Piche Elodie
email: piche.e@chu-nice.fr

Chorin Frédéric

Gerus Pauline

Jaafar Amyn

Reneaud Nicolas

Guerin Olivier

Zory Raphaël

Affiliations

Validity of a simple sit-to-stand method for assessing force-velocity profile in older adults

Keywords: Force-velocity relationship, simple method, clinical evaluation, sit-to-stand, ageing

Background: Lower limb muscle strength is an important determinant of physical function in older adults.

However, its measure in clinical settings is limited because of the requirement for large-scale and costly equipment. A new simple protocol based on sit-to-stand test (STS) is developed to measure force velocity (F-v) and power velocity (P-v) profile in the community-dwelling older adults. Objective: The objective of this study was to assess the validity of this new methodology for measuring F-v and P-v profile compared to the gold standard isokinetic BIODEX. Participants: 46 older people aged 65-85 years (M = 73.7; SD = 7.7). Methods: F-v and P-v profiles were assessed in participants on their dominant leg. The concurrent validity of STS was tested using Spearman's rank correlation coefficient and Passing Bablok: maximal power output Pmax, optimal velocity and force Vopt and Fopt, maximal force at null velocity F0, maximal unloaded velocity V0 and coefficient of F-v (

) and P-v equation (a_poly, b_poly). Results: No proportional difference for F0 and b_poly and a low significant correlation for Pmax (r = 0.314), (r = 0.229), a_poly (r=0.335) and b_poly (r= 0.226) whereas the other parameters were non correlated significantly. Conclusion: STS method is moderately reliable on force and power parameters whereas further improvements are needing for velocity parameters. However, its feasibility, portability and lower cost compared to other methods makes it very affordable in clinical context and will allow easy investigation of aging population.

INTRODUCTION

The "power-force-velocity" approach is based on force and power-velocity (F-v and P-v) relationships characterizing maximal mechanical capabilities of lower limbs' neuromuscular systems [START_REF] Morin | Interpreting power-force-velocity profiles for individualized and specific training[END_REF][START_REF] Samozino | Force-velocity profile: Imbalance determination and effect on lower limb ballistic performance[END_REF]. For now, individual F-v profile is widely used in sport performance as it give an optimal individual F-v profile that maximizes lower limb ballistic performance which allows individual determination of force or velocity deficit. These profiles could be very interesting to replicate in geriatrics and aging research as measures of muscle function could allow to screen for an age-related loss of muscle strength and power. Indeed, a loss of muscle function is related to frailty [START_REF] Batista | Relationship between lower-limb muscle strength and frailty among elderly people[END_REF], risk of disability [START_REF] Rantanen | Midlife hand grip strength as a predictor of old age disability[END_REF], and morbidity in older people, all of which can lead to a loss of independence in older people.

To date, Gold Standard for measuring muscle function in clinical practice and research settings is isokinetic dynamometry [START_REF] Drouin | Reliability and validity of the Biodex system 3 pro isokinetic dynamometer velocity, torque and position measurements[END_REF]. Nevertheless, those systems that measure force and velocity with accuracy are very expensive and time-consuming which limit their spread in clinics. The recent technological progress involving wearable sensors opens the field of possibilities and allows to replace those costly systems with convenient, simple and accurate analysis of the biomechanical kinematic variables. Samozino et al [START_REF] Samozino | A simple method for measuring force, velocity and power output during squat jump[END_REF] developed their own simple method for having power-force-velocity profiles which can be determined from a series of 2 to 6 loaded vertical squat jumps. In geriatrics, a more adapted exercise is the sit-to-stand (STS) test which is widely used to assess muscle strength [START_REF] Bohannon | Reference values for the five-repetition sit-to-stand test: A Descriptive metaanalysis of data from elders[END_REF][START_REF] Bohannon Rw | Test-retest reliability of the five-Repetition sit-to-stand test : a systematic review of the literature involving adults[END_REF]. Although time to complete STS is the primary measure of function, leg velocity and muscle power also contribute to understanding physical performance but require more sophisticated, time-consuming and expensive assessment tools such as force plates [START_REF] Lindemann | Measuring power during the sit-to-stand transfer[END_REF] and/or motion capture system. To avoid those systems, many studies have attempted to evaluate STS movements with either single or multiple accelerometers, both embedded and in smartphones, placed over different regions of the human body [START_REF] Hellec | Sit-to-stand movement evaluated using an inertial measurement unit embedded in smart glasses-a validation study[END_REF][START_REF] Millor | Gait Velocity and Chair Sit-Stand-Sit Performance Improves Current Frailty-Status Identification[END_REF][START_REF] Galán-Mercant | Differences in Trunk Accelerometry Between Frail and Nonfrail Elderly Persons in Sit-to-Stand and Stand-to-Sit Transitions Based on a Mobile Inertial Sensor[END_REF]. Rojas et al [START_REF] Rojas | Time measurement characterization of stand-to-sit and sit-to-stand transitions by using a smartphone[END_REF] characterize velocity profile during STS transitions with an embedded smartphone placed on L2/L3 vertebrae region while Lepetit et al [START_REF] Lepetit | Optimized scoring tool to quantify the functional performance during the sit-to-stand transition with a magneto-inertial measurement unit[END_REF] used a single magneto-inertial measurement unit fixed on the chest to quantify velocity, acceleration and kinematic data. All these studies only measure one parameter over leg power, force and velocity which does not allow the assessment of F-v and P-v profiles. To our knowledge, the only study that combines everything is Ruiz-Cardenas et al [START_REF] Ruiz-Cárdenas | Validity and reliability of an iPhone App to assess time, velocity and leg power during a sit-to-stand functional performance test[END_REF]. They developed an App technology to objectively measure time, velocity and leg power during a single STS test. The App was designed for analysing STS via highspeed video recording (240 frames-per-second) allowing the calculation of time between two frames selected by the user and subsequent calculation of mean vertical velocity and vertical power relative to body weight.

Nowadays, the spread of technology enable the development of simple method by only using smartphones or simple devices. Recently, an open library for real-time multi-person key point detection was proposed by Carnegie Mellon University named OpenPose. It is a real-time system to jointly detect 130 human body, hand and facial key points on single images. In addition, its computational performance on body key point estimation is invariant to the number of detected people in the image with high accuracy. OpenPose is freely available for all kinds of free non-commercial use [START_REF] Cao | OpenPose: Realtime Multi-Person 2D Pose Estimation Using Part Affinity Fields[END_REF]. This library enable to have every single joint of the body, angular velocities of the joint, and lower limb forces. In our knowledge, yet, no study has used OpenPose to calculate lower limb force and velocity on STS.

As STS has already proved its worth as an effective tool for assessing F-v profiles in clinical geriatric and as OpenPose seems to be a promising one also, combining the two could result in an effective simple method. In this context, the aim of this study was to propose and validate a new simple protocol for determining the F-v and P-v profiles in the hope of using it to detect decline in physical abilities in older adults. The concurrent validity of this method was tested by comparison to reference isokinetic protocol.

We hypothesized that force and power variables will be moderately correlated, velocity variables will be poorly correlated with linear association between them reflecting a sufficient concurrent validity.

MATERIALS AND METHODS

PARTICIPANTS

A total of 109 community-dwelling older adults aged between 65 and 85 years old were recruited. Exclusion criteria were pathologies that prohibit a maximal strength test, such as severe cardiovascular disease, artificial hip or knee, acute hernia, infection, or tumour. All participant gave written informed consent. Thirty acquisition were not well recorded during STS due to some improper Openpose detection caused by the camera angle, leading to missing data at some velocities (0, 5 or 10kgs). Also, Grubbs test [START_REF] Ghosh | Outliers: An Evaluation of Methodologies[END_REF] detected 33 recurrent outliers (abnormal STS parameters) resulting in their removal. Finally, 46 participants (6men, 40women) were included in this study (mean age = 73.7 +/-7.7 years; mean height = 161.8 +/-8.2 cm; mean weight = 61.5 +/-11.4 kgs; mean BMI = 23.5 +/-4.2 kgs/m²).

EXPERIMENTAL PROTOCOL

Participants were drawn from the Frailty project, which aims to identify multidisciplinary markers of aging.

Approval of the study protocol was obtained (n° 2015-A01188-41). The protocol consists of evaluating Fv profiles with Biodex dynamometer system 4 (Biodex Medical Systems, Inc., Shirley, NY) and with STS simple method. The side of healthy or dominant leg for F-v profile evaluation was chosen to fit with the unilaterality of BIODEX. Thus, hip and knee prosthesis, knee arthrosis and other pathologies were avoided.

The protocol included two standardized tests in the following order: STS protocol and Biodex protocol.

During each test, participants were clearly instructed by the test leader to perform with maximal effort.

Sit-to-stand protocol

The tool used was an adjusted chair without backrest. The seat height was visually adjusted to allow the participant to start with a 90° knee angle. A visual line was also placed on the ground to limit feet's displacement during the test. Participants were initially seated on the chair with their back in an upright position wearing their own shoes (in an ecologically situation) (Figure 1). Then, they were instructed to stand up as fast as they can with their arms folded across their chest in 3 conditions: with no additional load, with 10kgs load and with 5kgs. All trials were performed using the same chair and with similar ambient conditions. They repeated it 3 times to assure to reach maximal velocity during STS and the best of the 3 trials was taken. Resting time between the 3 conditions was sufficient to prevent participant from fatigue.

A camera was positioned to follow STS movement on their dominant leg unless there was a medical contraindication.

Analysis procedure and data recording to sit to stand protocol

Acquisitions were recorded with an Ipad (30hz, 1280x720p). The videos were processed on Python with OpenPose library [START_REF] Cao | OpenPose: Realtime Multi-Person 2D Pose Estimation Using Part Affinity Fields[END_REF]. This library can detect multiple people in real time which is the first time a library can jointly detect key points on human body, face, and foot. The protocol setup was to collect measurements from iPad, to retrieve key points by applying OpenPose "Body25 model" on videos, to preprocess (denoising, calculating kinematic parameters) and finally to calculate F-v and P-v profiles. To determine the time point for starting and ending the movement during STS test, plateau were detected on knee flexion extension kinematics. A 0° plateau corresponded to the end of the STS (full extension and a 90° to the start of the movement. The starting point of the STS was determined when a plateau at 90° on knee flexion kinematics was detected. The ending point was defined when full extension of knee was achieved in an upright stance corresponding to a 0° plateau. All the F-v and P-v parameters were calculated for one leg to allow comparison with the unilateral single joint BIODEX. With OpenPose, the coordinates (X, Y) over time for shoulder, hip and knee were taken allowing the calculation of their velocities. The associate thigh and body force (eg reaction force) for one leg were calculated as followed as well as power P: = 0.1416*p*g _ = 0.6028*(p+charge)*g With g the gravitational force (9,81 N/kg), p the weight of the participant (kg), charge could be 0, 5 or 10kgs depending on the wearing jacket and 0.1416 and 0.6028 were taken from the anthropometric De Leva's table [START_REF] Leva | Adjustments to ZATSIORSKY-SELUYANOV'S segment inertia parameters[END_REF].

Isokinetic protocol

Measurements were performed unilaterally on the same side of STS protocol. Participants were seated on a backward-inclined (5°) and adjustable chair, which is part of the dynamometer (Figure 1). A strap was applied across thigh and hips and shoulders were stabilized with safety belts to avoid additional movements.

The rotational axis of the dynamometer was aligned with the transversal knee-joint axis and connected to the point of force application at the distal end of the tibia (i.e. 5cm above the lateral malleolus) using a length-adjustable rigid lever arm. Range of motion was set from a knee joint angle of 90° to 160°, with a full extended leg corresponding to a 180° knee angle. Participants were evaluated at six different velocities:

180°/s, 150°/s, 120°/s, 90°/s, 60°/s, 30°/s to reach an accurate F-v profile. Before starting the evaluation, participants performed repetitions to familiarize with knee extension movement and to warm-up.

Analysis procedure and data recording for isokinetic protocol

Sampling was performed at 100 Hz using an electronic interface card (Biodex Medical Systems Inc., X2151, Shirley, NY, USA). The evaluation on the six different velocities allows to obtain a set of parameters such as the ratio between agonist and antagonist muscles as well as curves representing F-v and P-v relationships.

Force (F) -velocity (V) relationship was described by a first order linear equation:

! • ! + #
with and b_linear the coefficients of the polynomial relation. Note that F0 is a force corresponding to the theoretical moment at 0° /s velocity value. The velocity value which corresponds to 0 Nm was the maximum theoretical velocity (VMAX).

Power (P) -velocity (V) relationship was described by a second order polynomial equation:

$ %_&'() • !² + +_&'() • !
with a_poly and b_poly coefficients of the polynomial relation (Figure 2).

Maximum power (PMAX), and optimal velocity (VOPT) were determined from this equation as: Concurrent validity between STS and BIODEX was assessed by determining Spearman's rank correlation coefficient (r) . The following correlation classification was used: negligible r=0-0.10, low r=0.10-0.39; moderate: r = 0.40-0.69; high: r = 0.70-0.89; very high: r = 0.90-1.0 [START_REF] Schober | Correlation coefficients: Appropriate use and interpretation[END_REF]. Passing-Bablok regression was used to assess the linear association between parameters that succeed to CUSUM test (p>0.05) [START_REF] Bilic-Zulle | Lessons in biostatistics Comparison of methods : Passing and Bablok regression[END_REF]. All statistical tests were regarded as significant at p<0.05 with a Benjamini-Hochberg correction for multiple associations and statistical analyses were performed using XLSTAT (Addinsoft, New York, USA).

! ,$-- +_& '

RESULTS

Mean ± Standard deviation for both STS and BIODEX methods for mean force, velocity and power output are presented in Table 1. Spearman's rank correlation showed no to very low (r= -0.024-0.226) and nonsignificant (p=0.08-0.77) for most of the parameters except for Pmax (r = 0.314 ), (r = 0.229), a_poly (r=0.335) and b_poly (r=0.226) (Table 1). Finally, for all parameters except for , p-value of the CUSUM test was more than 0.05 (Table 1) concluding in a linear relation between the two methods. We were thus able to use Passing Bablok regression analysis presented in Figure 3.

The intercept value in Table 2 is a measure of systematic differences between the two methods. The 95% confidence interval containing the value 0 means no systematic difference between the two methods which is not the case for Pmax, Vopt, F0, V0, and a_poly. Then, it is concluded that the difference between STS and the reference BIODEX method differs at least by a constant amount. For Fopt and b_poly, the confidence interval is respectively [-111.9; 0.3] and [-123.1; 1.04] concluding in no systematic difference between STS and BIODEX.

The slope coefficient in Table 2 is a measure of proportional differences between the two methods. The 95% confidence interval for the slope including 1 means no proportional difference is found. For Pmax, Fopt, Vopt, V0, and a_poly, any of the confidence interval contains 1 concluding in a proportional difference between the two methods whereas for F0, and b_poly, no proportional difference between the two methods was found.

DISCUSSION

The objectives of this study were to assess the concurrent validity of a new method to assess F-v and P-v profiles in older adults based on STS protocol comparing it with the well-known isokinetic protocol.

Results obtained highlighted a linear relationship between most of the F-v and P-v parameters between the two methods, no proportional difference for F0 and b_poly and a low significant correlation for Pmax (r = 0.314), (r = 0.229), a_poly (r=0.335) and b_poly (r=0.226) whereas the other parameters were non correlated significantly. Results are, thus, encouraging showing a fairly low concurrent validity for parameters of force, power and slope of F-v and P-v relationship ( , a_poly and b_poly respectively).

Spearman's rank correlation was significant but quite low for Pmax (r = 0.314, p = 0.017), (r = 0.229, p=0.065), a_poly (r=0.335, p=0.012) and b_poly (r=0.226, p =0.065) whereas incongruent and nonsignificant correlation were found for the other parameters. Impellizzeri et al. also found a quite low correlation for peak force by comparing a jump test with an isokinetic leg extension (r = 0.48, 95%CI: 0.26-0.66) [START_REF] Impellizzeri | A vertical jump force test for assessing bilateral strength asymmetry in athletes[END_REF]. But when compared to a more similar test (pneumatic leg extension), the correlation is stronger (r=0.83, 95%CI = 0.7-0.91) [START_REF] Impellizzeri | A vertical jump force test for assessing bilateral strength asymmetry in athletes[END_REF][START_REF] Balachandran | Validity, reliability, and measurement error of a sit-to-stand power test in older adults: A pre-registered study[END_REF][START_REF] Alcazar | The sit-to-stand muscle power test: An easy, inexpensive and portable procedure to assess muscle power in older people[END_REF]. In view of these results, results are consistent with the literature and the difference between the two protocols, BIODEX and STS, may strongly impact on the agreement between F-v and P-v parameters. Indeed, when one method (BIODEX) recruited one leg in a closed kinematic chain (isolate knee joint), the other (STS) recruited two legs in open kinematic chain (multiple joints participating to the movement). Studies have demonstrated that mechanical output in leg multi-joint movement is different from that in a single-joint exercise [START_REF] Yamauchi | Force-velocity, force-power relationships of bilateral and unilateral leg multi-joint movements in young and elderly women[END_REF]. However, even if methods are quite different, the agreement in Pmax, thus the STS calculation, is quite consistent with previous results which highlight the capacity of the STS method to determine the mean peak force in community-dwelling older adults. Indeed, the range of magnitude of Pmax between BIODEX and STS were very similar (M= 143.9W, SD= 49.9W; M=187.5 W, SD=147.3W for BIODEX and STS respectively). The 187.5W found during the STS was for one leg which gave approximately 375W for the entire power developed by the two legs during the STS movement. This value was compared to other studies that used different methods for Pmax calculation to check the consistency of this study's protocol. Regarding the range of magnitude, the study's Pmax appeared to be very closed to the 400W value found in other studies highlighting the consistency of this value [START_REF] Cheng | Can sit-to-stand lower limb muscle power predict fall status?[END_REF][START_REF] Zijlstra | A body-fixed-sensor-based analysis of power during sit-to-stand movements[END_REF].

The high variability in Pmax represented by a SD of +/-147.3W (more than 75% of variability)

should also be noticed as, for other studies using STS test, the variability was lower (around 40% in Pmax variability) [START_REF] Alcazar | The sit-to-stand muscle power test: An easy, inexpensive and portable procedure to assess muscle power in older people[END_REF]. The high variability obtained here, which is also extended to all STS parameters, can be partly attributed to the STS design protocol used in this study. Indeed, a reasonable variability was found for BIODEX parameters when the range of movement was individually adjusted allowing participants to start with a precise 90° knee angle. Also, when Alcazar et al [START_REF] Alcazar | The sit-to-stand muscle power test: An easy, inexpensive and portable procedure to assess muscle power in older people[END_REF] used a standardized armless chair (0.49 m height), a 40% variability was found. In this study, knee angle was not precisely measured even if the start position during STS was visually corrected by the investigator. The starting position visually adjusted was at 85° as a mean in all the participants which highlight that the visual placement at 90° is not perfect.

To improve precision, future studies should use the video recording and Openpose to directly correct knee angle and re-adjust chair height to allow participant to begin STS with a precise 90° knee angle which can be done in live. The high STS variability is also explained by the protocol itself where the STS parameters were very dependent to the patient stand up velocity whereas for the BIODEX, the patient had to simply follow the pace and the fixed velocity. This contributes to more technical variation in STS execution leading to a high intra-variability thus a lower reliability.

The high variability is also participating in the lower reliability results found especially in velocity parameters (Vopt and V0) where no proportional and systematic difference was found in Passing Bablok.

Several studies in young and older adults have also reported lower agreement values for V0 [START_REF] Lindberg | Force-velocity profiling in athletes: Reliability and agreement across methods[END_REF][START_REF] García-Ramos | Force-Velocity Relationship in the Countermovement Jump Exercise Assessed by Different Measurement Methods[END_REF][START_REF] Giroux | What is the best method for assessing lower limb force-velocity relationship?[END_REF][START_REF] Alcazar | The Force-Velocity Relationship in Older People: Reliability and Validity of a Systematic Procedure[END_REF][START_REF] Janicijevic | Feasibility of the two-point method for assessing the force-velocity relationship during lower-body and upper-body isokinetic tests[END_REF]. This could be attributable partially to the STS method itself to assess F-v profiles. Indeed, while the F-v profile was assessed by using multiple point (i.e. 6 velocities), the STS was using three points (0, 5, 10kgs). Janicijevic et al [START_REF] Janicijevic | Feasibility of the two-point method for assessing the force-velocity relationship during lower-body and upper-body isokinetic tests[END_REF] have recently shown a low concurrent validity of the two-point method, which is quite similar to the three-point, for V0 and F0. When moving from multiple-point methods (usually 4 to 6 velocities or loads) to a two-point method (or a three-point), the extrapolation between the two-points to construct F-v and P-v slopes is wider than the multiple points providing a larger range of force and velocity data thus a large extrapolation error risk [START_REF] Janicijevic | Feasibility of the two-point method for assessing the force-velocity relationship during lower-body and upper-body isokinetic tests[END_REF]. Also, the model used to fit the F-v relationship was the linear model over all the sample (from 0 to 100% of F0) which was largely used in other studies [START_REF] Samozino | A simple method for measuring force, velocity and power output during squat jump[END_REF][START_REF] Samozino | Optimal force-velocity profile in ballistic movements-Altius: Citius or Fortius?[END_REF].

This can be debatable as the linear equation was found to exceptionally fit the F-v shape only between 45 to 100% of F0 [START_REF] Alcazar | Comparison of linear, hyperbolic and double-hyperbolic models to assess the force-velocity relationship in multi-joint exercises[END_REF]. Many models have been proposed to fit the F-v relationship with a hyperbolic equation as Hill et al proposed [START_REF] Hill | The heat of shortening and the dynamic constants of muscle[END_REF] when others demonstrated that the relationship was deviated from this rectangular hyperbola at forces close to the maximal isometric force and that a double-hyperbolic equation was better [START_REF] Edman Kap | DOUBLE-HYPERBOLIC FORCE-VELOCITY RELATION IN FROG MUSCLE FIBRES[END_REF]. It is understandable that the misconception of the shape of the F-v relationship can lead to serious errors in the estimation of several parameters derived from linear equations, such as F0 and V0. However, despite this questionable choices, the shape of the STS F-v and P-v equation (Sfv and a_poly respectively) was found to be correlated (r=0.229-0.335) while other studies found inconsistent results for them (p>0.05) [START_REF] Giroux | What is the best method for assessing lower limb force-velocity relationship?[END_REF][START_REF] Janicijevic | Feasibility of the two-point method for assessing the force-velocity relationship during lower-body and upper-body isokinetic tests[END_REF].

Although results are similar to previous works, this method presents some limitations. First, participants were filmed in a side view bounding Openpose on one side to calculate kinematics. This could have increase the risk of losing data or abnormal recording if the subject was not well detected. When missing data appeared, it results in extrapolation when calculating F-v parameters that could partially explained the incongruent correlation obtained [START_REF] Lindberg | Force-velocity profiling in athletes: Reliability and agreement across methods[END_REF][START_REF] García-Ramos | Force-Velocity Relationship in the Countermovement Jump Exercise Assessed by Different Measurement Methods[END_REF]. An improvement could be to place the participant at 45° to the camera to allow Openpose to refer to both right and left sides in order to improve detection thus reliability on calculation. Another limit is that absolute fixed loads were used (5 and 10kgs) instead of relative to body weight to simplify the protocol. It appeared not to correspond to some patients which were removed from the study as the hypothesis of reducing velocities with heavier loads was not always respected. It was for only 30% of the participants but to reduce exclusion, a real time feedback on movement velocity should be given during STS to be sure the hypothesis is well respected [START_REF] González-Hernández | Movement velocity can be used to estimate the relative load during the bench press and leg press exercises in older women[END_REF]. Finally, the study design did not allow for a test-rest reliability on STS but previous results seem promising. Indeed, many studies have tested reliability of a STS power test which appears to be excellent (ICC=0.96) [START_REF] Balachandran | Validity, reliability, and measurement error of a sit-to-stand power test in older adults: A pre-registered study[END_REF][START_REF] Alcazar | The sit-to-stand muscle power test: An easy, inexpensive and portable procedure to assess muscle power in older people[END_REF]. Future studies should, therefore, test reliability on other F-v parameters than power. It is therefore likely that the observed limited validity in the F-v parameters especially in velocity parameters are caused by extrapolation error and the combination of technical/instrumental and biological variations that could be easily rectified.

However, for a measurement tool to be widely accepted, factors such as feasibility, safety, and cost are just as, if not more, important than measurement properties. Several tools have been used to assess F-v and P-v profiles in older adults but are very expensive and time-consuming. Indeed, an isokinetic ergometer is feasible with older adults but will take long time (at least 20mn for warm up and protocol) and is overpriced (around 80 000$). A method based on loaded and unloaded squat jump was developed by Samozino et al [START_REF] Samozino | A simple method for measuring force, velocity and power output during squat jump[END_REF] to assess F-v and P-v profiles with low-cost materials. It has shown to have sufficient validity and reliability but there remains safety and feasibility concerns, especially in low functioning older adults. This study was, to our knowledge, the first to propose and validate STS protocol to assess F-v and P-v profiles in older adults and follows in the footsteps Balachandran et al [START_REF] Balachandran | Validity, reliability, and measurement error of a sit-to-stand power test in older adults: A pre-registered study[END_REF] who succeed to develop STS power test in older adults. This method is portable, quicker (around 10mn for the all protocol) and less expensive (estimated at less than 300$) than the existed methods.

As a conclusion, we found better correlation in variables of force and power but not significant in velocities and we shared the same conclusion with better results as other studies that have reported lower or no reliability values for V0, Pmax or Sfv [START_REF] Giroux | What is the best method for assessing lower limb force-velocity relationship?[END_REF][START_REF] Janicijevic | Feasibility of the two-point method for assessing the force-velocity relationship during lower-body and upper-body isokinetic tests[END_REF]. The simple computation method proposed here might offer an inexpensive and easy alternative to assess and individualize F-v profiles without the need of expensive technologies in community-dwelling older adults. Future research should investigate if the method is consistent to detect frailty among older adults or future fallers based on F-v and P-v profiles.

  v the angular velocity of the upper body and thigh respectively.

Figure 1 :

 1 Figure 1: Isokinetic BIODEX positioning of the participant (a) and STS protocol at 0kgs (b), 10kgs (c) and 5kgs with OpenPose skeleton fitting specific body point (d).

Figure 2 :

 2 Figure 2: Force-velocity (F-v) and power-velocity (P-v) profiles

Figure 3 :

 3 Figure 3: Passing Bablok regression analysis of the two methods, STS and the reference BIODEX, for Pmax, Vopt, Fopt, F0, V0, Sfv, Fmax, a_poly and b_poly. Scatter diagram with regression line and confidence bands. Identity line is dash

Table 1 :

 1 Mean ± Standard deviation and concurrent validity between both methods for mean force, velocity, and power output in one leg.

		Biodex (reference	STS	Spearman's r	CUSUM test for
		measures)		coefficient	linearity
	Pmax (W)	143.9 +/-49.9	187.5 +/-147.3	0.314*	0.124
	Vopt (rad/s)	3.6 +/-0.8	3.3 +/-3.3	-0.043	0.621
	Fopt (N)	39.5 +/-14.9	36.3 +/-37.9	0.161	0.05
	F0 (N)	83.1 +/-28.5	62.3 +/-42.6	0.199	0.387
	V0 (rad/s)	7.1 +/-2.1	6 +/-7	-0.024	0.109
	34 (N.s. 6 78 )	-12.7 +/-5.1	-4.4 +/-11.6	0.229*	0.021**
	a_poly	-12.7 +/-5.1	-4.1 +/-11.7	0.335*	0.05
	b_poly	81.7 +/-28.7	61.8 +/-43.3	0.226*	0.621
	34 : slope of the F-v relationship
	a_poly, b_poly: coefficient of the P-v relationship
	* Significant correlation (p<0.05) (p_value < B-H adjusted p_value)
	** non-linear relation between the two methods (p < 0.05) (Passing Bablok not applicable)

Pmax: Maximal power output against different loading conditions Vopt: optimal velocity to reach maximal power Fopt: optimal force to reach maximal power F0: the theoretical maximal force at null velocity, V0: the theoretical maximal velocity at which lower limbs can extend during one extension under zero load

Table 2 :

 2 Passing Bablok model coefficient for Pmax, Vopt, Fopt, F0, V0, , a_poly and b_poly

Intercept value (95% confidence interval) Slope coefficient (95% confidence interval)

	b_poly	-73.4 [-123.1 ; 1.04] a	1.7 [0.7 ; 2.4] b
	Pmax (W)	-182 [-408 ; -45.1]	2.4 [1.5 ; 4.2]
	Vopt (rad/s)	75.4 [0.7 ; 8.08]	-20.9 [-1.6 ; 0.4]
	Fopt (N)	-42.9 [-111.9 ; 0.3] a	2.3 [1.1 ; 4.1]
	F0 (N)	-50.7 [-155.2 ; 0.7]	1.4 [0.7 ; 2.8] b
	V0 (rad/s)	42.3 [5.1 ; 19.3]	-5.8 [-2.3 ; -0.13]
	34 (N. s. 6 78 )	23 [47.8 ; 9.3]	2.2 [1.1 ; 4.4]
	a_poly	22.6 [41.1; 12.2]	2.4 [1.4 ; 3.2]

a No systematic difference between the two methods b No proportional difference between the two methods


Philip H Crowley
email: pcrowley@uky.edu

Cédric Tentelier
email: cedric.tentelier@univ-pau.fr

Resolving Conflict Over Within-Pair Mating Rate in External Fertilizers

Keywords: Game theory, male care, monogamy, sexual conflict, sperm supply

Besides egg fertilization, females of many taxa obtain direct fitness benefits from male mates, such as food, protection or paternal care. But males often increase their own fitness by mating with several females, among which they distribute sperm along with the above-mentioned benefits, reducing the benefits to individual females. These diverging interests lead to a conflict in which each female may try to ensure male fidelity and get exclusive access to male-provided benefits. Here, we use a theoretical model to show how a female of an externally fertilizing species may achieve mate fidelity by soliciting copulations at such a rate that the male has insufficient sperm left to increase his fitness with additional females. We show that three alternative condition-dependent evolutionarily stable mating relationships emerge in this scenario, based on whether one mate's preference for mating rate dominates, or the conflict is resolved by what amounts to negotiation. We demonstrate how these outcomes depend on some features of physiology, ecology, and behavior. In particular, a greater reproductive benefit to a female from exclusive access to a male partner-or the occasional tendency of females to withhold eggs during mating-can increase male fidelity; and continuous sperm regeneration rather than an initially-set stock of sperm allows for multiple within-pair mating across all three mating patterns.

Introduction

Females often mate with multiple males to gain additional resources [START_REF] Arnqvist | The evolution of polyandry: multiple mating and female fitness in insects[END_REF][START_REF] Hosken | Benefits of polyandry: A life history perspective[END_REF] or to obtain good genes for offspring [START_REF] Halliday | Multiple mating by females: a perspective from quantitative genetics[END_REF][START_REF] Hunter | Why do females mate multiply? A review of the genetic benefits[END_REF]. This may even be true for some socially monogamous females [START_REF] Birkhead | Sperm competition in birds: Evolutionary causes and consequences[END_REF]. But why do permanently or sequentially monogamous females often mate repeatedly with the same mate, despite possible energetic or risk-related mating costs [START_REF] Daly | The cost of mating[END_REF]? While multiple mating by females (polyandry) has been the focus of many theoretical models, within-pair repeated mating has received much less attention. [START_REF] Petrie | Copulation frequency in birds: why do females copulate more than once with the same male?[END_REF] and [START_REF] Hunter | Why do females mate multiply? A review of the genetic benefits[END_REF] reviewed empirical work on repeated mating (mainly in birds) and proposed several hypotheses pertaining to fertilization insurance, mate choice or mate guarding. In particular [START_REF] Petrie | Copulation frequency in birds: why do females copulate more than once with the same male?[END_REF] suggested that a female may mate repeatedly with her partner in order to manipulate him into monogamy, thereby securing exclusive access to his gametes, his good genes or the direct benefits he provides. In the context of sexual conflict [START_REF] Chapman | Sexual conflict[END_REF], this hypothesis can be conceptualized as a negotiation game between mates for access to gametes and support, where females would trade copulation, hence opportunity for fertilization, for male-provided dilutable benefits (i.e. benefits which are shared among the male's partners; [START_REF] Tazzyman | Fixed and dilutable benefits: female choice for good genes or fertility[END_REF]).

Petrie's hypothesis is both appealing and supported by some observations [START_REF] Hunter | Why do females mate multiply? A review of the genetic benefits[END_REF][START_REF] Petrie | Copulation frequency in birds: why do females copulate more than once with the same male?[END_REF] but the evolutionary stability of such a strategy has not been assessed. In particular, whether repeated mating may resolve sexual conflict over mate fidelity could depend not only on the magnitude of male-provided benefits, but also on the relative capacity of both males and females to sustain a high mating rate. The patterns of gamete production and release are therefore likely to play a central role in the evolution of within-pair repeated mating. The aim of the model presented here is to explore how gamete management may affect the evolutionarily stability of within-pair repeated mating as a female strategy to secure exclusive access to male provided benefits.

The role of gamete management in the evolution of conflict over mating rate has been addressed mainly in the context of sperm competition in polygynandrous internal fertilizers [START_REF] Abe | Sperm economy between female mating frequency and male ejaculate allocation[END_REF][START_REF] Parker | Sperm competition and the evolution of ejaculates: Towards a theory base[END_REF]Parker, , 1970a;;[START_REF] Parker | Sperm competition and ejaculate economics[END_REF], and to a lesser extent in the context of fertilization efficiency in external fertilizers with no sperm competition [START_REF] Kiflawi | Adaptive gamete allocation when fertilization is external and sperm competition is absent: Optimization models and evaluation using coral reef fish[END_REF][START_REF] Shapiro | Mating tactics in external fertilizers when sperm is limited[END_REF]. Models developed in both contexts indicate that under the risk of sperm limitation, males may maximize reproductive success by adjusting ejaculate size to the risk of sperm competition and to the number of eggs released by the female, which she sets with her own reproductive success at stake. In response to ejaculate size, high mating rate could be a way for females to ensure fertilization of their eggs by sperm of males that prudently allocate their ejaculates between partners [START_REF] Alonzo | Selection on female remating interval is influenced by male sperm competition strategies and ejaculate characteristics[END_REF][START_REF] Kiflawi | Adaptive gamete allocation when fertilization is external and sperm competition is absent: Optimization models and evaluation using coral reef fish[END_REF][START_REF] Wedell | Sperm competition, male prudence and spermlimited females[END_REF].

Sperm economics mainly accounts for the total amount of sperm available to males (assessed by testis size; [START_REF] Lüpold | How sperm competition shapes the evolution of testes and sperm: a meta-analysis[END_REF] and the way this stock is parceled among mating occasions. Models addressing the evolution of mating rate do not clearly account for the dynamics of sperm production, although the pattern of sperm production is likely to constrain the evolution of mating rate. In some species, males start the mating season with a finite stock of sperm that cannot be replenished [START_REF] Boivin | Spermatogeny as a life-history index in parasitoid wasps[END_REF][START_REF] Michalik | Male reproductive system of spiders[END_REF][START_REF] Wootton | Reproductive biology of teleost fishes[END_REF]. In other species, males can replenish their sperm stock after a refractory period that may vary among and within species [START_REF] Dewsbury | Ejaculate cost and male choice[END_REF][START_REF] O'dea | Male body size and condition affects sperm number and production rates in mosquitofish, Gambusia holbrooki[END_REF].

In this article, we set sperm competition aside and focus on external fertilizers, which pair in nests where males provide dilutable benefits to females and offspring. Many fish species correspond to these characteristics [START_REF] Wootton | Reproductive biology of teleost fishes[END_REF]. Although alternative reproductive tactics exist [START_REF] Taborsky | Alternative reproductive tactics in fish[END_REF], the incapacity of females to store sperm limits the scope for sperm competition in external fertilizers [START_REF] Fitzpatrick | Sperm competition and fertilization mode in fishes[END_REF]. On the other hand, external fertilization often allows females to control mating rate and the number of eggs released per mating. On some occasions females may even solicit mating but withhold eggs, thereby increasing mating rate (and sperm expenditure by males) but not fertilization [START_REF] Jones | The spawning behaviour of brown trout and salmon[END_REF][START_REF] Esteve | Two examples of fixed behavioural patterns in salmonines: female false spawning and male digging[END_REF][START_REF] Schneider | SCUBA observations of spawning small-mouth bass[END_REF][START_REF] Ridgway | Courtship and spawning behavior in smallmouth bass (Micropterus dolomieui)[END_REF][START_REF] Roy | Quantitative and qualitative analysis of spawning behaviour of Heteropneustes fossilis (Bloch.) (Siluridae) in laboratory aquaria[END_REF][START_REF] Yamazaki | High frequency of mating without egg release in highly promiscuous nonparasitic lamprey Lethenteron kessleri[END_REF][START_REF] Petersson | False orgasm in female brown trout: Trick or treat?[END_REF]. Moreover, males can either start the mating period with a finite stock of sperm or replenish it throughout the season [START_REF] O'dea | Male body size and condition affects sperm number and production rates in mosquitofish, Gambusia holbrooki[END_REF], and they often provide direct benefits such as paternal care in the nest they build [START_REF] Gross | The evolution of male and female parental care in fishes[END_REF]. Within-pair repeated mating is sometimes observed in such species [START_REF] Johnson | Reproductive Ecology of Lampreys[END_REF].

Our goal is therefore to predict how gamete management affects the resolution of malefemale conflict over mating rate in external fertilizers with no sperm competition but with maleprovided benefits (i.e. to build a "resolution" model, sensu [START_REF] Parker | Sexual conflict over mating and fertilization: an overview[END_REF]. To do this, we derive and analyze a simple game theoretic model based on finding the ESS mating rate for a single mating season and the use of all available gametes, nest building, and external fertilization (Figure 1), from which we derive the expected mating patterns. This enables us to address the following four questions:

(1) How can sexual conflict over within-pair mating rate be resolved for external fertilizers?

(2) How does this resolution depend on whether or not males continually regenerate sperm?

(3) How do the mating patterns depend on the efficacy of male help at the nest?

Methods

As a biological grounding for the derivation, we envision an externally fertilizing, semelparous, nest building, pair bonding species, such as many fishes [START_REF] Wootton | Reproductive biology of teleost fishes[END_REF]. Our model is deterministic, using fixed parameter magnitudes to obtain what can be

RELEASE OF RECHARGED SPERM γ, τ, ρ, smax RELEASE OF STOCKED SPERM σ, τ, ρ RECHARGED FERTILIZATION ø, x STOCKED FERTILIZATION ø, x MATING cm, μ, μmax MATING cf, μ, μmax PATERNAL CARE θ FITNESS, F EGG RELEASE ε, β, τ MALES FEMALES MALES & FEMALES
considered expectations from an underlying stochastic formulation. We emphasize that a stochastic version of the dynamics we depict may yield somewhat different results.

We begin by assuming that constant numbers of eggs and sperm, specifically E and S, are released by mates in each mating bout. We assume that the ratio of sperm released to eggs released during a mating bout determines the fertilization efficiency λ, but with two important caveats.

(1) A larger number of eggs released by a mating female should generally provide a denser target for sperm to encounter them more efficiently; in other words, each individual sperm has a chance of encountering an egg that increases per egg with greater numbers of eggs released [START_REF] Vogel | Fertilization kinetics of sea urchin eggs[END_REF]. We represent this effect in the sperm-to-egg ratio by raising the number of eggs E to the power x, where 0 < x < 1. (2) Fertilization efficiency λ should increase with the sperm-to-egg ratio (as modified in caveat (1)) but with diminishing returns, so that very high ratios approach the maximum efficiency of 1 [START_REF] Lehtonen | Mathematical models of fertilization-An eco-evolutionary perspective[END_REF]. We represent this effect with the exponential function

= 1 - ø = 1 -ø , ( 1
)
where ø is the coefficient determining how rapidly the efficiency approaches its upper limit with increasing sperm-to-egg ratio. See Table 1 for a list of all model parameters, definitions, standard or default magnitudes, and units. Next we determine the number of eggs released during mating. Suppose a female at maturity contains her full complement of ε eggs that can be released for external fertilization during mating events. We assume that she spends the duration τ of her reproductive interval with a single male partner. With the help of this partner, she constructs and maintains a nest where eggs are released, fertilized, tended, and protected to benefit their survival and initial development. We assume that mating happens at some consistent frequency μ to use up all eggs during τ, which would imply the release of E = ε/ (μτ) eggs at each mating. However, we also allow for the possibility that females fail to release any eggs in the proportion β of matings, as documented in some fishes [START_REF] Esteve | Two examples of fixed behavioural patterns in salmonines: female false spawning and male digging[END_REF][START_REF] Jones | The spawning behaviour of brown trout and salmon[END_REF][START_REF] Ridgway | Courtship and spawning behavior in smallmouth bass (Micropterus dolomieui)[END_REF][START_REF] Roy | Quantitative and qualitative analysis of spawning behaviour of Heteropneustes fossilis (Bloch.) (Siluridae) in laboratory aquaria[END_REF][START_REF] Schneider | SCUBA observations of spawning small-mouth bass[END_REF][START_REF] Yamazaki | High frequency of mating without egg release in highly promiscuous nonparasitic lamprey Lethenteron kessleri[END_REF]. As a result, the number of eggs released when eggs are not withheld, E, would need to be

= / 1 - , (2)
so that all eggs are released during time τ. To address whether egg withholding could be beneficial to the female (question 4 in the introduction), we assessed the effect of variation in β on the mating rate at equilibrium and on fitness. However, for the sake of simplicity, β was not allowed to evolve in the model, and the mating rate, µ, was the only evolving variable. We assume that mating rate has an upper limit at μmax, determined by physiological and behavioral constraints on the initiation, continuation, and termination of mating events. As often observed in externally fertilizing fishes, the female is assumed to solicit matings by adopting a specific posture, and the male is assumed both to accept these solicitations to the extent allowed by his sperm supply, and to be unable to force matings.

In the next two sections below, we address alternative ways that sperm release at each mating, S, is determined.

Sperm Stocked at Maturation

Analogous to the stock of eggs, a maturing male may have a fixed amount of sperm σ available to expend during the entire mating interval of duration τ [START_REF] Boivin | Spermatogeny as a life-history index in parasitoid wasps[END_REF][START_REF] Docker | The lamprey gonad[END_REF][START_REF] Michalik | Male reproductive system of spiders[END_REF][START_REF] Wootton | Reproductive biology of teleost fishes[END_REF]. He allocates S sperm per bout, dispensing all available sperm during τ. For males with a single mate, we allow for withholding the fraction ρ from each sperm release to allow for opportunistic mating or for the recruiting of an additional mate [START_REF] Cornwallis | Social status and availability of females determine patterns of sperm allocation in the fowl[END_REF]. This means that = .

(3)

Here, S1 has the subscript 1 to indicate that the male has a single mate.

In some cases, males could benefit from attracting and securing exclusive access to two mates simultaneously, in the same or in different nests, despite the possible reduction in sperm available to each mate [START_REF] Warner | Sexual conflict: males with highest mating success convey the lowest fertilization benefits to females[END_REF]. The male with two mates must mate twice as often as each female, with the mating interval 1/(2μ), to prevent the females from abandoning the nest in search of the higher mating rate consistent with better overall fertilization of her egg supply, given sperm limitation. We assume available alternative mating opportunities here. In this twofemale case, we assume for simplicity that the male uses his entire reserve at each mating to address the additional demand (i.e. ρ = 0), shifting sperm accumulation and thus the number of sperm released in each mating by a male with two mates S2 to

= , ( 4
)
where the subscript 2 indicates that the sperm supply is shared between the two females.

We emphasize the contrasting levels of sperm release in S1 and S2, where the subdivided sperm supply reducing sperm release per mating bout in S2 and possible sperm withholding (ρ > 0) in S1 determine their relative magnitudes.

Sperm Recharge

Here we assume that males, rather than having a fixed stock, produce sperm continuously over time t between matings at a rate that declines as the sperm storage limit smax is approached [START_REF] O'dea | Male body size and condition affects sperm number and production rates in mosquitofish, Gambusia holbrooki[END_REF]. This process is characterized by the exponential function ! " =

! #$% 1 -&'
, where s(t) is sperm availability after accumulation over time t, and γ is the sperm production coefficient (Figure 2A). Males maximize sperm production when they use all of their available stock at each mating. This is because each cycle of sperm production and release in this case is based on the interval of highest sperm production in Figure 2A. So, with 1/μ as the interval between mating bouts, the number of sperm released per bout S with sperm recharge becomes = ! = ! #$% (1 -) * +. But in this case, as for a fixed stock of sperm, males may benefit by maintaining a sperm reserve after each mating, despite a reduced sperm production efficiency and sperm release per mating bout. Let the parameter ρ be the proportion of s(1/μ) held in reserve and not released. Then the single-mate sperm release with sperm recharge becomes

= 1 -, ! #$% 1 -&/ .
(5)

See Figure 2B. Following the assumptions that having two mates doubles the male's frequency of mating, and that the male in this case expends all available sperm in each mating bout (i.e. β = 0), we have

= ! 1/ 2 = ! #$% (1 - ) .* +. (6)
See Figure 2C.

To summarize this section, we have developed mathematical relationships to represent sperm release in a mating bout as a function of mating frequency when sperm are continually recharged. These relationships depend on whether a male is mating regularly with one or with two females, and whether males store some sperm between mating bouts. are produced with diminishing returns as the stock approaches the storage limit smax. The

= ! 1/ = ! #$% 1 -&/ 0 1/μ, Time Between Matings S, Sperm Per Mating smax A s, Sperm Stock Dynamics Mating Time 1/μ 2/μ 3/μ 0 B s Sperm Stock Dynamics Mating Time 1/(2μ) 1/μ 3/(2μ) 2/μ 5/(2μ) 3/μ 0 C ρsmax ! 1/ 2 = ! #$% 1 -&/ ! #$% , / 1 -, 1 -&/ SPERM RELEASE PER MATING SPERM RELEASE PER MATING smax smax ! 1/ = ! #$% 1 -&/
coefficient γ is the rate constant for sperm accumulation. B. Sperm availability over time, when mating with a single female while maintaining the proportion ρ of smax in reserve at each mating time. C. Sperm availability over time, when a male alternates regular mating with two females, each at frequency μ; his interval between mating bouts is 1/(2μ). We assume that no reserves are retained in this case.

Overall, mating rate influences the advantage or disadvantage to a male of having a second mate through the implications for gamete availability and fertilization efficiency. Other influences on the optimal number of mates arise from benefits of male help at the nest and costs of mating, as noted in the next section.

Fitness

For simplicity and clarity, we assume that both partners' reproductive success is determined during the mating interval τ and that fitness is maximized when reproductive success during τ is maximized, as is typical in semelparous species [START_REF] Bell | The costs of reproduction and their consequences[END_REF]. Note that fitness maximization would still hold for the male with subsequent partnering, if the same conditions apply to each partnership.

When the male has a single mate, eggs are fertilized at the efficiency λ1, obtained by substituting E (from equation ( 2)) and S1 (from equation ( 3) or ( 5), depending on the mode of sperm production) into equation ( 1). Incorporating the total number of eggs released ε and the time interval τ during which these are released produces an overall single-mate fertilization rate of λ1ε/τ. Similarly, when the male has 2 mates, S2 (from equation ( 4) or ( 6)) is substituted into equation ( 1) to obtain λ2, yielding a 2-mate fertilization rate of λ2ε/τ. Figure 1 illustrates how the parameters and key variables link mating to fitness (here, number of offspring successfully produced) for the two different types of sperm production: sperm recharge and a total stock of sperm set at maturation.

Suppose that a male with a single female mate provides more direct benefits (e.g. better care for fertilized eggs and developing offspring in the nest) than he could by attending two females. We express this additional increment of reproductive benefit per mate with one mate rather than two by the fitness multiplier θ > 1. (This effect could instead be equivalently expressed as a cost, i.e. a reduction of care per mate when he attends two females, from having a second mate.) The features and implications of paternal care have been the focus of much research [START_REF] Goldberg | The costs and benefits of paternal care in fish: a meta-analysis[END_REF][START_REF] Gross | The evolution of male and female parental care in fishes[END_REF][START_REF] Stockley | Paternal care and litter size coevolution in mammals[END_REF], but for clarity here we must ignore these details and address only the net effect of dilutable benefits [START_REF] Tazzyman | Fixed and dilutable benefits: female choice for good genes or fertility[END_REF], including care, and how male-provided benefits may increase through the fitness multiplier θ with a single female mate rather than two.

Moreover, we express any fitness costs per mating, arising for example from physical damage to mates or enhanced predation risk [START_REF] Daly | The cost of mating[END_REF]Parker, 1970b;[START_REF] Rowe | The costs of mating and mate choice in water striders[END_REF] with negative exponential functions of the mating rate µ and sex-specific coefficients cf and cm.

Putting these features together, based on the total number of eggs ε provided by each female, allows us to express male and female fitness F via expected total offspring production over the reproductive interval τ, depending on the number of females attended by the male, as

0 1 = 2 3 4 , ( 7
)
0 # = 2 3 5 , 0 1 = 3 4 , and
0 # = 2 3 5 ,
where the subscripts of fitness indicate the relevant sex and the number of females attended by the male.

We now have the fitness functions for each sex corresponding to one or two female mates per male, based on fertilization efficiency, egg number, male-provided dilutable benefits, and mating costs. These functions provide a basis for each mate to achieve the best mating rate contingent on the other mate's preference.

Solving the Game Between Mates

The result is a game to resolve the sexual conflict between mates, which we solve under the assumption that the mates can accurately predict their own and their partner's (or partners') costs and benefits in relation to mating rate. In the special case when β > 0 and thus females occasionally withhold eggs, we assume that both partners behave as if they know the magnitude of β, but males are unaware of egg withholding in any particular instance [START_REF] Petersson | False orgasm in female brown trout: Trick or treat?[END_REF][START_REF] Yamazaki | High frequency of mating without egg release in highly promiscuous nonparasitic lamprey Lethenteron kessleri[END_REF]. For example, physiological and environmental cues may inform females of ε and τ, whereas males may estimate them from female body size and environmental cues. The partners may respond to a male's θ and γ, σ, or smax based on his body size and environmental cues. The parameter ø may tend to be relatively independent of body sizes and environment. Mating costs, linked to increased predation and disease risk or energetic expenditure, may be partly determined by observable body size and environmental conditions.

We assume that the mates use heuristics [START_REF] Gigerenzer | Heuristics: The foundations of adaptive behavior[END_REF] to integrate these data and their implications for the fitness functions and thus their decision making. An implication of this line of thinking is that the responses should be plastic, depending for example on body sizes (e.g. ε, θ, σ, smax, γ, cf, cm, μmax) and mortality risk (τ, cf, cm).

The game between mates can be resolved by considering the relationships among these fitness functions in response to adjustments in mating frequencies, as we illustrate graphically.

The logic resulting in solutions of these games between mates is summarized in Appendix A.

This is based on the game theoretic perspective that each mate acts to maximize their own fitness subject to constraints imposed by the partner seeking its own fitness maximization. To put this more formally but without the need for mathematics, we identify evolutionarily stable (ESS)

strategies that correspond here to sex-specific strong Nash equilibria (Mesterton-Gibbons, 2000;

Weibull, 1995): mating rates from which neither sex can shift and improve its fitness under the constraints imposed by the other sex. In particular, although the mating rate that maximizes male and female fitnesses may differ, neither male nor female can force its partner to mate. However, the female can decide to desert the nest and seek another mate if her partner chooses instead to obtain a second mate when this generates a severe cost to the female. Because mating rate, the sole evolving variable, is continuous and unimodally related to fitness, the ESS strategies are pure-no mixed strategies are possible. To evaluate the sensitivity of results to the model's parameter magnitudes, we determine the resulting mating rates and fertilization efficiencies across ranges of magnitudes of each parameter, placing the results less central to our questions of interest in Appendix B.

Results

We summarize the results in the following sections by answering the four questions posed in the Introduction.

How can sexual conflict over within-pair mating rate be resolved for external fertilizers?

In this section, we focus on the results for sperm recharge to characterize and compare examples of the three outcomes that emerge across the full range of parameter magnitudes (Figure 3). These are found by comparing the fitness curves for a male with a single mate (blue lines), a female whose mate has no other female (red lines), a male with two female mates (black lines), and a female whose mate has a second mate (cyan lines). Corresponding fertilization efficiencies are for males with one mate (red lines) or two mates (black lines). We assume that each sex will attempt to choose the mating pattern and mating rate that maximize their fitness except when constrained by the choice of the other sex. fitness when the male is mating with a single female. The black line (k) is the male fitness when he is mating with two females; the cyan line (c) shows the female's fitness in that case.

Fertilization efficiencies (right-hand panels) are shown for when the male mates a single female (red line, r) vs 2 females (black line, k). In panels A and B, the male fitness multiplier θ = 2.1.

Both the male and female maximize their fitness with a single mate, which results in consensus monogamy, outcome 1. A Male and female fitness are indicated by the red dashed line, where females maximize their fitness at μ*. B Fertilization efficiencies are maximized near the optimal mating rate. Unsurprisingly, the efficiency is consistently higher for a male with one mate than for a male with two. C. In this case, the male fitness multiplier θ = 1.2. The male is always better off with a second female, which is the female's only option. Both fitness curves of the polygynous male (black) and his mate (cyan) are maximized at μ*, corresponding to the vertical cyan line. This pattern is polygyny, outcome 2. D. As in B, with very similar efficiencies. E.

Here, the male fitness multiplier θ = 1.5, the default magnitude. The negotiated settlement is a mating frequency just above the black dashed line, indicating where blue and black lines intersect at μ*. The male is restricted to a single mate. This pattern is referred to here as negotiated monogamy, outcome 3. F. Efficiencies are very similar to those in B and D.

The first outcome, which we refer to as "consensus monogamy", or outcome 1, corresponds to a situation where the male benefitted from having a single mate at all mating frequencies, involving no conflict between mates over sperm allocation to an additional female.

In this situation (Figure 3A), male and female fitness peak near the same mating rate. In this example we assume that females could maximize their fitness instead of the male's; their slightly lower optimal mating rate than that of males resulted from higher mating costs under our assumed standard parameter magnitudes, and a lower mating rate is easily enforced by slowing down the mating process (see Appendix A), allowing female preference to dominate (red dashed line). Fertilization efficiencies (Figure 3B) are moderately high but higher for the male with a single mate.

The second outcome, we refer to as "polygyny", or outcome 2. Outcome 2 resulted when the male was always better off with a second female, and females had no way of negotiating a better deal (Figure 3C). The male and female fitness maxima coincided at the vertical dashed cyan line. Any possibility of outcome 3 (see below) was precluded by the upper limit on mating rate μmax. The female's only recourse in this case is to attempt to drive her rival away [START_REF] Wittenberger | The evolution of monogamy: Hypotheses and evidence[END_REF], which the male might try to prevent; we ignored these potential aggressive dynamics in the present analysis. Fertilization efficiencies (Figure 3D) were very similar to those for outcome 1. This is because the male fertilizes about the same number of eggs per mating bout with each female at the lower optimal mating rate with slightly more sperm released per bout (none held in reserve).

The third outcome, which we refer to as "negotiated monogamy", or outcome 3, is depicted in Figure 3E. In this case, the male benefited from having a second mate below a pivotal mating frequency μ*, above which he was better off with a single mate. Because the female's fitness was much lower regardless of mating rate if she shared her mate, she would solicit mating from the male [START_REF] Briskie | Copulation patterns and sperm competition in the polygynandrous Smith's longspur[END_REF]as often observed;[START_REF] Hatch | Copulation and mate guarding in the northern fulmar[END_REF][START_REF] Hatchwell | An experimental study of mating competition in monogamous and polyandrous dunnocks, Prunella modularis: II. Influence of removal and replacement experiments on mating systems[END_REF] to a level just beyond the frequency at which he benefited from having a single mate. If the male would not mate that frequently, the female had a strong incentive to seek another mate to avoid the severe fitness cost of sharing, providing strong encouragement for the male to go along. Thus, the negotiated mating frequency was just above that at the pivotal frequency; fitness declined for mates above this cutoff, eliminating any incentive to mate more frequently. Fertilization efficiencies (Figure 3F) were similar to those for the other two outcomes, here based on the higher mating rate at μ*.

In Figure 3, the three different outcomes resulted from different magnitudes of the fitness coefficient accounting for male help at the nest, which we address in more detail below. With increasing magnitude of each parameter of the model (Table 1), the outcomes appear in the sequence 1 -> 3 -> 2 for the parameters x (egg number exponent), smax (maximum recharged sperm per mating bout), ϕ (sperm:egg fertilization coefficient), ρ (sperm recharge reserve), and τ (reproductive interval duration); and the outcomes appear in the order 2 -> 3 -> 1 for parameters ε (eggs per female), β (sham mating proportion), γ (sperm recharge production coefficient), θ (fitness multiplier for male help), and μmax (maximum possible mating rate). Outcomes are insensitive to the magnitudes of parameters cf and cm (mating cost coefficients for females and males, respectively). We illustrate these results in Appendix B.

How does this conflict resolution depend on whether or not males continually regenerate sperm?

We answer this question by comparing the examples in Figures 3 (sperm recharge) and 4

(stocked sperm). The same three outcomes (Figures 4A, 4C, and4E) in same sequence with respect to parameter magnitudes are found for stocked sperm as for sperm recharge, following the same logic in accounting for each outcome. Outcome 3 (negotiated monogamy) yields a similar mating rate μ* (Figure 4E vs 3E) but slightly higher fertilization efficiencies with stocked sperm relative to sperm recharge (Figure 4F vs 3F). A key distinction arises with the other two outcomes. In both outcome 1 (Figure 4A) and outcome 2 (Figure 4C), the optimal mating frequencies are shown as zero, which implies biologically that the highest fitness is achieved with a single mating over the entire mating interval-in other words, fertilization of the female's entire egg load happens all at once with all of the male's sperm. The single mating bout (i.e. no repeated mating) maximizes fitness in this case because fertilization efficiency is maximized at the minimal mating rate, as a result of higher likelihood of sperm-to-egg contact for a given sperm:egg ratio with more eggs per mating bout. This negative relationship between efficiency and mating frequency results from a fertilization exponent x < 1, consistent with trend anticipated in the Methods. In contrast, sperm recharge results in a sharp decline in fertilization efficiency as the mating rate becomes very small, because sperm production per mating bout is constrained by the upper limit smax, while egg number per bout increases with a declining mating frequency. This effect ensures that the optimal mating rate for any of the three outcomes is positive, generally implying multiple mating episodes, for sperm recharge. he is mating with two females; the cyan line (c) shows the female's fitness in that case.

Fertilization efficiencies (right-hand panels) are shown for when the male mates a single female (red line) vs 2 females (black line). In, panels A and B, the male fitness multiplier θ = 2.1. The male has a single mate. Male and female both maximize their fitness at μ* = 0. This pattern is consensus monogamy, outcome 1. B. Fertilization efficiency declines with increasing mating rate, because the fewer eggs per mating are fertilized less efficiently. C and D. Here, the male fitness multiplier θ = 1.2. The male has a single mate.; the male is always better off with a second female, which is the female's only option. Both fitness curves are maximized at μ* = 0.

This pattern is polygyny, outcome 2. The fertilization efficiencies here are similar to those in B.

E and F. Here, the male fitness multiplier θ = 1.5, the default magnitude. The negotiated settlement is a mating frequency just above the black dashed line, indicating where blue and black lines intersect at μ*. The male is restricted to a single mate. This is referred to negotiated monogamy, outcome 3. Fertilization efficiencies are very similar to those in B and D.

To summarize the key result, for stocked sperm, only the negotiated mating frequency (outcome 3) allowed repeated within-pair mating, whereas with continuous recharge, fitnesses are always maximized at intermediate mating frequencies.

How do the mating patterns depend on the dilutability of male-provided benefits?

With the parameter θ, we represent the additional contribution to reproductive success in a nest that a male makes by restricting himself to a single mate, rather than two. Figure 5 shows how the mating rate, outcome, and fitness depend on the magnitude of θ. With θ = 1, there is no reproductive advantage per nest of the male's fidelity to one mate regardless of the sperm production mechanism, and outcome 2 with the male attending two mates is the stable outcome (Figures 5A and5C). For intermediate magnitudes of θ, the result is outcome 3 (negotiated monogamy), with mating rate steeply declining for larger θ. Magnitudes above about 1.65 (sperm recharge, Figure 5A) or 1.9 (stocked sperm, Figure 5B), the result is outcome 1 (consensus monogamy). Except in the zone of outcome 3, mating frequencies are low and independent of θ (sperm recharge) or minimal (i.e. once for the female's entire egg load, stocked sperm). Fitnesses are constant within sex and differ in favor of males by about a factor of 2 in the outcome-2 zone, converging to nearly identical magnitudes for the other outcomes, with magnitudes rising from the increased levels of male help. Overall, the magnitude of male help and the extent to which this reproductive contribution can become diluted by a males's divided loyalties between females strongly influences the mating patterns expected to arise. Why might females sometimes withhold eggs during mating bouts?

Consider first the sperm recharge case. With default parameter magnitudes (Table 1), females are better off not withholding eggs (Figure 6A, red line)-their highest fitness

C D B A

corresponds to β = 0. However, changes in some of the parameter magnitudes yields magnitudes of β between 0 and 1 with higher female fitnesses than for β = 0. Figure 6 contains five examples with a range of β magnitudes producing increased female fitness. In each case, the relevant range of β is between a pair of vertical dashed lines on the figure, with the maximal-fitness magnitude of β indicated by a solid vertical line. These are for a total number of eggs per female ε = 10,000 (default ε = 50,000) (Figure 6B); raising either the sperm:egg fertilization coefficient ϕ from 0.003 to 0.007 or the maximum sperm recharge smax from 50,000 to 117,000 (Figure 6C); increasing the sperm production coefficient γ from 3 to 7 (Figure 6D); increasing the total reproductive interval duration τ from 96 hours to 480 hours (Figure 6E); and decreasing the fitness multiplier for male help from 1.5 to 1.3 (Figure 6F). maximizing (optimal magnitude) of β. A All parameters are at default magnitudes (Table 1), and outcome 3 (negotiated monogamy) prevails over the whole range of β. In all other cases here, the patterns arise with outcome 2 (polygyny) at β = 0, shifting to outcome 3 at some higher magnitude of β. B: Total eggs per female ε = 10,000. C Either ϕ = 0.007 or smax = 117,000. D γ = 7. E τ = 480. F θ = 1.3.

For stocked sperm, raising β above zero did not increase female fitness for any of the alternative magnitudes we checked of the male help coefficient θ or the reproductive interval duration τ. However, substantial increases in the female's egg load ε or the sperm:egg fertilization coefficient ϕ did increase female fitness for ranges of β between 0 and 1, similar to sperm recharge.

These results show that optimal magnitudes of the frequency of egg-withholding by females during mating can in some cases exceed zero. So in some situations females may actually increase their own fitness by not releasing eggs in every mating bout.

Discussion

In this analysis, we addressed the possibility that repeated within-pair mating could be a way to resolve the sexual conflict over fertilization efficiency and dilutable benefits in external fertilizers. We showed that repeated within-pair mating can be a mate guarding tactic used by females to manipulate males into monogyny through increased fertilization efficiency and increased benefits provided by males to their joint offspring. Despite mating costs that favor low mating frequency for some parameter values, other magnitudes (discussed below) allowed females to raise mating frequency up to a level that secured monogamy. Under the default parameter values, based on lamprey reproductive biology, the ESS mating frequency reached 10 matings per hour, similar to what is observed in these animals [START_REF] Applegate | Natural history of the sea lamprey (Petromyzon marinus) in Michigan[END_REF][START_REF] Malmqvist | Breeding behaviour of brook lampreys Lampetra planeri: Experiments on mate choice[END_REF][START_REF] Manion | Spawning behavior and fecundity of lampreys from the upper three Great Lakes[END_REF].

While repeated within-pair mating was initially thought to result from males trying to maximize their paternity share in the context of sperm competition [START_REF] Møller | Copulation behaviour of birds[END_REF], [START_REF] Petrie | Copulation frequency in birds: why do females copulate more than once with the same male?[END_REF] proposed that frequent copulation, possibly with solicitation, could be a way for a female to guard her mate by depleting his time, energy, sperm supply, and opportunities to mate with others. She predicted that females paired with high-quality males (providing more direct benefits, in the form of paternal care or territory quality) in species with more intense female-female competition for high-quality males should tend to solicit mating repeatedly (e.g. [START_REF] Davies | Female control of copulations to maximize male help: a comparison of polygynandrous alpine accentors, Prunella collaris, and dunnocks, P.modularis[END_REF][START_REF] Saether | Male mate choice, sexual conflict and strategic allocation of copulations in a lekking bird[END_REF]. Likewise, females in such species are expected to solicit mating throughout the period when their mates can provide direct benefits, such as paternal care or territory quality [START_REF] Doran-Sheeny | The strategic use of sex in wild female western gorillas[END_REF] and possibly outside the fertile period; e.g. [START_REF] Wysocki | The frequency and timing of courtship and copulation in blackbirds, Turdus merula, reflect sperm competition and sexual conflict[END_REF], and more intensively when the risk of extra-pair copulation or desertion is high [START_REF] Davies | Female control of copulations to maximize male help: a comparison of polygynandrous alpine accentors, Prunella collaris, and dunnocks, P.modularis[END_REF]e.g. Eens and Pinxten, 1995;[START_REF] Sandell | Already mated females constrain male mating success in the European starling[END_REF].

In place of the classical direct/indirect categorization for male-provided benefits [START_REF] Andersson | Sexual selection[END_REF], we used the alternative concept of dilutable benefits proposed by [START_REF] Tazzyman | Fixed and dilutable benefits: female choice for good genes or fertility[END_REF]. They emphasized that the distinction between fixed benefits whatever the number of mates and those that are dilutable among the male's mates could shape the form of female preference. Instead of a binary categorization, we modelled benefit dilutability as a continuous feature through the parameter θ, which quantified the benefit to the offspring of a monogamous father relative to the offspring of a polygamous one. This quantitative approach showed that the dilutability of benefits could affect the evolutionarily stable outcome (monogamy, polygyny, negotiated monogamy) and the mating rate at equilibrium (Figure 5). For example, when a male fish accommodates the eggs of all his mates in a single nest [START_REF] Jones | How cuckoldry can decrease the opportunity for sexual selection: Data and theory from a genetic parentage analysis of the sand goby, Pomatoschistus minutus[END_REF][START_REF] Kraak | Sexual selection in sticklebacks in the field: correlates of reproductive, mating, and paternal success[END_REF], the size of the nest can set the dilutability of the benefits it provides: egg survival would depend less on the number of females which oviposited in it (i.e. less dilutable benefit) in a large nest than in a small nest. In such a system, our model predicts that negotiated monogamy and repeated mating should be found in nests of intermediate size, while monogamy and polygyny with low mating rate should be observed in small and large nests, respectively.

A central role in our model was played by gamete management from both the male and female perspective. Females of species with marked seasonal reproduction usually start the season with a finite stock of oocytes that may mature all at once or in batches, a feature known as determinate fecundity in fishes [START_REF] Ganias | Understanding the determinateindeterminate fecundity dichotomy in fish populations using a temperature dependent oocyte growth model[END_REF] and pro-ovigeny in insects (particularly in parasitoids, [START_REF] Jervis | Life-history strategies in parasitoid wasps: a comparative analysis of 'ovigeny[END_REF]. Moreover, females of externally fertilizing species can control the frequency with which they release their eggs to have them fertilized by their mates' sperm during mating [START_REF] Kiflawi | Adaptive gamete allocation when fertilization is external and sperm competition is absent: Optimization models and evaluation using coral reef fish[END_REF][START_REF] Shapiro | Mating tactics in external fertilizers when sperm is limited[END_REF]. They can also occasionally withhold eggs despite adopting the usual mating posture that in this case triggers ineffective ejaculation [START_REF] Jones | The spawning behaviour of brown trout and salmon[END_REF][START_REF] Esteve | Two examples of fixed behavioural patterns in salmonines: female false spawning and male digging[END_REF][START_REF] Schneider | SCUBA observations of spawning small-mouth bass[END_REF][START_REF] Ridgway | Courtship and spawning behavior in smallmouth bass (Micropterus dolomieui)[END_REF][START_REF] Roy | Quantitative and qualitative analysis of spawning behaviour of Heteropneustes fossilis (Bloch.) (Siluridae) in laboratory aquaria[END_REF][START_REF] Yamazaki | High frequency of mating without egg release in highly promiscuous nonparasitic lamprey Lethenteron kessleri[END_REF][START_REF] Petersson | False orgasm in female brown trout: Trick or treat?[END_REF]. This deceptive behavior can be quite frequent, reaching for example 65% of matings in the Arctic lamprey [START_REF] Yamazaki | High frequency of mating without egg release in highly promiscuous nonparasitic lamprey Lethenteron kessleri[END_REF]. Outcome 3 (negotiated monogamy), emerging under some magnitudes of our model's parameters, indicates that females may adjust their rate of egg release, and hence the rate of within-pair mating. This can maximize female fitness based on the sperm supplied and the dilutable male-provided benefits. We also showed that under some conditions, withholding eggs for a substantial proportion of matings could be adaptive for the females because all other parameters being equal, it could result in a shift from polygyny to negotiated monogamy and an associated increase in female fitness. The proportion of eggs withheld having a quite weak effect on female fitness (flat curve around optimum in Figure 6) and a negative effect on male fitness, one could expect males to evolve counteradaptations. However, efficient external fertilization often requires males to ejaculate synchronously or even slightly before egg release, especially under sperm competition [START_REF] Fitzpatrick | Ejaculate expenditure and timing of gamete release in rainbow trout Oncorhynchus mykiss[END_REF][START_REF] Yeates | A two-second delay confers first-male[END_REF], despite the risk of getting tricked by females. Our model indicates that egg withholding could favor the evolution of mating rate as a male-guarding strategy in females, but the probability of withholding (β) was not allowed to evolve. On the other hand, we suggest that future attempts to understand the evolution of egg withholding in external fertilizers should account for its possible role as a mate guarding strategy.

The male-female settlement on mating rate obtained under outcome 3 resembles the logic of reproductive skew (e.g. see [START_REF] Buston | Reproductive skew and the evolution of conflict resolution: a synthesis of transactional and tug-of-war models[END_REF], except that the negotiations here are between sexes rather than within sex. As in transactional models of reproductive skew, where a dominant individual grants just enough reproduction to group members so that they benefit more by staying and helping than by leaving, here the female ensures a mating rate just sufficient to incentivize the male to stay, and to secure exclusive access to dilutable benefits. In transactional models of reproductive skew, the access of subordinates to reproduction is limited to what the dominant consents to give. Yet one subtlety of our model arises from physiological processes such as sperm recharge rate or amount of stored sperm that impose a limit on the staying incentives the male can accept from the female. Sperm management, and in particular the dynamics of spermatogenesis, had major effects on the evolutionarily stable outcome of the sexual conflict addressed by our model. First, repeated within-pair mating was predicted to some extent under all three outcomes (1: consensus monogamy, 2: polygyny, 3: negotiated monogamy) when sperm was recharged after ejaculation, whereas it was predicted to occur only under negotiated monogamy in the case of finite stock (Figures 34). Second, within each scenario of spermatogenesis, parameters pertaining either to the maximum sperm storage, to the reserve held at each ejaculation, or to the rate of sperm production affected the outcome (1, 2 or

3) and the mating rate (see Appendix B).

Interspecific variation exists in the dynamics of sperm stocking, with males of many species being able to recharge their sperm stock during a refractory period [START_REF] Dewsbury | Ejaculate cost and male choice[END_REF], and males of other species relying on a finite sperm stock once mature [START_REF] Boivin | Spermatogeny as a life-history index in parasitoid wasps[END_REF][START_REF] Docker | The lamprey gonad[END_REF][START_REF] Michalik | Male reproductive system of spiders[END_REF][START_REF] Wootton | Reproductive biology of teleost fishes[END_REF]. Within species, interindividual variation is observed either in the maximum sperm storage or in the rate of recharge [START_REF] O'dea | Male body size and condition affects sperm number and production rates in mosquitofish, Gambusia holbrooki[END_REF], and selection could operate on this variability, especially when sperm limitation impacts a male's fitness [START_REF] Dewsbury | Ejaculate cost and male choice[END_REF]. Nevertheless, most models of gamete management focus on gamete release (so-called ejaculate economics) without explicitly representing the dynamics of gamete synthesis, whether the stake is fertilization efficiency in external fertilizers without sperm competition [START_REF] Kiflawi | Adaptive gamete allocation when fertilization is external and sperm competition is absent: Optimization models and evaluation using coral reef fish[END_REF][START_REF] Shapiro | Mating tactics in external fertilizers when sperm is limited[END_REF] or sperm competition in polyandrous internal fertilizers [START_REF] Parker | Sperm competition and ejaculate economics[END_REF][START_REF] Wedell | Sperm competition, male prudence and spermlimited females[END_REF]. Our results suggest that repeated within-pair mating should be observed in species where males recharge their sperm only under conditions favoring negotiated monogamy, and that female repeated solicitation for mating should coevolve with rapid sperm recharge.

Our model focused on polygyny and did not consider polyandry or polygynandry, despite the "polyandry revolution" [START_REF] Parker | Polyandry: the history of a revolution[END_REF][START_REF] Pizzari | The polyandry revolution[END_REF] initiated by Parker (1970a). Our point is not to downplay the major role of sperm competition in the evolution of mating behavior (although external fertilizers may be less exposed to sperm competition; [START_REF] Fitzpatrick | Sperm competition and fertilization mode in fishes[END_REF], but to identify another process which may be at play. Many models of gamete allocation exist in the context of polygynandry [START_REF] Parker | Sperm competition and ejaculate economics[END_REF][START_REF] Wedell | Sperm competition, male prudence and spermlimited females[END_REF], some of which allow gamete allocation among matings to evolve in both males and females [START_REF] Abe | Sperm economy between female mating frequency and male ejaculate allocation[END_REF][START_REF] Alonzo | Selection on female remating interval is influenced by male sperm competition strategies and ejaculate characteristics[END_REF]. These models address multiple mating without explicitly distinguishing between mating with multiple partners and repeated within-pair mating. But repeated within-pair mating could also be a way for polyandrous females to express male preference [START_REF] Andrade | Male condition, female choice, and extreme variation in repeated mating in a scaly cricket, Ornebius aperta (Orthoptera: Gryllidae: Mogoplistinae)[END_REF][START_REF] Simmons | Sperm competition as a mechanism of female choice in the field cricket, Gryllus bimaculatus[END_REF] or to limit harassment by other males [START_REF] Wysocki | The frequency and timing of courtship and copulation in blackbirds, Turdus merula, reflect sperm competition and sexual conflict[END_REF]. Theoretical work incorporating sperm competition to the kind of model described here would therefore be a further step toward the understanding how sexual conflicts over mating rate are resolved.

Figure 1

 1 Figure 1 Overview diagram of mating and reproductive success in the model. Variables (bold

Figure 2

 2 Figure 2 Dynamics of continuous sperm recharge. A. Sperm per mating S = s(1/μ), where males

Figure 3

 3 Figure 3 Fitness and fertilization efficiency as functions of mating frequency in 3 examples,

Figure 4

 4 Figure 4 Fitness and fertilization efficiency as functions of mating frequency in 3 examples,

Figure 5

 5 Figure 5 Mating frequencies and fitnesses over a range of the male fitness multiplier θ. Vertical

Figure 6

 6 Figure 6 Fitnesses of males (blue lines) and females (red lines) vs the blank or withheld-fraction

D

  Figure legends for Appendix B

Figure B- 2 B

 2 Figure B-2 Mating frequencies and fitnesses over ranges of parameter magnitudes, with

Figure B- 3

 3 Figure B-3 Mating frequencies and fitnesses over ranges of parameter magnitudes, with sperm

Figures

  Figures from Appendix B

  Figure B1

Table 1 .

 1 Parameters of the Model

	Default

Acknowledgments

We acknowledge grant support from the Fulbright Foundation to PHC funding his 2019-2020 year in residence at the INRAE field station in Saint Pée sur Nivelle, France; he greatly appreciates the warm hospitality and enthusiastic collaboration of the Ecobiop research group.

PHC also acknowledges sabbatical support from the University of Kentucky.

This work was supported by a Research Scholar grant to PHC from the Fulbright Foundation, covering his 2019-2020 year in residence at the INRAE field station in St. Pée-sur-Nivelle, France.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Availability of data and material: Not applicable, this project generated no empirical data.

Code availability: The MATLAB ® code used to generate the results is available on request from the authors.

Author Contributions: Conceptual development (PHC, CT); modeling, computer coding, figure preparation (PHC); writing and editing (PHC, CT); final approval (PHC, CT) Appendix A. Logic resulting in solutions to the games between mates There are three possible qualitatively different outcomes (1, 2, and 3), based on the four mating-frequency-related fitness magnitudes (the fitness of a male with one mate [M-1], a male with two mates [M-2], a female when her mate has no other mate [F-1], and a female when her mate has an additional mate [F-2]). The circumstances in which each outcome arises are as follows:

1: Consensus monogamy. Here the male or the female can maximize fitness-whichever has the lower optimal mating frequency. This requires that M-1 and F-1 are greater than M-2 and F-2 across the entire possible range of mating rates. Because either mate can slow the mating rate in the absence of forced mating, the sex with the lower optimum will prevail-which amounts to manipulation rather than negotiation to achieve an evolutionarily stable mating rate that neither sex can improve on. 2: Polygyny. When M-2 exceeds M-1 for all mating frequencies, the male will attempt to obtain and retain a second mate, and the first female has no recourse. This evolutionarily stable outcome generally requires a fertilization efficiency almost as high for each female in the twomate case as for a single mate. In the present analysis, we assume that the second mate is readily available when desired, but in nature a lag before a second mate can be acquired benefits the first mate. In this case, one female strategy might be to stay with a male only until a second mate arrives, a possibility we do not pursue further here. Whether this could provide sufficient protection for the eggs already released would depend on the particular circumstances.

3: Negotiated monogamy. There is some mating frequency at which M-1 = M-2. In this case, letting the mating rate of equality be μ*, at μ*, with both slopes negative. The intersection requires that male fitness is higher with two females than with one at low mating rates because of efficient use of sperm; but fitness is lower with two females than with one at high mating rates because of less efficient sperm use. In this case, females need to avoid sharing their mate with an additional female, because F-2 is generally much lower than F-1. Females seeking to accomplish this may solicit mating. Males will comply because in this situation females will avoid or abandon males with a second female. The compromise reached at M-1 = M-2 can be considered the result of "negotiation". This is an evolutionarily stable outcome, because neither male nor female can increase fitness by imposing a different mating rate.

Appendix B. Additional Analysis of Parameter Effects

Mating rates and outcomes across plausible ranges of all parameters from Table 1 (except θ, for which mating rates were explored in text Figure 5) are presented here in Figures B1-B4. sensitivity analysis, from which we conclude that mating rates for both types of sperm production are very strongly sensitive to the magnitude of θ (text Figure 5 and Figure B4-B), strongly sensitive to the magnitudes of the parameters ε, x, smax, ϕ, γ, ρ, τ (for sperm recharge, but only moderately sensitive for stocked sperm), and cm, moderately sensitive to β, and incentive to μmax and cf. This overall pattern suggests that mating rates may respond extensively to the physiological and ecological features that can influence parameter magnitudes under a given set of conditions. Particular attention must be paid to understanding how the relative contributions from male help θ may be expressed depending on whether the male is able to retain a second mate. Outcomes are generally much less sensitive to parameter magnitudes, especially for stocked sperm, but again with θ as an exception.


Paulo Pereira

Irene Maria Bollati

Isabelle Rouget

Manu Monge-Ganuzas

Manu Monge

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The ProGEO SW Europe Regional Working Group was founded in 2010, gathering the ProGEO members from France, Italy, Portugal and Spain. Since then, the group met in five occasions to discuss the main topics regarding geoconservation initiatives in each country and to set strategies for collaboration between geoheritage researchers and practitioners.

IN FRANCE

Geoconservation origins date back to the mid-nineteenth century, and the recognition and protection of geoheritage are today integrated in French environmental policies and in the law since 2002. The national inventory programme was launched in 2007.

• 3455 inventoried geosites available in a WebGIS database • Geotope Protection Order was established in 2015 • Increased awareness among the general public and planners

• 7 UNESCO Global Geoparks

IN ITALY

The debate on geoheritage began in the early 1990s in the academic community. The national inventory of geosites was published online by the Italian Institute for Environmental Protection and Research, starting in the 2000s.

IN PORTUGAL

Geoheritage and geoconservation are topics mostly discussed In academic forums, aiming at the engagement of the nature conservation official entities in their geoconservation duties established by law since 2008.

IN SPAIN

Geoconservation policies have been developed over the last 40 years thanks to the work of the Spanish universities, the Geological Heritage Commission of the Spanish Geological Society and the Spanish Geological Survey.

• 4544 inventoried geosites available in a WebGIS database

• National legislative geoconservation framework since 2007

• Wide range of geotourism offer

• 16 UNESCO Global Geoparks
The main aim of the regional group is the collaboration between the four countries, through the organization of scientific meetings and courses, the promotion of geoheritage and geoconservation amongst university students, the comparison and discussion of legislation regarding geoheritage protection and the development of a SW Europe regional inventory of geosites based on the national inventories and on a geological frameworks model common to the four countries.

• 325 inventoried geosites (of national and international relevance)

• Ongoing fusion of geosite inventories for management purposes

  Paulo Pereira Institute of Earth Sciences, University of Minho, Portugal Irene Maria Bollati Earth Sciences Department, University of Milan, Italy Isabelle Rouget Natural History National Museum, Paris, France Manu Monge-Ganuzas Service of Urdaibai Biosphere Reserve. Basque Government, Spain

•

  Official nature policies starting to include geoconservation• 5 UNESCO Global Geoparks• 2100 inventoried geosites available in a WebGIS database • Geoheritage is protected only through some regional laws• 4 geosites in the UNESCO World Heritage

	• 11 UNESCO Global Geoparks
	PROGEO SW EUROPE

REGIONAL WORKING GROUP