge) can be computed (see Fig. 1b). From the wing envelope, the aerodynamic chord and the lifting line are computed. The lifting line is the line passing through the quarter of chord, which is itself defined as the segment connecting the leading edge to the trailing edge and orthogonal to the lifting line (Fig. 1b). This extraction algorithm is explained in detail in15.In order to calculate the aerodynamic forces, the angle of attack of the wing profile has to be evaluated. Each wing element defines a plane containing the lifting line and the aerodynamic chord as pictured in Fig. 1c. The orientation of the plane is identified by the orthogonal unit vectors ((hat{textbf{e}}_n, hat{textbf{e}}_t, hat{textbf{e}}_b)), where (hat{textbf{e}}_n) is the vector perpendicular to the plane and (hat{textbf{e}}_t) is the tangent to the lifting line. To compute the effective angle of attack, the velocity perceived by the wing profile is computed as the sum of the velocities due to the body and wing motion, and the velocity induced by the wake. The first contribution, (textbf{U}), accounts for$$begin{aligned} textbf{U} = textbf{U}_{infty } – textbf{v}_{kin} – textbf{v}_{q}end{aligned}$$where (textbf{U}_{infty } = u hat{textbf{e}}_{x’} + w hat{textbf{e}}_{z’}) is the actual flight velocity, (textbf{v}_{kin}) is the relative velocity of the wing due to its motion, and (textbf{v}_{q}) is the component induced by the angular velocity of the body q and calculated as$$begin{aligned} textbf{v}_{q} = qhat{textbf{e}}_{y’} wedge (textbf{P}_{i} – textbf{G})end{aligned}$$This velocity vector (textbf{U}) defines the angle (alpha), as pictured in Fig. 1d.The second contribution is due to the induced velocity field by the wake, i.e. the downwash velocity (w_{d}), and acting along the normal unit vector (-w_{d}hat{textbf{e}}_n). The resulting effective angle of attack, (alpha _{r}), is thus$$begin{aligned} alpha _{r} = alpha – frac{w_{d}}{|textbf{U}|}end{aligned}$$The downwash velocity (w_d) is computed according to the Biot-Savart law23, assuming the wake being shed backwards in the form of straight and infinitely long vortex filaments at each time step of the simulation13,15. This quasi-steady approximation is justified a posteriori by ensuring that our reduced frequency, inversely proportional to the unknown airspeed, never exceeds the value of 0.2, below which the effects of time-dependent wake shapes on wing circulation are negligible (e.g. see discussion in27). Once the downwash is evaluated, it is possible to evaluate the circulation, and consequently the aerodynamic force and moment acting at the element (P_i), i.e. (F_{x’, i}(textbf{x}(t), t), F_{z’, i}(textbf{x}(t), t), M_{y’, i}(textbf{x}(t), t)), as explained in detail in13. We use the thin airfoil theory for the estimation of the lift coefficient, with a slope of (2pi) that saturates at an effective angle of attack (alpha _{r}) of (pm 15^{circ }).Drag production by body and wingThe main body and the wings induce drag that should be accounted for in a model aiming at characterizing energetic performance. Body-induced drag is named parasitic because the body itself does not contribute to lift generation, and only induces skin friction and pressure drag around its envelope28. The total body drag is$$begin{aligned} D_{b} = frac{1}{2}rho C_{d, b} S_{b}|textbf{U}_{infty }|^{2} end{aligned}$$
(3)
where (rho) is the air density. We implemented the model described by Maybury28 to compute the body drag coefficient (C_{d, b}). This depends on the morphology of the bird and the Reynolds number Re according to$$begin{aligned} C_{d,b} = 66.6m_{b}^{-0.511}FR_{t}^{0.9015}S_{b}^{1.063}Re^{-0.197} end{aligned}$$
(4)
with (S_{b}) and (FR_{t}) are respectively the frontal area of the body and the fitness ratio of the bird, and both of them can be estimated from other allometric formulas i.e.28,29.$$begin{aligned} S_{b}= & {} 0.00813m_{b}^{2/3} end{aligned}$$
(5)
$$begin{aligned} FR_{t}= & {} 6.0799m_{b}^{0.1523} end{aligned}$$
(6)
The Reynolds number (Re = rho |textbf{U}_{infty }| overline{c} / mu) is calculated with the reference length of the mean aerodynamic chord (overline{c}), with (mu) being the dynamic viscosity. This model is found to be suitable for Reynolds number in the range (10^{4}-10^{5})28. Another source of drag is the profile drag due to friction between the air and the feathers on the wings. It is the sum of the profile drag at each section along the wingspan, i.e.$$begin{aligned} D_{p,w} = frac{1}{2} rho C_{d, pro} sum _{i=1}^{n} c_{i}|textbf{U}_{r,i}|^{2} ds_{i} end{aligned}$$
(7)
with (c_{i}) the chord length, (ds_{i}) the length of the lifting line element along the wingspan, and (textbf{U}_{r,i}) the velocity at the wing section i accounting for the body velocity, the kinematics velocity of the wing and the downwash velocity (Fig. 1c,d). We used a value of profile drag of (C_{d, pro} = 0.02) and this is assumed to be constant over the wingspan and throughout the flapping cycle30. In reality, due to the wing motion, this value should be gait dependent. However, the aforementioned assumption has been largely used in previous works31,32.Tail modelSince the span of the tail is of the same magnitude as its aerodynamic chord, here the lifting line approach cannot be used23. Therefore, the tail is modelled according to the slender delta wing theory, as a triangular planform33. Its morphology is illustrated in Fig. 1b and characterized by the opening angle (beta) and the chord (c_t). This latter parameter is kept constant, thus the tail span is controlled via (beta) from the trigonometrical relationship$$begin{aligned} b_{t} = 2c_{t}tan frac{beta }{2}end{aligned}$$The main limitation of this framework is the low range of angles of attack ((alpha _{tail} More

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