A E Barzakh
email: barzakh_ae@pnpi.nrcki.ru

A N Andreyev

D Atanasov

J G Cubiss

R D Harding

D Atanasov D

M Al Monthery

N A Althubiti

B Andel

S Antalic

J Ballof

K Blaum

T E Cocolios

P Van Duppen

T Day Goodacre

A De Roubin

C Duchemin

G J Farooq-Smith

D V Fedorov

V N Fedosseev

D A Fink

L P Gaffney

L Ghys

M Huyse

N Imai

J Johnson

S Kreim

D Lunney

K M Lynch

V Manea

B A Marsh

Y Martinez Palenzuela

P L Molkanov

D Neidherr

V N Panteleev

M Rosenbusch

R E Rossel

S Rothe

L Schweikhard

M.D Seliverstov

S Sels

C Van Beveren

E Verstraelen

A Welker

F Wienholtz

R N Wolf

K Zuber

Producing Gold at ISOLDE-CERN

Keywords: ISOLDE, yield, FLUKA, production efficiency

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Introduction

Isotope separation on-line (ISOL) facilities exploit thick targets to produce radioactive isotope beams which are used for studies of fundamental nuclear properties and for other purposes such as material science and nuclear medicine applications [START_REF] Blumenfeld | Facilities and methods for radioactive ion beam production[END_REF].

Typically in the thick-target ISOL method which is the subject of this paper, high-energy (hundreds of MeV and more) proton beams impinge on targets to induce a wide range of nuclear reactions, producing a large variety of stable and radioactive isotopes. Those isotopes are subsequently extracted from the target matrix by diffusion at high temperature, transported to the ion source (effusion process), ionized, accelerated to energies of 10 -100 keV, and finally separated by their mass-tocharge ratio through a dipole magnet. The range of elements is crucially dependent on specific chemical and thermodynamic properties, which define the release pattern from the target. In the best case, the isotopes with half-lives down to a few milliseconds can be delivered to experiments [START_REF] Borge | ISOLDE past, present and future[END_REF].

The knowledge of the yields of the radioactive isotopes at ISOL installations plays a key role in preparing and planning future experiments. It is therefore important to develop ade-quate models that enable the yields of isotopes with unknown production rates to be estimated. To check the reliability of these models and to improve their predictability, they need to be validated with experimental production yields. Comprehensive yield databases exist for more than 70 elements from two thick-target ISOL facilities: ISOLDE (CERN) [START_REF] Ballof | The upgraded ISOLDE yield database -a new tool to predict beam intensities[END_REF][START_REF]The ISOLDE Yield Database[END_REF] and ISAC (TRIUMF) [START_REF] Kunz | ISAC Yield Database[END_REF].

The ISOLDE database also provides in-target production estimation by means of the ABRABLA [START_REF] Schmidt | Production of neutron-rich heavy residues and the freeze-out temperature in the fragmentation of relativistic 238 U projectiles determined by the isospin thermometer[END_REF][START_REF] Kelić | Abla07 -towards a complete description of the decay channels of a nuclear system from spontaneous fission to multifragmentation[END_REF] and FLUKA-CERN [START_REF]FLUKA-CERN Website[END_REF][START_REF] Böhlen | The FLUKA code: Developments and challenges for high energy and medical applications[END_REF][START_REF] Battistoni | Overview of the FLUKA code[END_REF] codes. At the same time, extensive efforts are underway to build a self-consistent approach for the description of the release from a thick target (see, for example, Refs. [START_REF] Bennett | Delays in thick targets[END_REF][START_REF] Lukić | Systematic comparison of ISOLDE-SC yields with calculated in-target production rates[END_REF][START_REF] Barzakh | Secondary neutrons as the main source of neutron-rich fission products in the bombardment of a thick U target by 1 GeV protons[END_REF][START_REF] Ballof | The upgraded ISOLDE yield database -a new tool to predict beam intensities[END_REF][START_REF] Cocolios | Resonant laser ionization of polonium at rilis-isolde for the study of ground-and isomer-state properties[END_REF][START_REF] Stora | A high intensity 6 he beam for the β-beam neutrino oscillation facility[END_REF] and references therein).

For non-volatile elements (such as Ti, Hf, Ir, Pt etc.), difficulties arise to produce radioactive ion beams using the ISOL technique because of their comparatively long release times from a thick target [START_REF] Köster | Im-)possible ISOL beams[END_REF]. As a result, there are marked gaps in the yield information for isotopes of these elements. Gold belongs to this category and until now the yields for only two gold isotopes, 201,202 Au, have been reported from ISOLDE [START_REF] Stora | Recent developments of target and ion sources to produce ISOL beams[END_REF]. However, to provide a benchmark for testing the release models which were applied primarily to the elements with relatively short release time [START_REF] Lukić | Systematic comparison of ISOLDE-SC yields with calculated in-target production rates[END_REF], experimental yields are required.

In the present work we report the yields for 18 radioactive gold nuclei produced in the thick uranium target at ISOLDE by 1.4-GeV protons. The measured yields allowed us to check the consistency of the in-target production predictions by the FLUKA-CERN code and to test the applicability of the different release models.

The investigation presented in this paper is a part of our recent experimental campaign at the ISOLDE facility aimed at the nuclear spectroscopy and shape coexistence studies of the neutron-deficient gold isotopes by means of laser ionization spectroscopy. Partial results were reported in Refs. [START_REF] Cubiss | Change in structure between the I = 1/2 states in 181 Tl and 177,179 Au[END_REF][START_REF] Harding | Laserassisted decay spectroscopy for the ground states of 180,182 Au[END_REF][START_REF] Barzakh | Hyperfine anomaly in gold and magnetic moments of I π = 11/2gold isomers[END_REF][START_REF] Cubiss | Laser-assisted decay spectroscopy and mass spectrometry of 178 Au[END_REF][START_REF] Barzakh | Shape coexistence in 187 Au studied by laser spectroscopy[END_REF].

Experimental details

A detailed description of the experiment can be found in Refs. [START_REF] Cubiss | Change in structure between the I = 1/2 states in 181 Tl and 177,179 Au[END_REF][START_REF] Barzakh | Shape coexistence in 187 Au studied by laser spectroscopy[END_REF]. The gold nuclei were produced in spallation reactions induced by the 1.4-GeV proton beam from the CERN PS Booster, impinging on a 50-g cm -2 -thick UC x target (#524) of the General Purpose Separator (GPS) of ISOLDE [START_REF] Catherall | The ISOLDE facility[END_REF]. The target and tantalum ionizer cavity were operated at 2150 • C and 2100 • C, respectively. The reaction products diffused through the target and effused as neutral atoms into the hot cavity of the Resonance Ionization Laser Ion Source (RILIS) [START_REF] Mishin | Chemically selective laser ionsource for the CERN-ISOLDE on-line mass separator facility[END_REF][START_REF] Fedosseev | Ion beam production and study of radioactive isotopes with the laser ion source at ISOLDE[END_REF], where gold atoms were ionized by the laser beams with frequencies tuned to the three-step gold ionization scheme presented in [START_REF] Marsh | Development of a RILIS ionization scheme for gold at ISOLDE, CERN[END_REF]. The ions were then extracted by a 30-keV electrostatic potential and separated according to their mass-to-charge ratio. Following the separation, the ions were transported to either the ISOLTRAP's multireflection time-of-flight mass separator (MR-ToF MS) [START_REF] Mukherjee | ISOLTRAP: An on-line Penning trap for mass spectrometry on short-lived nuclides[END_REF][START_REF] Wolf | ISOLTRAP's multi-reflection time-of-flight mass separator/spectrometer[END_REF], or to the Windmill (WM) system [START_REF] Andreyev | New type of asymmetric fission in proton-rich nuclei[END_REF][START_REF] Seliverstov | Electromagnetic moments of odd-A 193-203,211 Po isotopes[END_REF] for photoion monitoring during the laser wavelength scans. The WM was used for weakly produced, α-decaying isotopes, whereas the MR-ToF MS was applied in the longerlived, β-decaying cases. The hyperfine structure (hfs) and isotope shift (IS) measurements were made upon the 267.7-nm atomic transition in gold (6s 2 S 1/2 → 6p 2 P 1/2 ), by scanning a frequency-tripled Titanium: Sapphire laser in a narrowband mode (bandwidth of ≈ 600 MHz before tripling). Two broadband dye lasers (bandwidth of ≈ 20 GHz) were used for the second and third excitation steps. Details of the scanning procedures can be found in Refs. [START_REF] Cubiss | Change in structure between the I = 1/2 states in 181 Tl and 177,179 Au[END_REF][START_REF] Barzakh | Shape coexistence in 187 Au studied by laser spectroscopy[END_REF][START_REF] Seliverstov | Electromagnetic moments of odd-A 193-203,211 Po isotopes[END_REF].

At the WM, the ion beam entered through the central hole of an annular silicon detector (Si1), and was implanted into one of ten, 20 µg cm -2 -thick carbon foils mounted on a rotatable wheel. A second silicon detector (Si2) was positioned a few millimeters behind the foil being irradiated. Together, Si1+Si2 were used to measure the short-lived α-decay activity at the implantation site. The typical Si1+Si2 α-detection efficiency was 25 -40% depending on the specific experimental conditions.

At the MR-ToF MS, separation of the gold ions from isobaric contaminants was achieved due to their mass difference, which resulted in their temporal separation during 1000 revolutions between the electrostatic mirrors of the device. The ions were then extracted from the cavity of the MR-ToF MS and detected with an electron multiplier. Details of the ISOLTRAP apparatus were extensively covered in Refs. [START_REF] Wolf | ISOLTRAP's multi-reflection time-of-flight mass separator/spectrometer[END_REF][START_REF] Barzakh | Shape coexistence in 187 Au studied by laser spectroscopy[END_REF]. The total transport efficiency between the ISOLDE GPS front end and the MR-ToF MS analysis detector was estimated to be 1% [START_REF] Barzakh | Shape coexistence in 187 Au studied by laser spectroscopy[END_REF]. With direct ion detection and mass-resolving powers in the order of 10 5 , this method offers the advantage being independent of the decay properties of the nuclei under investigation.

Yields of short-lived isotopes measured at the end of the run, coincide in the limit of uncertainties with that obtained at the beginning of the experiment. Thus, we did not observe the target aging.

Experimental results

In Fig. 1 and Table 1 the yields of the investigated gold isotopes are shown. They are linearly normalized to a proton current of 1 µA (the proton current during the measurements was varied between 1.6 -2 µA). Note, that all measurements were done in the narrow-band mode of the RILIS installation. This means that the yields can be increased by a factor of 2 -3 when using RILIS in broad-band mode [START_REF] Andel | Van Duppen, β-delayed fission of isomers in 188 Bi[END_REF]. We preferred not to compare directly the 201,202 Au yields from [START_REF] Stora | Recent developments of target and ion sources to produce ISOL beams[END_REF] with our data as the experimental conditions in the earlier experiment differ substantially from those of the present work (broad-band RILIS mode, different target thickness, different ionizer type etc.)

For 178,177,176 Au g,m the data obtained by the WM decay station were used. The yields were deduced from the α-decay count rate (Si1+Si2) at laser frequencies tuned to the maxima of the hfs, accounting for the α-decay branching ratio, relative intensity of the chosen α line and efficiency of the α-decay detection at the WM (see Refs. [START_REF] Harding | Laserassisted decay spectroscopy for the ground states of 180,182 Au[END_REF][START_REF] Harding | Laser-assisted nuclear decay spectroscopy of[END_REF][START_REF] Cubiss | Laser-assisted decay spectroscopy and mass spectrometry of 178 Au[END_REF] and references therein). For the gold nuclei with A ≥ 178 the hfs spectra measured by the MR-ToF MS were analyzed accounting for the aforementioned transport and detection efficiency. Yields of 178 Au g,m were deduced by both procedures. The results obtained from the WM data are in agreement with those from MR-ToF MS measurements within a factor of two (see Fig. 1). This indicates the reliability of the applied procedures and gives an estimation of the yield systematic uncertainties. In Table 1 the yields of 178 Au g,m from MR-ToF MS measurements are presented. The yield of 190 Au was determined by direct photoion-current measurement with a Faraday cup.

The irregularities in the yields at A = 187, 191, 193 (see Fig. 1) are explained by the comparatively short half-lives of the isomers studied at these mass settings in comparison with the half-lives of the corresponding ground states (we did not measure the long-lived ground states of 187,191,193 Au since their IS and hfs were investigated previously, see Ref. [START_REF] Passler | Quadrupole moments and nuclear shapes of neutrondeficient gold isotopes[END_REF] and references therein). For example, T 1/2 ( 191 Au g ) = 3.18 h and T 1/2 ( 191 Au m ) = 0.92 s.

Calculations with the FLUKA model were made in the present work with the "FLUKA CERN version 4.1" code [8, 9, 10] (see Fig. 1). For that purpose, a full ISOLDE target unit was modelled and exposed to a proton beam of 1.4 GeV with a Gaussian spatial distribution (with standard deviation of 3.5 mm). The beam and target parameters were based on the experimental values. The irradiated container (cylinder of 0.7 cm radius and 19.5 cm length) of the target unit was filled with UC x material (with the following weight fractions: 235 U -0.28%, 238 U -82.7%, Carbon -17.0%) with an apparent density of 3.5 g cm -3 as used during the experiment. During the simulation, the residual nuclei yield was computed taking into account the full evolution of the secondary particle cascade. For the production of isomers, FLUKA uses equal-sharing between the ground and the excited states as the first-order approximation. However, this approximation is very crude: we know that sometimes isomer ratio can reach 10 -10 2 . Therefore, we use total in-target production values, without division on existing isomers. This approach along with the procedure outlined be-low (see Sec. 4.5), enables one in some cases to determine the experimental values of an isomer ratio rather than to rely upon its theoretical predictions.

The difference between the measured yields and predicted in-target production values reflects the efficiency of the laserionization scheme which is the same for all gold isotopes, and the decay losses during the release process which depend on the nuclear half-life. a For isotopes with unknown spin the designation ls (low spin) and hs (high spin) was applied (see Refs. [START_REF] Harding | Laser-assisted nuclear decay spectroscopy of[END_REF][START_REF] Cubiss | Laser-assisted decay spectroscopy and mass spectrometry of 178 Au[END_REF]). Spins indicated between brackets are tentative.

Description of production efficiency

The ratio of the experimentally observed yield, Y exp ( A Au), and the in-target production, Y in-target ( A Au), will be denoted as a production efficiency, ǫ tot = Y exp ( A Au)/Y in-target ( A Au). Several models exist to describe the release process and, in particular, the production efficiency of the radioactive species from the target (see also [START_REF] Ballof | The upgraded ISOLDE yield database -a new tool to predict beam intensities[END_REF] and references therein).

Model A

The diffusion-effusion (DE) model is based on the solution of the diffusion equations (see Refs. [START_REF] Ravn | New techniques at ISOLDE-2[END_REF][START_REF] Fujioka | Diffusion of radioisotopes from solids in the form of foils, fibers and particles[END_REF][START_REF] Bennett | Delays in thick targets[END_REF][START_REF] Barzakh | Secondary neutrons as the main source of neutron-rich fission products in the bombardment of a thick U target by 1 GeV protons[END_REF]). In this model the production efficiency of an isotope with a half-life T 1/2 can be presented as follows:

ǫ tot (T 1/2 ) = ǫ diff (T 1/2 )ǫ eff (T 1/2 )ǫ ioniz , (1)
where ǫ diff , ǫ eff , ǫ ioniz are the diffusion, effusion and ionization efficiencies.

Diffusion and effusion efficiencies depend on the diffusion and effusion time constants t d and t e [START_REF] Ravn | New techniques at ISOLDE-2[END_REF][START_REF] Fujioka | Diffusion of radioisotopes from solids in the form of foils, fibers and particles[END_REF]:

ǫ eff = T 1/2 T 1/2 + t e , (2)
ǫ diff = 3δ 1/2 coth(δ -1/2 ) -δ 1/2 , (3)
where δ = T 1/2 /(π 2 t d ).

Model B

In Ref. [START_REF] Lukić | Systematic comparison of ISOLDE-SC yields with calculated in-target production rates[END_REF] an empirical function for the description of the dependence of the production efficiency on the isotope half-life was proposed and successfully applied to a large number of isotopic chains:

ǫ tot (T 1/2 ) = ǫ ioniz 1 1 + T 1/2 t 0 -α . (4)
where α and t 0 are adjustable parameters. This function may be considered as a simplified version of the DE relations: at α = 1 it coincides with ǫ eff ǫ ioniz , at α ≈ 1.5 it closely approximates ǫ diff ǫ eff ǫ ioniz . Its advantage is the decrease of the number of free parameters which is essential in the case of limited experimental information.

Model C

The third approach is based on the empirical delay function p(t) describing the probability that a stable isotope produced by the protons at t = 0 is released at time t [START_REF] Lettry | Pulse shape of the ISOLDE radioactive ion beams[END_REF] :

p(t) = 1 N 1 -e -λ r t γe -λ f t + (1 -γ)e -λ s t , (5)
where N is a normalizing coefficient, λ r , λ f , λ s are related to the rise, fast-fall and slow-fall time constants t r , t f , t s [λ r = ln(2)/t r etc.] and γ is a weighting factor between the slow and fast release components. This four-parameter function satisfactorily describes the experimental release curves whereby the yield was measured as a function of time elapsed after the proton beam pulse impinged on a target. In our analysis we neglect the fast component (i. e. we assume that γ ≡ 0) since the fluctuations of the experimental points and the lack of the data for isotopes with T 1/2 <1 s do not allow a reasonable fit of a function with so many free parameters. Then, the production efficiency for radioactive isotope is given by:

ǫ tot = ǫ ioniz ∞ 0 p(t)e -λt dt = ǫ ioniz λ s (λ r + λ s ) (λ + λ s )(λ + λ s + λ r ) , (6)
where λ = ln(2)/T 1/2 . In the data fitting by various models we used chisquared minimization procedure performed with a Levenberg-Marquardt algorithm.

Accounting for the parent in-target decay

The production efficiency ǫ tot (T 1/2 ) depends on the ionization efficiency ǫ ioniz , which is the same for all isotopes and on the decay losses due to the finite decay time. Evidently, ǫ tot should be a smooth function of T 1/2 .

However, there are several factors which might cause irregularities in the experimental data. First of all, the neutrondeficient gold isotopes studied in this work can be produced in the target by the β + /EC decay of the isobaric mercury precursors. Indeed, the in-target production of mercury isotopes is comparable with or larger than that of the isobaric gold isotopes at A > 183. Thus, in these cases the influence of the parent intarget decay might be, in principle, non-negligible. The fraction of the mercury isotopes which decay in the target, depends on the release efficiency for these isotopes. The observed yield of the gold isotope A Au, Y( A Au) obs , consists of two parts: ions of gold, produced by the direct reaction in the target, Y( A Au) direct , and ions of gold, produced by the in-target β decay of the isobaric mercury isotopes, Y( A Au) decay :

Y( A Au) obs = Y( A Au) direct + Y( A Au) decay = Y( A Au) in-target ǫ tot ( A Au)+ 1 - ǫ tot ( A Hg) ǫ ioniz (Hg) Y( A Hg) in-target ǫ tot ( A Au). ( 7
)
In order to estimate the possible contribution of the mercury in-target decay, we evaluated the production efficiency for mercury isotopes. Yield information for the selected mercury isotopes (A = 177, 178, 180, 181) produced in the UC x target was taken from Ref. [START_REF] Sels | Shape staggering of midshell mercury isotopes from in-source laser spectroscopy compared with density-functional-theory and Monte Carlo shell-model calculations[END_REF], and the in-target production was calculated by the FLUKA-CERN code. Note, that temperature parameters during the yield measurements for mercury isotopes were the same as in the present study.

The production efficiencies for mercury isotopes are shown in Fig. 2. The dependence of ǫ tot on T 1/2 was fitted by Eq. [START_REF] Blumenfeld | Facilities and methods for radioactive ion beam production[END_REF] and Eq. (4) (Models A and B, respectively) with the ionization efficiency fixed according to the off-line measurements in Ref. [START_REF] Goodacre | RILIS-ionized mercury and tellurium beams at ISOLDE CERN[END_REF], ǫ ioniz = 0.06. As seen in Fig. 2 the diffusion-effusion model noticeably underestimates the speed of decrease of the production efficiency for short-lived isotopes, whereas empirical Model B satisfactorily describes the experimental data.

We estimated Y( A Au) decay with deduced release parameters [Model B: t 0 = 45(14) s, α = 2.0(2)]. It contributes ≈ 30% of the total yield for 181,182,183 Au, 10% for 180 Au and less than 1% for other gold isotopes. The corresponding corrections were taken into account in the efficiency-curve analysis. The possible variation of the mercury ionization efficiency ǫ ioniz in the limits of a factor of two, will lead to only a negligible change of this contribution (less than 1%).

Another source of the in-target decay is the α decay of thallium isotopes with 4 more mass units. However, this would only be possible for the lightest isotopes in our data set, for which the production of thallium is at most comparable to that of gold. Considering the small α-decay branching ratio for these thallium isotopes, the impact of these in-target decays will be even less than in the case of mercury. Thus, one can neglect the influence of the parent in-target α decays on the production efficiency of the studied gold isotopes.

Accounting for the presence of the long-lived isomers

As seen in Table 1, at A = 176, 177, 178 the yields of both ground and long-lived isomeric states were independently determined. However, calculation of the in-target production gives the sum of the production rates for all long-lived states of the nucleus in question. One cannot simply add the measured yields of the ground and isomeric states, since they have different half-lives and, correspondingly, different production efficiencies ǫ tot (T 1/2,g ) and ǫ tot (T 1/2,m ):

Y( A Au g ) = Y in-target ( A Au)βǫ tot (T 1/2,g ), (8)
Y( A Au m ) = Y in-target ( A Au)(1 -β)ǫ tot (T 1/2,m ), ( 9
)
where β is the (unknown) fraction of A Au nuclei in ground state, produced by spallation reaction in the UC x target. Thus, the respective production efficiency can be determined by the iterative procedure with the relation derived from Eq. ( 9) after substitution of β from Eq. ( 8):

ǫ tot (T 1/2,g ) = Y( A Au g ) + Y( A Au m ) ǫ tot (T 1/2,g ) ǫ tot (T 1/2,m ) /Y in-target .
(10) Namely, the production efficiency dependence on T 1/2 should be fitted without taking into account the yield for the A Au nucleus. Then the ratio ǫ tot (T 1/2,g ) ǫ tot (T 1/2,m ) should be calculated with the obtained release parameters. Inserting this ratio in Eq. [START_REF] Battistoni | Overview of the FLUKA code[END_REF], one obtains ǫ tot (T 1/2 , g). This value should be included in the next iteration of the fitting procedure and so on until a self-consistent result is obtained. Note, that Eqs. [START_REF]FLUKA-CERN Website[END_REF][START_REF] Böhlen | The FLUKA code: Developments and challenges for high energy and medical applications[END_REF][START_REF] Battistoni | Overview of the FLUKA code[END_REF] are valid only when internal transitions are absent. This is the case for 176,177,178 Au m .

Production efficiency for neutron-deficient gold isotopes

General considerations

In Fig. 3 the production efficiency of the studied ground and isomeric states of gold isotopes is presented as a function of the half-life of the corresponding nuclei. 187,191,193,196,198 Au were not included in the fits (see text for details). Fits by the Model A, B and C are shown by dotted, full and dashed lines, respectively.

One can see that in Fig. 3 all points but those for 187,191,193,196,198 Au follow the same systematic trend. The deviation for 196,198 Au g can be explained by the presence of longlived high-spin isomers in these nuclei ( 196 Au m , T 1/2 = 9.6 h, I = 12; 198 Au m , T 1/2 = 2.3 d, I = 12). The behavior of the 196,198 Au g production efficiencies may be regarded as an indication that the yields of the corresponding long-lived high-spin isomers are larger than those for the low-spin ground states by at least a factor of ten. In this case one can expect that after adding the production efficiencies of these isomers to the measured efficiencies of the ground states (taking into account the difference in the half-lives as shown in Sec. 4.5) the drop of the 196,198 Au g points will disappear. Note, that 190 Au also has a high-spin isomer (T 1/2 = 125 ms, I = 11), but due to its short half-life and the predominance of the IT branch, it decays almost completely to the low-spin ground state in the target and the observed production efficiency of 190 Au g comprises the sum of the production efficiencies of both isomer and ground state. Correspondingly, there is no marked deviation from the common trend for 190 Au g . In the fitting procedures, the (incomplete) data for 196,198 Au g were ignored.

However, there are three additional points outside the common trend (A = 187, 191, 193) in the efficiency curve (see Fig. 3). The deviation of these points from the systematic trend amounts to a factor of 220 for 187 Au, a factor of 290 for 191 Au and a factor of 40 for 193 Au. Note, that accounting for the yield of the ground states of these nuclei (which were not measured in our experiment) would increase this deviation. The additional experimental and theoretical studies are needed in order to explain the observed irregularities.

Comparison with theoretical models

Despite the clear irregularities in the efficiency curve at A = 187, 191, 193 (see Fig. 3), the data for other isotopes (apart from 187,191,193,196,198 Au) are described well by the Model B with the following parameters: ǫ ioniz = 0.04(2), t 0 = 73(21) s, α = 2.5(2) (see Fig. 3). The obtained ionization efficiency matches well with the off-line estimation ǫ ioniz (Au) > 0.03 [START_REF] Marsh | Development of a RILIS ionization scheme for gold at ISOLDE, CERN[END_REF]. At the same time Model A (ǫ ioniz = 0.06, t r = 100 s, t s = 600 s) noticeably underestimates the slope of the efficiency curve for the shortlived gold isotopes, whereas Model C (ǫ ioniz = 0.06, t diff = 400 s, t eff = 300 s) poorly describes the "intermediate" isotopes with 10 s < T 1/2 <100 s.

Conclusions

To summarize, the production yield of radioactive ion beams for 18 gold nuclei produced in the thick uranium target at ISOLDE by 1.4-GeV protons and ionized by RILIS, were measured. The production-efficiency dependence on the half-life was established by using the FLUKA-CERN in-target production values. The deviations from the common trend of this dependence for 196,198 Au were explained by the assumption of the stronger production of the long-lived high-spin isomers in comparison with their low-spin ground states.

On the other hand, the observed irregularities in the ǫ tot half-life dependence for the long-lived high-spin gold isomers (A = 187, 191, 193) demand additional experimental and theoretical efforts in order to explain this deviation and establish reliable release model. First of all, yield measurements for gold should be continued with special emphasis on the isotopes having long-lived isomer states. It would be important to fill the gaps in our data (183 < A < 198) in order to check the consistency of the theoretical prediction in the specific mass region. It would be instructive also to compare the FLUKA results with the predictions of other transport codes. Measuring the release curve could further improve our understanding of the processes involved and predictions for other nuclides.

Figure 1 :

 1 Figure 1: Yields of gold isotopes measured in this study in narrow-band RILIS mode. Upward triangles and squares represent the WM and MR-ToF measurements, respectively. Full and hollow symbols correspond to ground states and isomers, respectively. Circles show the in-target production predicted by the FLUKA-CERN model.

Figure 2 :

 2 Figure2: Production efficiency of the neutron deficient mercury isotopes from the UC x target. Yields were taken from[START_REF] Sels | Shape staggering of midshell mercury isotopes from in-source laser spectroscopy compared with density-functional-theory and Monte Carlo shell-model calculations[END_REF]. In-target production was calculated by the FLUKA-CERN code. Fits by Models A (diffusion and effusion) and B are shown by dashed and full lines, respectively.

Figure 3 :

 3 Figure3: Production efficiency of the gold isotopes released from the thick UC x target, when the in-target production was calculated with the FLUKA-CERN code. The data for187,191,193,196,198 Au were not included in the fits (see text for details). Fits by the Model A, B and C are shown by dotted, full and dashed lines, respectively.

Table 1 :

 1 Yields of gold isotopes at ISOLDE measured in this study.

	A	T 1/2	I a	Yield (ions/µC)
	176(m1)	1.36 s	hs	1.5E+01
	176(m2)	1.05 s	ls	7.2E+00
	177(g)	1.5 s	1/2	4.1E+01
	177(m)	1.19 s	(11/2)	2.5E+01
	178(g)	3.4 s	ls	4.4E+02
	178(m)	2.7 s	hs	2.9E+03
	179	7.1 s	1/2	2.6E+04
	180	8.1 s	(1)	2.9E+05
	181	13.7 s	(3/2)	1.5E+06
	182	15.5 s	(2)	8.8E+06
	183	42.8 s	(5/2)	2.2E+07
	187(m)	2.3 s	9/2	2.2E+06
	190	42.8 min	1	1.4E+07
	191(m)	0.92 s	(11/2)	4.1E+04
	193(m)	3.9 s	11/2	5.7E+04
	195(m)	30.5 s	11/2	1.7E+05
	196(g)	6.2 d	2	1.7E+04
	198(g)	2.7 d	2	1.4E+03

Preprint submitted to Nuclear Instruments and Methods in Physics ResearchDecember[START_REF] Stora | A high intensity 6 he beam for the β-beam neutrino oscillation facility[END_REF] 2021

Acknowledgments

We thank A. Kelić-Heil for the discussion on the ABRABLA and FLUKA calculations. This work was done with support from the European Union's Horizon 2020 Framework research and innovation programme under Grants No. 654002 (ENSAR2) and No. 665779 (CERN-COFUND) and No. 861198 (LISA), by RFBR according to the Research Project No. 19-02-00005, by grants from the UK Science and Technology Facilities Council (STFC), by FWO-Vlaanderen (Belgium), by Contracts No. GOA/2010/010 and No. STG/15/031 (BOF KU Leuven), by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office (BriX network P7/12), by the Slovak Research and Development Agency (Contract No. APVV-18-0268), by the Slovak Grant Agency VEGA (Contract No. 1/0651/21), and by the German Federal Ministry of Education and Research (BMBF Contracts No. 05P15HGCIA, and No. 05P18HGCIA). This project has received funding through the European Union's Seventh Framework Programme for Research and Technological Development under Grants No. 262010 (ENSAR), No. 267194 (COFUND), and No. 289191 (LA 3 NET).


François Guillot

Alexandre Guille

Sandra Ventalon

par

est destinée au dépôt

1. Laboratoire d'Océanologie et de Géosciences (UMR8187 Univ. Lille-Univ. Littoral Côte d'Opale-CNRS-IRD), Univ.Lille, bâtiment Sciences de la Terre-SN5, 59650 VILLENEUVE D'ASCQ (France) -2. Atelier de litholamellage, Département des Sciences de la Terre, Univ. Lille, même adresse *francois.guillot@univ-lille.fr

Six échantillons de roche ont été prélevés, le 21 août 2021 (tabl. 1), situés (fig. 1) sur un extrait de la carte IGN et sur la vue de satellite correspondante. Ceux signalés comme "en place" ont été dégagés au marteau, les autres ramassés sur un terril. Leur poids ne dépasse pas 3 kg par échantillon. Ils ont été traités (sciage, plaques minces, analyses) au laboratoire et les premières constatations faites sur ces échantillons sont résumées dans ce qui suit.

Site étudié

Ces prélèvements concernent la mine du Carro et ses abords. Il s'agit d'une installation minière abandonnée, non répertoriée sur les cartes, semblant avoir exploité du minerai de fer sous forme de sidérite (carbonate de fer, FeCO 3 ). Le site comprend un ancien puits aujourd'hui comblé, à côté de ruines de bâtiments en pierre, le tout à la source d'un épandage de déblais de couleur rouille le long du vallon voisin. Ces déblais forment même un terril d'une dizaine de mètres de haut entre les ruines et le puits comblé.

Situé à 2850 m d'altitude, ce site minier n'a pas été signalé à notre connaissance dans la littérature géologique [START_REF] Bertrand | Étude géologique de la partie française du massif du Grand-Paradis (région de Bonneval-sur-Arc, Vanoise orientale)[END_REF][START_REF] Deville | Feuille Tignes. Carte géol. Fr[END_REF] et n'est pas mentionné sur la carte IGN. Le massif du Grand Paradis (au sens géologique), auquel ce site appartient, recèle plusieurs anciennes excavations minières abandonnées depuis environ un siècle, qui étaient exploitées comme gisements de fer. Les anciennes mines situées en Italie ont été recensées par [START_REF] Cevales | I giacimenti minerari del Gran Paradiso[END_REF], la plus proche étant celle de la Cuccagna (2750 m), dans une montagne dominant Ceresole Reale. On peut supposer, en attente de recherches d'historiens dans les archives minières de Savoie et du Piedmont, que la mine du Carro faisait partie de ces exploitations et aurait été active notamment avant le rattachement de la Savoie à la France en 1860.

Description des échantillons

L'échantillon G2109 est conservé à titre de témoin, semblant représentatif du gisement par sa minéralogie et sa structure (fig. 2). Avec les cinq autres échantillons nous avons confectionné des plaques minces (fig. 3 à fig. 8) pour l'étude microscopique de ces roches. G2110, G211 et G2112 (fig. 3 à 5) ont été prélevées sur la croupe partant au SSW de la mine, le long d'un affleurement de roches à patine rouille prononcée (visible sur la photo satellite, fig. 1). L'examen microscopique des trois plaques minces révèle des amphibolites. Leur patine rouille semble localisée dans des fissures tardives, parfois denses, postérieures au litage fin de la roche, nommé foliation, qu'elles recoupent et décalent.

G2113 (équivalent du témoin G2109) est du minerai (fig. 6), blanc et rouge sombre, couleurs correspondant respectivement à de la barytine (sulfate de baryum) et à de la sidérite (carbonate de fer). Au microscope (fig. 7) des cristaux de quartz inframillimétriques, bien formés, sont fréquents ainsi que de la pyrite et des paillettes de mica blanc magnésien.

Fig. 1 .

 1 Fig. 1.-Localisation des échantillons. Fonds cartographique et photographique copiés sur https://www.geoportail.gouv.fr . La photographie satellite montre des traînées colorées, déblais miniers rougeâtres le long du ruisseau au SE du puits, et niveaux rouille échantillonnés au S de la mine.

Fig. 2 .

 2 Fig. 2. Échantillon-témoin du minerai de la mine du Carro. Des yeux brun-rouge de sidérite cristalline sont soulignés par des bandes blanches de barytine. Ces yeux semblent résulter d'une déformation par aplatissement et/ou cisaillement.

Fig. 3 .

 3 Fig. 3. Plaque mince de l'échantillon G2110. Foliation (vert pâle) recoupée par fractures postérieures (rousses) à oxydes de fer.

Fig. 4 .

 4 Fig. 4. Plaque mince de l'échantillon G2111. Foliation subhorizontale (verte) avec fentes (rousses) à oxydes de fer.

Fig. 5 .

 5 Fig. 5. Plaque mince de l'échantillon G2112. Comme en fig. 3 et 4, avec des gerbes vert pâle d'amphibole dans les fentes (à g. du centre)

Fig. 7 .

 7 Fig. 7. Aspects microscopiques du minerai G2113. Site observé localisé en fig. 6. A, vue en lumière transmise polarisée. B, vue en lumière polarisée analysée. C, vue en lumière réfléchie. D, carte électronique en électrons rétrodiffusés, en couleurs choisies pour individualiser les minéraux par leurs chimie: qz, SiO2, quartz (bleu sombre). ba, BaSO4, barytine (rose pâle). py, FeS2, pyrite (vert bleuté). si, FeCO3, sidérite (vert sombre). mi, K2Al4MgSi7O20(OH)4, mica blanc magnésien (rose tyrien). La vue B montre que la sidérite (transparente) est en partie transformée en oxyde de fer (hématite, Fe2O3), rouge sang sous cet éclairage.

Fig. 8 .

 8 Fig. 8. Plaque mince de l'échantillon G2114. Sur un fond à grain fin blanc (de quartz) et vert pâle (d'albite), taches sombres de grenat et chlorite. Des carbonates sont présents dans les fentes visibles en bas.

  Tabl. 1.-Échantillons collectés le 21/08/2021 en zone centrale du Parc, commune de Bonneval-sur-Arc.

	numéro	destination altitude (m)	lat. decim. (°)	lon. decim. (°)	qualité	localisation formation	description
	G2109	témoin	2852	45,424194 7,121818	terril minier	terril, ancienne mine de fer du Carro	minerai	yeux 5 cm de sidérite étirés dans alternances cm sidérite/barytine poecillitique en cristaux pluricentimétriques
	G2110	plaque mince polie	2820	45,421694 7,121252	en place	300 m à N325 du refuge du Carro	niveau minéralisé rouille proche de la mine	brèche massive ferrugineuse de fragments vert pâle, à géodes de chlorite, mica blanc, et mouches rouillées (de FeCO3?)
	G2111	plaque mince polie	2832	45,422416 7,121673	en place	400 m à N345 du refuge du Carro	niveau minéralisé rouille mine proche de la	schiste vert-bleuté fin à patine noire ou rouille, mouches rouillées
	G2112	plaque mince polie	2842	45,422989 7,121742	en place	450 m à N350 du refuge du Carro	niveau minéralisé rouille proche de la mine	lentille bréchoïde à patine rouille à noir, à rosettes fibreuses vert pâle cm
	G2113	plaque mince polie	2852	45,424194 7,121818	terril minier	terril, ancienne mine de fer du Carro	minerai	yeux 2 cm de sidérite étirés dans alternances cm sidérite/barytine
	G2114	plaque mince polie	2854	45,424199 7,121719	en place	toit de l'ancienne mine de fer du Carro	micaschiste sombre épais de 2 m au toit de la mine	micaschiste roussâtre et pulvérulent, légèrement effervescent à HCl


Fernanda S Valdovinos
email: fvaldovinos@ucdavis.edu

Kayla R S Hale

Sabine Dritz

Paul R Glaum

Kevin S Mccann

Sophia M Simon

Elisa Thébault

William C Wetzel

Kate L Wootton

Justin D Yeakel

A bioenergetic framework for aboveground terrestrial food webs

Keywords: Fast-slow continuum, plant defenses, tissue-specialist herbivores, plant mutualists, consumer-resource, stage structure

Bioenergetic approaches have been greatly influential for understanding community functioning and stability and predicting effects of environmental changes on biodiversity. These approaches use allometric relationships to establish species' trophic interactions and consumption rates, and have been most successfully applied to aquatic ecosystems. Terrestrial ecosystems, where body mass is less predictive of plant-consumer interactions, present inherent challenges that these models have yet to meet. Here, we discuss the processes governing terrestrial plant-consumer interactions and develop a bioenergetic framework integrating those processes. Our framework integrates bioenergetics specific to terrestrial plants and their consumers within a food-web approach. It also considers mutualistic interactions, advancing understanding of terrestrial food webs and predictions of their responses to environmental changes.

Ecology needs a terrestrial bioenergetic approach

Bioenergetic food web approaches (see Glossary) [START_REF] Yodzis | Body Size and Consumer-Resource Dynamics[END_REF][START_REF] Brose | Allometric scaling enhances stability in complex food webs[END_REF] have fueled an industry of ecological research [START_REF] Schneider | Animal diversity and ecosystem functioning in dynamic food webs[END_REF][START_REF] Martinez | Allometric Trophic Networks From Individuals to Socio-Ecosystems: Consumer-Resource Theory of the Ecological Elephant in the Room[END_REF][START_REF] Curtsdotter | Ecosystem function in predator-prey food webs-confronting dynamic models with empirical data[END_REF][START_REF] Ávila-Thieme | Alteration of coastal productivity and artisanal fisheries interact to affect a marine food web[END_REF][START_REF] Glaum | Integrating economic dynamics into ecological networks: The case of fishery sustainability[END_REF][START_REF] Wootton | Towards a modular theory of trophic interactions[END_REF][START_REF] Gauzens | Biodiversity of intertidal food webs in response to warming across latitudes[END_REF][START_REF] Brose | Predator traits determine food-web architecture across ecosystems[END_REF]. However the inherent focus on body size has resulted in an approach less suitable for exploring empirical patterns in terrestrial systems [START_REF] Potapov | Trophic Position of Consumers and Size Structure of Food Webs across Aquatic and Terrestrial Ecosystems[END_REF][START_REF] Shurin | All wet or dried up? Real differences between aquatic and terrestrial food webs[END_REF], especially plant-consumer (herbivore, mutualist) interactions, which are often determined by factors other than body size [START_REF] Schneider | Animal diversity and ecosystem functioning in dynamic food webs[END_REF][START_REF] Brose | Predator traits determine food-web architecture across ecosystems[END_REF][START_REF] Shurin | All wet or dried up? Real differences between aquatic and terrestrial food webs[END_REF]. With the increased use of bioenergetic approaches to understand complex outcomes of global change [START_REF] Martinez | Allometric Trophic Networks From Individuals to Socio-Ecosystems: Consumer-Resource Theory of the Ecological Elephant in the Room[END_REF][START_REF] Ávila-Thieme | Alteration of coastal productivity and artisanal fisheries interact to affect a marine food web[END_REF][START_REF] Glaum | Integrating economic dynamics into ecological networks: The case of fishery sustainability[END_REF][START_REF] Gauzens | Biodiversity of intertidal food webs in response to warming across latitudes[END_REF][START_REF] Donohue | Loss of predator species, not intermediate consumers, triggers rapid and dramatic extinction cascades[END_REF], there is increasing need for a holistic bioenergetic framework that addresses the challenges introduced in terrestrial above-ground ecosystems. Here we review previous efforts to capture the mechanistic processes governing aboveground plant-consumer interactions, and develop a conceptual and mathematical guide for integrating these processes into a framework that is established on bioenergetic constraints.

Terrestrial plant-consumer interactions are mostly determined by traits external to body mass -a problematic characteristic to apply to many plant species -such as phytochemistry [START_REF] Thaler | Cross-talk between jasmonate and salicylate plant defense pathways: effects on several plant parasites[END_REF][START_REF] Rasmann | Induced Responses to Herbivory and Jasmonate in Three Milkweed Species[END_REF][START_REF] Mallinger | Methyl Salicylate Attracts Natural Enemies and Reduces Populations of Soybean Aphids (Hemiptera: Aphididae) in Soybean Agroecosystems[END_REF] and morphology of physical structures such as flowers [START_REF] Eklöf | The dimensionality of ecological networks[END_REF][START_REF] Albrecht | Reward regulation in plant-frugivore networks requires only weak cues[END_REF][START_REF] Pichler | Machine learning algorithms to infer traitmatching and predict species interactions in ecological networks[END_REF]. These characteristics, rather than body size, matter most to consumers that range from leaf galling arthropods to large mammal grazing, as well as mutualists consuming floral rewards and fruits [START_REF] Herrera | Plant Animal Interactions: An Evolutionary Approach[END_REF]. Terrestrial plants also exhibit large variation in tissue growth and turnover to build structures that not only attract and repel herbivores and mutualists [21], but that serve to fight gravity in a race for space and light, relationships that defy traditional bioenergetic approaches. Consequently, our understanding of community stability and ecosystem functioning that is obtained through the use of bioenergetic models is by definition biased toward aquatic systems, where trophic interactions and consumption rates tend to scale with organismal body mass [START_REF] Brose | Predator traits determine food-web architecture across ecosystems[END_REF][START_REF] Shurin | All wet or dried up? Real differences between aquatic and terrestrial food webs[END_REF]. Additionally, food web theory has traditionally emphasized the consumer perspective, reflected in, for example, the greater detail in the functional responses of consumers compared to those of primary producers, or the focus on consumer adaptive foraging rather than the adaptive response of resources against consumption [START_REF] Valdovinos | Consequences of adaptive behaviour for the structure and dynamics of food webs[END_REF]. These emphases have resulted in over-simplistic models of plant growth and the trait-mediated responses of plants to herbivore attack [START_REF] Yodzis | Body Size and Consumer-Resource Dynamics[END_REF][START_REF] Curtsdotter | Ecosystem function in predator-prey food webs-confronting dynamic models with empirical data[END_REF][START_REF] Ávila-Thieme | Alteration of coastal productivity and artisanal fisheries interact to affect a marine food web[END_REF], potentially biasing our understanding of food web dynamics from the bottom up. This complexity of aboveground terrestrial plant-consumer interactions requires a deeper consideration of their unique processes giving rise to communities. We review the literature on the bioenergetics of plant-animal interactions, and discuss extensions to traditional food web frameworks.

These extensions integrate advances in network analyses, bioenergetics, and the biological mechanisms underlying interactions between plants, their consumers and mutualists.

Terrestrial bioenergetic framework

The dimensional reduction offered by allometric scaling has a rich history in ecology, but harnessing it to analyze species interactions was not seriously examined until the seminal bioenergetic model by Yodzis and Innes [START_REF] Yodzis | Body Size and Consumer-Resource Dynamics[END_REF]. This framework was expanded to communities of interacting species with the Allometric Trophic Network (ATN) model [START_REF] Brose | Allometric scaling enhances stability in complex food webs[END_REF][START_REF] Donohue | Loss of predator species, not intermediate consumers, triggers rapid and dramatic extinction cascades[END_REF] (see Box 1), which models food web dynamics with a minimal number of parameters, namely, the body sizes of the consumer and resource species and a handful of allometric constants [START_REF] Curtsdotter | Ecosystem function in predator-prey food webs-confronting dynamic models with empirical data[END_REF][START_REF] Ávila-Thieme | Alteration of coastal productivity and artisanal fisheries interact to affect a marine food web[END_REF][START_REF] Boit | Mechanistic theory and modelling of complex food-web dynamics in Lake Constance[END_REF][START_REF] Jonsson | Predictive power of food web models based on body size decreases with trophic complexity[END_REF]. This model has demonstrated particular success with respect to aquatic systems, where the presence/absence of trophic interactions and rates of consumption are assumed to scale allometrically (see Box 1, Figure 1), largely due to the gape limitations that constrain so many aquatic consumer interactions [START_REF] Brose | Predator traits determine food-web architecture across ecosystems[END_REF][START_REF] Shurin | All wet or dried up? Real differences between aquatic and terrestrial food webs[END_REF][START_REF] Petchey | Size, foraging, and food web structure[END_REF].

A central tenant of the new perspective we propose (Figure 1) is the notion that plant species can be organized along a fast-slow growth axis [START_REF] Reich | The world-wide 'fast-slow' plant economics spectrum: a traits manifesto[END_REF][START_REF] Salguero-Gómez | Fast-slow continuum and reproductive strategies structure plant life-history variation worldwide[END_REF][START_REF] Adler | Functional traits explain variation in plant life history strategies[END_REF], determining plant mass-specific metabolic rates and ultimately the flow of energy from primary production to higher trophic levels via consumption.

Fast-growing plants invest in photosynthetic machinery at the expense of defenses and structural tissue [START_REF] Jacobsen | Growth rate and life history shape plant resistance to herbivores[END_REF], which itself is defensive because it is difficult to digest. Because of these investments, fast growing plants tend to be leafier, more nutritious for consumers (lower C:N,P ratios and higher tissue digestibility), and less resistant but more tolerant to herbivory. This increased tolerance arises both because their ability to grow quickly allows them to quickly replace lost tissue and because their lack of investment in structure and defenses lowers the per unit cost of their tissue [START_REF] Jacobsen | Growth rate and life history shape plant resistance to herbivores[END_REF]. Slowgrowing plants, in contrast, invest more heavily in structure and defenses that promote the longevity of their tissues and therefore tend to be larger, woodier, less nutritious for consumers, and more resistant but less tolerant of herbivory. This lack of tolerance arises because each bite of tissue is more valuable and more costly to replace [START_REF] Reich | The world-wide 'fast-slow' plant economics spectrum: a traits manifesto[END_REF][START_REF] Salguero-Gómez | Fast-slow continuum and reproductive strategies structure plant life-history variation worldwide[END_REF][START_REF] Adler | Functional traits explain variation in plant life history strategies[END_REF][START_REF] Stowe | The Evolutionary Ecology of Tolerance to Consumer Damage[END_REF]. A fast-slow plant axis thus affects key parameters governing food web dynamics, including herbivore ingestion ( f ji) and assimilation ( e ji ) of plant biomass, foraging effort ( p ji (B)), attack rate ( a ji), and handling time ( h ji) (see Box 1) [START_REF] Thaler | Cross-talk between jasmonate and salicylate plant defense pathways: effects on several plant parasites[END_REF][START_REF] Rasmann | Induced Responses to Herbivory and Jasmonate in Three Milkweed Species[END_REF][START_REF] Mallinger | Methyl Salicylate Attracts Natural Enemies and Reduces Populations of Soybean Aphids (Hemiptera: Aphididae) in Soybean Agroecosystems[END_REF][START_REF] Miller | Plant Reproductive Allocation Predicts Herbivore Dynamics across Spatial and Temporal Scales[END_REF].

Plant structural complexity

Plants have evolved different tissue types to address their simultaneous needs to acquire water and nutrients, photosynthesize, and reproduce [START_REF] Lambers | Plant Physiological Ecology[END_REF], and diverse guilds of herbivores have in turn evolved to consume and at times specialize on them (Figure 2). These tissues include leaves, stems, wood, roots, underground storage organs, seeds, nectar, pollen, and sap/phloem, all of which vary in terms of plant investment, nutritional value, and the cost (or benefit) to the plant if the tissue is consumed [START_REF] Lambers | Plant Physiological Ecology[END_REF] (Figure 2). These differences influence both the biomass available to herbivores and the effect of biomass loss on plant maintenance, growth, and reproduction [START_REF] Thaler | Cross-talk between jasmonate and salicylate plant defense pathways: effects on several plant parasites[END_REF][START_REF] Rasmann | Induced Responses to Herbivory and Jasmonate in Three Milkweed Species[END_REF][START_REF] Mallinger | Methyl Salicylate Attracts Natural Enemies and Reduces Populations of Soybean Aphids (Hemiptera: Aphididae) in Soybean Agroecosystems[END_REF][START_REF] Miller | Plant Reproductive Allocation Predicts Herbivore Dynamics across Spatial and Temporal Scales[END_REF]. Plants are indeterminate growers, such that their allocation to different organs or tissues are often plastic in response to both internal and external factors [START_REF] Lambers | Plant Physiological Ecology[END_REF]. Internal factors include life stages and phenology [START_REF] Cope | The role of timing in intraspecific trait ecology[END_REF], whether the plant has a fast or slow growth strategy [START_REF] Salguero-Gómez | Fast-slow continuum and reproductive strategies structure plant life-history variation worldwide[END_REF][START_REF] Adler | Functional traits explain variation in plant life history strategies[END_REF], and how resistant or tolerant the plant is to herbivory [START_REF] Stowe | The Evolutionary Ecology of Tolerance to Consumer Damage[END_REF][START_REF] Crawley | Herbivory The Dynamics of Animal-Plant Interactions[END_REF]. In contrast, external factors include the effects of environmental pressures (e.g. water and nutrient availability [START_REF] Mundim | Whole-Plant Metabolic Allocation Under Water Stress[END_REF][START_REF] Eziz | Drought effect on plant biomass allocation: A meta-analysis[END_REF]), competition with other plants [START_REF] Turcotte | Phenotypic Plasticity and Species Coexistence[END_REF], and herbivory [START_REF] Pastore | Insect herbivore effects on resource allocation to shoots and roots in Lespedeza capitata[END_REF]. For example, in resource-poor environments, plants may exhibit slower growth rates, altering energetic allocation to different organs [START_REF] Lambers | Plant Physiological Ecology[END_REF] in response to the total energy available to the ecosystem. As a result, profiles of organ proportions differ across environments or seasons [START_REF] Laughlin | Climatic constraints on trait-based forest assembly[END_REF][START_REF] Abramoff | Are above-and below-ground phenology in sync?[END_REF], potentially driving substantial changes in the herbivore community [START_REF] Miller | Plant Reproductive Allocation Predicts Herbivore Dynamics across Spatial and Temporal Scales[END_REF].

The effects of plant structural complexity can be integrated into a bioenergetic food web framework by incorporating the chemical and physical constraints governing the interactions between herbivores and particular plant tissues -as opposed to interactions with plant species or functional groups (Figure 2A). This can be accomplished using either fixed (Figure 2B) or dynamic pool (Figure 2C) approaches. The fixed pool approach assumes the plant biomass is composed of fixed fractions of each tissue, with herbivore groups limited to feeding on that fraction of biomass. Alternative tissues vary in their nutrient composition and thus provide different yields to herbivores. This fixed pool approach incorporates the topological complexity of different animal guilds feeding on different plant tissues without increasing the complexity of the dynamic models. In contrast, the dynamic pool approach allows dynamic allocations to growth and maintenance for each tissue [21,[START_REF] Valdovinos | Adaptive foraging allows the maintenance of biodiversity of pollination networks[END_REF][START_REF] Valdovinos | Niche partitioning due to adaptive foraging reverses effects of nestedness and connectance on pollination network stability[END_REF] at the cost of additional model complexity. Dynamic pools allows for feedbacks between consumption and production of each tissue, such that adaptive foraging behaviors among herbivores [START_REF] Valdovinos | Consequences of adaptive behaviour for the structure and dynamics of food webs[END_REF][START_REF] Valdovinos | Adaptive foraging allows the maintenance of biodiversity of pollination networks[END_REF][START_REF] Valdovinos | Niche partitioning due to adaptive foraging reverses effects of nestedness and connectance on pollination network stability[END_REF][START_REF] Valdovinos | Mutualistic networks: moving closer to a predictive theory[END_REF], in response to the relative availability, cost, and benefit from different tissues, may promote coexistence even when diets are similar.

Herbivore ingestion and assimilation of plant biomass

The proportion of plant biomass (B i ) available for consumption by herbivores (that is, the fraction ingested, B i / f ji, see Box 1) is constrained by plant and herbivore traits. Indeed, much plant-herbivore research (especially for insects) examines how plant defenses, including the vast diversity of phytochemicals and physical traits such as toughness and spinescence, influence f ji and rates of herbivory in general. We review literature on constitutive and inducible defenses in Box 2 and propose to integrate those defenses in our framework as affecting herbivores' consumption parameters ( a ji , e ji , f ji , h ji , p ji) (see Boxes 1 and 2). For ground-based mammals (e.g., ungulates), the proportion of plant biomass available will also depend on the relative height of the plant and the herbivore, because these herbivores can only access tissue within a vertical range roughly spanning ground level to shoulder-height [START_REF] Larramendi | Shoulder Height, Body Mass and Shape of Proboscideans[END_REF]. While plant height might place a physical limit on access, these mammals tend to partition their diets across a relatively lowdimensional plant trait access correlating with nutritional quality [START_REF] Potter | Mechanisms of dietary resource partitioning in large-herbivore assemblages: A plant-trait-based approach[END_REF]. Key to understanding plant-herbivore interactions is that not all green tissue is equally available -physically or biochemically -to herbivores.

Once plant biomass is ingested, plant-consumer interactions are constrained by the efficiency with which herbivores can transform ingested food into new biomass. That is, the assimilation efficiency ( e ji in Eq. 1 of Box 1). We propose expressing this efficiency as yield from the perspective of bulk requirements of consumer-resource interactions [START_REF] Yeakel | Dynamics of starvation and recovery predict extinction risk and both Damuth's law and Cope's rule[END_REF]. The consumer yield (grams of consumer produced per grams of resource consumed) is given by Y ji =M j E D i / E L j , where M j is the body mass of consumer j (g), E D i is the energy density of resource i (Joules/g) and E L j represents the lifetime energetic requirements of a consumer j that reaches maturity (Joules). The resource removed by the consumer is then proportional to the efficiency e ji =d ji B i /Y ji, where d ji is the proportion of digested plant biomass, which must be a function of both plant biochemistry and the herbivore's digestive abilities (see Box 2 and Online Supplemental Information Appendix S1).

Herbivores exhibit species-specific behaviors designed to optimize e ji within particular communities and habitats [START_REF] Simpson | The geometric analysis of feeding and nutrition: a user's guide[END_REF], the result of unique evolutionary trajectories driven by local fitness gradients [START_REF] Pansu | The generality of cryptic dietary niche differences in diverse large-herbivore assemblages[END_REF].

Adaptive behavior of plants and herbivores

Both consumers and resources interact dynamically, adapting the energy allocated to searching for and consuming, attracting, and/or defending against those species with which they interact. Following [START_REF] Valdovinos | Consequences of adaptive behaviour for the structure and dynamics of food webs[END_REF], we define adaptive behavior as the fitness-enhancing changes in individuals' feeding-related traits due to variation in their trophic environment. This includes adaptive foraging of consumers, as well as the adaptive responses of resources, which we define as changes in resource behavior and other traits in response to consumers and environmental cues. Box 3 details a method by which herbivore adaptive foraging and plant adaptive responses can be introduced into a comprehensive bioenergetic ATN framework.

Herbivores adaptively forage in a multi-scale manner [START_REF] Carrasco | Insect host plant selection in complex environments[END_REF] by first searching the landscape for a promising foraging habitat, and then locating particular plant individuals using multiple sensory modalities [START_REF] Dominy | How chimpanzees integrate sensory information to select figs[END_REF], after which a decision to eat them or keep searching is made [START_REF] Pansu | The generality of cryptic dietary niche differences in diverse large-herbivore assemblages[END_REF]. Insects use a diverse array of cues to find their hosts, including habitat context and plant odor and color, after which many sense tissue quality and make feeding decisions using specialized chemoreceptors [START_REF] Carrasco | Insect host plant selection in complex environments[END_REF][START_REF] Bruce | Insect host location: a volatile situation[END_REF]. Ovipositing females also search for places to lay eggs by sensing the leaves with their ovipositor [START_REF] Gripenberg | A meta-analysis of preferenceperformance relationships in phytophagous insects[END_REF].

And while it is clear that herbivores respond to a complex constellation of plant traits and conditions to maximize profitability, plants demonstrate equally dynamic responses to both repel and attract their herbivores.

Chemical defenses are central to the adaptive response to herbivory by plants, with many species upregulating the production of toxins following detection of herbivore damage or other herbivore cues (Box 2). The fast-slow trait axis (T i ) may also affect the response to herbivory of plant species i by influencing its average adaptation rate ( s ' i) or the benefit in per-capita growth rate obtained by its response to herbivore j ( ∂ G i /∂ v ij). Plant adaptive responses also involve mutualistic interactions in terms of attracting the consumers of their herbivores (i.e., indirect defenses) or attracting pollinators and seed dispersers. For example, some plants respond to herbivore cues or attack by releasing volatiles that attract predators or even reward predators with nectar or pearl bodies (concentrations of protein) [START_REF] Pearse | Generalising indirect defence and resistance of plants[END_REF].

Other plants produce chemicals that, after being ingested by herbivores, volatilize from their feces and guide predators to them [START_REF] Weinhold | Trichome-derived O-acyl sugars are a first meal for caterpillars that tags them for predation[END_REF]. Inducible extrafloral nectaries attract ants that then remove herbivores from the plant [START_REF] Marazzi | The diversity, ecology and evolution of extrafloral nectaries: current perspectives and future challenges[END_REF]. Plants also provide predator shelters (domatia; e.g., leaf pits, swollen thorns), the production of which can be upregulated following herbivory [START_REF] Huntzinger | Relaxation of Induced Indirect Defenses of Acacias Following Exclusion of Mammalian Herbivores[END_REF].

Finally, floral rewards and fruits produced by plants to attract pollinators and seed dispersers, respectively, can also be formalized as adaptive responses.

They are resource traits where investment responds directly to consumers and environmental cues, though their role is to attract, rather than repel, the consumer (pollinator, seed disperser) with a potentially positive effect on plant fitness [START_REF] Eklöf | The dimensionality of ecological networks[END_REF][START_REF] Albrecht | Reward regulation in plant-frugivore networks requires only weak cues[END_REF][START_REF] Pichler | Machine learning algorithms to infer traitmatching and predict species interactions in ecological networks[END_REF].

Frequency dependence can play an important role in plant-herbivore adaptive responses. For some species of caterpillars, survival is low when they attack a plant in small groups and high when they attack in larger groups, apparently because plant responses depend on herbivore density [START_REF] Fordyce | Aggregative feeding of pipevine swallowtail larvae enhances hostplant suitability[END_REF]. For example, herbivores that overcome plant defenses by attacking en masse, such as bark beetles, often have aggregation hormones that help them reach high local densities [START_REF] Raffa | The Role of Host Plant Resistance in the Colonization Behavior and Ecology of Bark Beetles (Coleoptera: Scolytidae)[END_REF]. In other systems, negative densitydependence drives dynamics. Herbivores avoid damaged plants because: (i) previously attacked plants are likely to have induced resistance traits [START_REF] Karban | Induced Responses to Herbivory[END_REF], ii) earlier attacking herbivores are likely to have removed the best quality tissue [START_REF] Craig | Intraspecific Competition and Facilitation by a Shoot-Galling Sawfly[END_REF], (iii) to avoid direct interference interactions with competing herbivores [START_REF] Roitberg | Insects that Mark Host Plants[END_REF].

Stage-structure dynamics

Organismal ontogeny can play a significant role in changing species' metabolic rates [START_REF] Maino | Ontogenetic and Interspecific Metabolic Scaling in Insects[END_REF] and interactions [START_REF] Yang | Phenology, ontogeny and the effects of climate change on the timing of species interactions[END_REF], especially for plants [START_REF] Huang | Water content quantitatively affects metabolic rates over the course of plant ontogeny[END_REF]. Species can either consume or be consumed by different species as they grow and mature [START_REF] Rudolf | Stage structure alters how complexity affects stability of ecological networks[END_REF]. Integrating ontogenetic structure into aquatic food web models has had varied effects on food web dynamics, with some showing increased stability [START_REF] De Roos | Dynamic population stage structure due to juvenile-adult asymmetry stabilizes complex ecological communities[END_REF] due to tradeoffs or emergent facilitation [START_REF] De Roos | Stage-specific predator species help each other to persist while competing for a single prey[END_REF], and others showing decreased stability through ontogenetic niche shifts [START_REF] Rudolf | Stage structure alters how complexity affects stability of ecological networks[END_REF]. Terrestrial food web models integrating plant ontogeny remain scarce, though preliminary work indicates the potential for emergent facilitation in certain food web motifs at the autotroph level [START_REF] Glaum | Stage-structured ontogeny in resource populations generates non-additive stabilizing and de-stabilizing forces in populations and communities[END_REF]. The relationship between plant individual growth and defenses [START_REF] Barton | The Ontogeny of Plant Defense and Herbivory: Characterizing General Patterns Using Meta-Analysis[END_REF] can be a way to incorporate plant ontogeny in a more comprehensive bioenergetic framework. Inducible defenses tend to be highest during seedling stages, while constitutive defenses take over with individual growth. In turn, this ontogenetic variability in plant defenses influences herbivores to prefer particular plant stages over others [START_REF] Quintero | Plant and herbivore ontogeny interact to shape the preference, performance and chemical defense of a specialist herbivore[END_REF][START_REF] Stiegel | Environment vs. Plant Ontogeny: Arthropod Herbivory Patterns on European Beech Leaves along the Vertical Gradient of Temperate Forests in Central Germany[END_REF].

Ontogeny interacts with phenology to affect plant-herbivore interactions. In semelparous monocarpic plant species, unique stages are differentially available across the growing season, potentially creating distinct phenological windows of interaction between consumers that would instead be static trophic links without considering ontogeny [START_REF] Yang | Seasonal windows of opportunity in milkweed-monarch interactions[END_REF] (compare Figure S1B with Figure S1C in Supplementary Information). In longer-lived, multi-season, iteroparous plants, seasonally specific growth for younger versus older stages can still open up distinct interaction windows (as in Figure S1C) but with potential cross-generational intraspecific competition.

For example, high adult density limits the survival or maturation rates of younger stages either by restricted access to necessary nutrients [START_REF] Harms | Pervasive density-dependent recruitment enhances seedling diversity in a tropical forest[END_REF] or increased exposure to soil pathogens [START_REF] Janzen | Herbivores and the Number of Tree Species in Tropical Forests[END_REF].

Structure of terrestrial networks

Our bioenergetic framework advocates for a broader definition of food web topology that includes both antagonistic and mutualistic trophic interactions (Figure 2). Food webs typically exclude mutualistic trophic interactions, which limits the analysis of terrestrial food web dynamics [21].

Of the few networks available [START_REF] Brose | Predator traits determine food-web architecture across ecosystems[END_REF], many include only a subset of the local taxa, with uneven levels of taxonomic resolution, often representing the specialties of the investigators. As a consequence, plants and insects tend to be less resolved than vertebrates [START_REF] Baskerville | Spatial Guilds in the Serengeti Food Web Revealed by a Bayesian Group Model[END_REF][START_REF] O'connor | Unveiling the food webs of tetrapods across Europe through the prism of the Eltonian niche[END_REF], potentially biasing our understanding of both structure and dynamics in these systems. Fortunately, recent advances in DNA barcoding from feces and stomach contents provides unprecedented opportunity to increase sampling resolution [START_REF] Schmidt | Interaction webs in arctic ecosystems: Determinants of arctic change?[END_REF].

Despite these challenges, that aquatic and terrestrial food webs reveal clear differences in topological and biomass structures is well understood (see panels iv of Figure 1). Aboveground terrestrial food webs have shorter food chains with more producer biomass and less herbivore biomass and consumption than aquatic food webs with similar net primary productivity, presumably due to the greater structural complexity and lower edibility of terrestrial plant tissues [START_REF] Potapov | Trophic Position of Consumers and Size Structure of Food Webs across Aquatic and Terrestrial Ecosystems[END_REF][START_REF] Shurin | All wet or dried up? Real differences between aquatic and terrestrial food webs[END_REF]. We suggest that a comprehensive bioenergetic framework that includes the unique relationships observed between plants and their consumers (including mutualists) may improve our understanding of where these differences arise.

Generative models of food web topologies [START_REF] Williams | Simple rules yield complex food webs[END_REF][START_REF] Allesina | A General Model for Food Web Structure[END_REF][START_REF] Williams | Success and its limits among structural models of complex food webs[END_REF] offer a powerful means by which food web bioenergetic dynamics can be explored, given the inherent difficulty of collecting food web data. These phenomenological models generate topologies with broadly similar properties compared to empirical food webs [START_REF] Allesina | A General Model for Food Web Structure[END_REF][START_REF] Williams | Success and its limits among structural models of complex food webs[END_REF], though typically contain too few herbivore species, too few plant-herbivore interactions, and accumulate too many trophic levels [START_REF] Gauzens | Biodiversity of intertidal food webs in response to warming across latitudes[END_REF][START_REF] Williams | Success and its limits among structural models of complex food webs[END_REF] compared to terrestrial communities. Together these differences substantially alter predicted biomass dynamics compared to empirical terrestrial topologies [START_REF] Gauzens | Biodiversity of intertidal food webs in response to warming across latitudes[END_REF]. We predict these deviations will be magnified when the traditionally unresolved plant taxa are better resolved and when the consumption of different plant tissues is incorporated (Figure 2).

To better accommodate these deviations, we propose a terrestrial extension to a class of topological models that use an empiricallymeasurable body size axis. These models, inspired by the Allometric Diet Breadth Model, specify an energetically optimal body mass ratio ¿) at which animals can most efficiently feed on a resource [START_REF] Schneider | Animal diversity and ecosystem functioning in dynamic food webs[END_REF][START_REF] Petchey | Size, foraging, and food web structure[END_REF]. Animals can feed on a range of resource sizes, but efficiency decreases away from the optimum until effectively no interaction occurs. Therefore, species' traits determine both the presence and rate of feeding interactions (Box 1). We propose that plant trait values ( T i) on a fast-slow axis determine the feeding efficiency of herbivores on plant tissues (Figure 1). Such a trait axis contains high dimensional information on the nutritiousness (stoichiometry) and defendedness of plant leaves. Herbivores of a given metabolic class (e.g., ectotherm invertebrates, endotherm vertebrates) likely have maximum feeding efficiency on optimally-matched plant traits ( M j /T i) but can tolerate a range of plant traits with diminishing feeding efficiency and yield (i.e., Eq. 4 is dependent on T i , see also Online Supplemental Information Appendix S1).

For example, larger-bodied herbivores can handle taller and less nutritious forage -'slower' plants in our framework -than can smaller-bodied herbivores, due to lower mass-specific metabolic needs and greater digestive capacity and efficiency [START_REF] Belovsky | Optimal foraging and community structure: The allometry of herbivore food selection and competition[END_REF]. Though empirical evidence connecting this pattern to herbivory network structure is sparse, some observations suggest this to be a good first hypothesis. For example, mammals of similar body sizes tend to have similar diets [START_REF] Kartzinel | DNA metabarcoding illuminates dietary niche partitioning by African large herbivores[END_REF], and resource partitioning among African savanna grazers is well-explained by their body sizes [START_REF] Kleynhans | Resource partitioning along multiple niche dimensions in differently sized African savanna grazers[END_REF]. Within a taxonomically diverse leaf-chewing community, smaller insect herbivores preferred younger, less defended, and more nutritious leaves of Ficus wassa than larger species [START_REF] Cízek | Diet composition and body size in insect herbivores: Why do small species prefer young leaves?[END_REF].

Different trait axes may be appropriate for different plant tissues, allowing different sub-networks for specific types of herbivory such as nectarivory (Figure 2). Therefore, we propose generating similar subnetworks for different types of terrestrial feeding interactions using animal body size as the trait axis for carnivory and the matching of the plant trait and animal body size axes for herbivory. These sub-networks can then be interlinked into multiplex topologies following plausible assembly rules (e.g., [21,[START_REF] Williams | Simple rules yield complex food webs[END_REF][START_REF] Allesina | A General Model for Food Web Structure[END_REF][START_REF] Thébault | Stability of Ecological Communities and the Architecture of Mutualistic and Trophic Networks[END_REF]).

Community stability and ecosystem functions

The weakening and diversification of consumer-resource interactions are well-known to stabilize food web dynamics [START_REF] Mccann | Weak trophic interactions and the balance of nature[END_REF]. We suggest that introducing a more accurate accounting of plant and herbivore communities and their associated constraints in food web structure and function will fundamentally alter the distribution of interaction strengths relative to current bioenergetic approaches. Specifically, incorporating plant defenses and herbivory on different plant tissues (as we propose in Figs. 1 and2 and Box 2) will diversify and weaken energy flows from plants to herbivores and, therefore, stabilize terrestrial in comparison to aquatic food webs. Further, weakening interactions generally will tend to reduce the strength of trophic cascades by constraining vertical energy flow through the food web [START_REF] Mccann | Weak trophic interactions and the balance of nature[END_REF][START_REF] Rip | Cross-ecosystem differences in stability and the principle of energy flux[END_REF].

Intriguingly, empirical evidence from terrestrial ecosystems are consistent with weak-skewed interactions [START_REF] Rip | Cross-ecosystem differences in stability and the principle of energy flux[END_REF].

A holistic terrestrial bioenergetic framework may be well-positioned to advance our understanding on the relationship between biodiversity and ecosystem functioning. Current efforts have shown that diversity loss can simultaneously affect multiple ecosystem functions and services, such as primary and secondary production, pollination, pest control, and carbon sequestration [START_REF] Manning | Redefining ecosystem multifunctionality[END_REF]. A key challenge is now to understand the trade-offs and synergies among these ecosystem functions and services. The classical bioenergetic approach has been used to analyze the processes affecting primary and secondary production, as well as their trade-offs and synergies (e.g. [START_REF] Schneider | Animal diversity and ecosystem functioning in dynamic food webs[END_REF][START_REF] Albert | The hidden role of multi-trophic interactions in driving diversity-productivity relationships[END_REF]). A holistic terrestrial bioenergetic framework may contribute tools to analyze other important ecosystem functions and services, including pollination, seed dispersal, and biological control within plant-herbivore interactions. It can also provide important insight into the mechanisms behind the relations among ecosystem services such as pollination and pest control, whose combined effects -either synergistic or antagonisticremain poorly understood [START_REF] Gagic | Additive and interactive effects of pollination and biological pest control on crop yield[END_REF].

Concluding Remarks

Bioenergetic approaches have promoted productive research in food web ecology because of their ability to model food web dynamics by estimating demographic and consumption rates of interacting species using allometric scaling. Because of these successes, there is a great demand for a more terrestrially focused bioenergetic approach to address key fundamental and applied questions in community ecology (see Outstanding Questions). By combining perspectives and approaches unique to terrestrial plant-animal interactions with traditional tools from network ecology, we provided a roadmap that will guide the integration of bioenergetics specific to terrestrial plants and their biotic interactions into those of traditional food web models. larger than their resources. (iii) Growth and consumption rates are also calculated from body masses, allowing (iv) high consumption and production by herbivores typical of aquatic ecosystems. (B) Our framework uses plant traits ( T i) representing the "fast-slow" axis to determine the structure and dynamics of herbivory interactions. Fast-growing plants are smaller, leafier, with more palatable leaves; while slow-growing plants are larger, woodier, with less nutritious and more defended leaves. (i) Animals' potential feeding given metabolic class and T i. (ii) This creates herbivory sub-networks with weaker size structure. (iii) Plant growth ( r i , x i) and herbivory ( a ji ( L ji ) , e ji , f ji , h ji , p ji) can also be calculated using plant traits. (iv) Allowing variation in plant size and stoichiometry breaks the dependence of herbivore attack rate on consumer-resource body size ratio. This results in lower herbivore consumption and production than aquatic ecosystems with similar net primary productivity (NPP) due to less nutritious and more defended plant tissues. Plant and herbivore growth and reproduction strongly depend on these different trophic interactions, which indirectly affects the full food web dynamics. Despite the importance of these different interactions, the traditional approach to food webs has focused only on antagonistic herbivory (e.g., folivory), excluding "mutualistic" feeding by pollinators and seed dispersers. We propose two new approaches to incorporate the network complexity of different animal guilds feeding on different plant tissues by assuming plant biomass as: (A) composed of fixed fractions of each tissue, with herbivore groups limited to feeding on a specific fraction, and (B) partitioned into coupled pools, allowing dynamic plant allocations to growth or maintenance for each tissue, feedbacks between consumption and production of each tissue, and herbivore adaptive foraging. In both approaches, the structure of herbivory interactions on different plant tissues can be derived from the matching between plant and animal traits (Fig. 1).

Illustrative food webs show grayscale nodes lighter in color with increasing trophic level. Colored nodes indicate different plant tissues matching the photo borders for different types of herbivory. Links represent bioenergetic couplings, due to feeding (gray) or dynamic feedbacks between the production and maintenance of different tissues (green).

TEXT BOXES

Box 1 -Allometric Trophic Network (ATN) model

Consists of two sets of governing equations [START_REF] Brose | Allometric scaling enhances stability in complex food webs[END_REF][START_REF] Donohue | Loss of predator species, not intermediate consumers, triggers rapid and dramatic extinction cascades[END_REF], one for primary producers (Eq. 1) and another for consumers (Eq. 2):

d B i dt =r i B i D i ( B ) ⏞ Autotrop h ic growt h gain -x i B i ⏞ Maintenance loss -∑ j 1 f ji e ji
x j y j B j F ji ( B ) ⏞ Herbivory loss (Eq. 1)

d B j dt =x j B j ∑ k y j F ji ( B ) ⏞ Resources consumption gain -x j B j ⏞ Maintenance loss -∑ k 1 f kj e kj x k y k B k F kj ( B ) ⏞ Predation loss (Eq. 2)
where B is vector of biomasses for every species in the food web and B i is biomass of species i. B i of producer species i changes over time according to the balance between gains from autotrophic growth and losses due to metabolic maintenance and herbivory by consumer species j. Autotrophic growth is determined by the producer's intrinsic growth rate ( r i), metabolic rate ( x i), and logistic growth: D i ( B ) =1 -¿ ¿ , with K as carrying capacity of all primary producers. Biomass loss to herbivory increases with mass-specific metabolic rate ( x j) and maximum consumption rate ( y j ) of consumer species j, and decreases with ingestion ( f ji) and assimilation ( e ji) efficiencies by consumer j on producer i. B j of consumer species j (Eq. 2) changes over time according to the balance between biomass gains by resource consumption and biomass loss from metabolic maintenance and predation. Functional response F ji ( B ) determines the consumption rate of each consumer species j on each resource species i, defined:

F ji ( B ) = p ji ( B) a ji h ji B i q 1+ c j B j + ∑ l=resources p jl ( B ) a jl h jl B l q (Eq. 3)
where p ji (B), a ji, and h ji are, respectively, the foraging effort, attack rate, and handling time of consumer i on resource j, c j is the intra-specific foraging interference of consumer j, and q controls the shape of Eq. 3.

Parameters in Eqs. 1-3 are constrained by average body masses of individuals of the consumer ( M j ) and resource species ( M i), as:

r i = R i R ref ¿ ( M i M ref ) -0.25 x i = X i R ref ¿ a x a r ( M i M ref ) -0.25 y j = Y j X j ¿ a y a x (Eq. 4)
a ji =a 0 M j 0.25 M i 0.25 L ji h ji = e ji a y M j 0.25

where a 0, a r , a x , and a y are allometric constants specific to species' metabolic categories (producer, invertebrate, ectotherm vertebrate, or endotherm vertebrate) and L ji quantifies potential feeding efficiency given the energetically optimum body size ratio. Subscript ref denotes the reference producer species, the smallest in the system. Primary production ( R i), metabolism ( X j), and maximum consumption ( Y j) in Eq. 4 follow negative power laws with each species' average body mass as: R i =a r M i -0.25 ,

X i =a x M i -0.2 5 , Y j =a j M i -0.25
. We propose to consider plant growth traits (T i ) on a "fast-slow" axis to determine parameters ( R i ¿ , ( X i) which then can be used with animal body mass to calculate ( a ji), ( e ji), ( f ji), ( h ji), and ( p ji) (see Figure 1 and the Online Supplemental Information Appendix S1).

Box 2 -Plant defense

Plant defenses are organized in two major categories, constitutive and inducible [START_REF] Karban | Induced Responses to Herbivory[END_REF]. Constitutive defenses are always expressed and more common in environments where herbivore pressure is consistently high and with low resource availability, in which it is challenging to replace lost tissue.

Inducible defenses develop in response to environmental cues or to herbivory, with plants responding to chemical cues [START_REF] Karban | Plant Communication[END_REF]. For example, when many species in the pine family (Pinaceae) are attacked by herbivores, they induce production of resin and phenolic compounds that resist herbivores, and these induced responses are stronger in faster growing, low-latitude and low-elevation species than in the slower-growing species found at higher latitudes and elevations [START_REF] Moreira | Trade-offs between constitutive and induced defences drive geographical and climatic clines in pine chemical defences[END_REF]. These responses can be transgenerational [START_REF] Agrawal | Transgenerational induction of defences in animals and plants[END_REF] but commonly happen throughout the lifespan of an individual, even at the scale of hours [START_REF] Gómez | Timing of induced resistance in a clonal plant network[END_REF]. We propose to include both types of defenses as affecting herbivores' consumption parameters ( a ji , h ji , e ji , p ji) (see Box 1). For instance, high levels of defenses are expected to decrease e ji(section 3) and p ji (section 4) while inducible responses should strongly determine plant adaptive response v ij (Box 3) especially for fast growing plants.

There are two main defensive pathways for plant inducible defenses, the jasmonic acid pathway, which responds to chewers, such as caterpillars, and the salicylic acid pathway, which responds to pathogens and sucking-insects, such as aphids [START_REF] Thaler | Cross-talk between jasmonate and salicylate plant defense pathways: effects on several plant parasites[END_REF]. For example, many milkweeds (Asclepias spp.) increase production of toxic cardenolides and exudation of sticky latex in response to feeding by monarch caterpillars (Danaus plexippus), which can reduce monarch survival and performance [START_REF] Rasmann | Induced Responses to Herbivory and Jasmonate in Three Milkweed Species[END_REF]. Research shows clear constraints in some plant species to induction of defenses between these two pathways [START_REF] Thaler | Cross-talk between jasmonate and salicylate plant defense pathways: effects on several plant parasites[END_REF]. When a chewing herbivore attacks, the jasmonic-acid pathway is upregulated and the plant suppresses the salicylic-acid pathway, so it becomes more susceptible to the attack of phloem suckers or pathogens.

Plant chemical receptors and metabolic pathways responding to chemical cues are so sophisticated that they can even sense an insect walking on them before biting [START_REF] Helms | Exposure of Solidago altissima plants to volatile emissions of an insect antagonist (Eurosta solidaginis) deters subsequent herbivory[END_REF]. For trees or highly sectorial plants (e.g., shrubs), localized responses can lead individuals to be defensive mosaics, which are likely adaptive in the face of heterogeneous herbivore attack [START_REF] Volf | Localized defense induction in trees: a mosaic of leaf traits promoting variation in plant traits, predation, and communities of canopy arthropods?: Induced defenses in forest canopies[END_REF][START_REF] Lämke | Phytochemical variation in treetops: causes and consequences for tree-insect herbivore interactions[END_REF]. Communication via volatiles or below-ground mycorrhizal connections can lead plant responses to herbivore attack to include large patches of plants [START_REF] Karban | Plant Communication[END_REF]. Plants change their odors when attacked, and neighboring plants can respond in a process of plant-plant communication or eavesdropping [START_REF] Karban | Plant Communication[END_REF]. where p ji is the foraging effort consumer j assigns to resource i and v ij is the anti-predator effort resource i assigns against consumer j, respectively. Note that in some versions of the ATN (e.g., [START_REF] Brose | Allometric scaling enhances stability in complex food webs[END_REF]), p ji =ω ji denoting fixed consumer preference (unitless), while here it denotes variable foraging effort (also unitless, [START_REF] Donohue | Loss of predator species, not intermediate consumers, triggers rapid and dramatic extinction cascades[END_REF]). These efforts define the fraction of individuals' energy or time allocated to consuming a particular resource species and avoiding a particular consumer species, respectively [START_REF] Valdovinos | Consequences of adaptive behaviour for the structure and dynamics of food webs[END_REF]. The higher the foraging effort invested in a particular resource, the higher the capture efficiency is of that resource and the larger Eq. 5 is. The higher the anti-predator effort of a resource against a consumer, the lower the capture efficiency of that consumer and the smaller Eq. 5 is. Adaptive foraging and inducible defenses are incorporated into Eq. 5 by allowing p ji and v jk, respectively, to adapt over time: 6) d v jk dt =s ' j ( ∂ G j ∂ v jk -γ j ) (Eq. 7)

d p ji dt =s j ( ∂ G j ∂ p ji -γ j ) (Eq.
Parameters s j and s ' j are the rates by which species j changes its foraging and anti-predator efforts, respectively. Function G j = 1 B j d B j dt is j's per-capita (per-biomass in this case) growth rate. If s j < 1 or s ' i< 1, changes in foraging and anti-predator efforts are slower than population dynamics and the shift of strategies reflects evolutionary changes, whereas s j > 1 or s ' j > 1 represents faster changes acquired through behavioral responses [START_REF] Guill | Emergence of complexity in evolving niche-model food webs[END_REF].

These efforts increase when they increase the per-biomass growth rate more than the average per-biomass growth rate obtained from assigning the effort to other consumers or resources, γ j , defined as:

γ j = ∑ l ϵ resources p jl ∂ G j ∂ p jl + ∑ k ϵ consumers v jk ∂ G j ∂ v jk (Eq. 8)
If a species only adaptively forages, then v jk =0 , Eq. 7 is zero, and γ j in Eq. 6

will only contain the first sum of Eq. 8. If a species only adaptively defends, then p ji =0 , Eq. 6 is zero, and γ j in Eq. 7 only contains the second sum.

Optimization of Eqs. 6 and 7 is constrained by allocation costs [START_REF] Futuyma | Ecological Specialization and Generalization[END_REF],

representing the impossibility of individuals infinitely and simultaneously assigning energy or time to every task, expressed as: ∑ l ϵ resources p jl + ∑ k ϵ consumers v jk =1 (Eq. 9)

FIGURES Figure 1 .

 1 FIGURES

Figure 1 .

 1 Figure 1. Terrestrial alternative to bioenergetic models. (A) The classical approach determines food web structure and dynamics from allometric patterns based on each species' average adult body mass ( M j ) and metabolic class (Box 1). (i) Species' potential feeding L ji efficiency depends on their body mass relative to their resources. Consumers can feed on resources within a range of sizes around an energetically optimum body mass ratio. (ii) This enforces strong size structure with producers (trophic level [ TL ] = 1) of similar size and consumers approximately Z = 10-100x

Figure 2 .

 2 Figure 2.

Figure 2 .

 2 Figure 2. Approaches to herbivory in aboveground terrestrial food webs. The structural complexity of terrestrial plants supports many groups

Box 3 -

 3 Adding plant-herbivore adaptive behavior to our bioenergetic frameworkHerbivore adaptive foraging and plant defenses can be incorporated into the functional response of the ATN model (Eq. 3 of Box 1) as:F ji ( B )= p ji (B) ( 1-v ij) a ji h ji B i q 1+ c j B j + ∑ l=resources p jl ( B) ( 1-v lj )a jl h jl B l q (Eq. 5)

Acknowledgements

This working group was funded by CAMPOS Jump-Start grant to F.S.V., who also acknowledges support from National Science Foundation grant DEB-1834497. Collaboration between F.S.V, K.R.S.H., and E.T. was supported by Transatlantic Research Partnership, a program of FACE Foundation and the French Embassy.