Dr Etienne Ducrot

Costantino Creton
email: costantino.creton@espci.fr

Prof. C Creton

Characterizing Large Strain Elasticity of Brittle Elastomeric Networks by Embedding them in a Soft Extensible Matrix

Keywords: elastomers, mechanical properties, double network, fracture, non-linear elasticity

 . In the present paper, we report in more detail how the increase in strain at break resulting from the toughening can be used to provide great insight in the large strain properties of otherwise brittle acrylic well crosslinked networks. The networks have been prepared by sequences of polymerization and swelling with monomers. The parameters that have been varied are the nature of the base monomers and the degree of crosslinking of the first network. We characterize the small strain properties, equilibrium swelling and large strain properties in uniaxial tension. We show that the large strain properties of the multiple networks are quantitatively controlled by the large strain properties of the stretched first network which acts as a percolating filler, while the small and intermediate properties are controlled by the entanglement density which can be largely superior to that of homogeneous networks.

Introduction

Elastomers are widely used because of their large reversible deformability up to strains of several hundred percent. Their elastic behavior stems from entropic elasticity of crosslinked flexible chains with a low glass transition temperature. [START_REF] Treloar | The physics of rubber elasticity[END_REF] In the linear regime, the elastic modulus is typically given by the number of elastic chains per unit volume, which in the absence of a swelling solvent and for a homogeneously crosslinked system, is inversely proportional to the average molecular weight between crosslinks M x . [START_REF] Rubinstein | Polymer Physics[END_REF] Yet in uniaxial extension the majority of unfilled elastomers soften slightly relative to the linear behavior and then break in a brittle fashion well before the chains reach the ir maximum extensibility. [START_REF] Gent | [END_REF] As a result, and with the notable exception of natural rubber, it is quite difficult to prepare an unfilled, well crosslinked elastomer, displaying a sufficiently high strain at break to characterize the distinctive strain stiffening in uniaxial elongation due to the finite extensibility of the chains. This intrinsic limitation of conventionally crosslinked elastomers is due to the very localized nature of the breakage of bonds near the interfacial plane. In the absence of viscoelastic dissipation, the threshold fracture energy  0 of the elastomers is quantitatively explained by the sequential fracture of highly stretched elastic strands in the plane of the interface as the crack propagates. [1,5] To avoid this damage localization process leading to early fracture, it is possible to modify the network architecture to create additional energy dissipation mechanisms. Many strategies have been tried to increase the toughness of unfilled elastomers by network design.. [6] [7] [1,8] Our group has recently reported the successful design of multiple interpenetrating network elastomers through the sequential polymerization of acrylic monomers, in such a way that the network synthesized first becomes progressively isotropically stretched by the swelling while a second and third polymerization result in nearly unstretched and loosely crosslinked chains [1]] . When the elastomer is highly stretched or broken, the prestretched network chains break in the bulk of the elastomer well before the unstretched ones, significantly delaying the nucleation and propagation of a macroscopic crack. The best of the reported acrylic elastomers are very stiff (up to E = 4 MPa) and combine extensibilities of 2-300% with true stresses at break in excess of 30 MPa and values of fracture energies of 9000 J/m 2 while maintaining residual deformations immediately after unloading of only a few percent of strain.

One of the key advantages of these interesting new materials is their very pronounced strain stiffening in uniaxial extension which now occurs before the material breaks. In some specific applications such as resistance to cavitation [9] , or dielectric elastomers [10] one expects this feature to be very useful to fine tune the properties. Furthermore the characterization of the strain stiffening of the material may be directly related to the progressive stiffening of the prestretched polymer chains as they approach their limiting extensibility.

In the current paper we explore how, by changing the monomer composition and the degree of crosslinking used in the first network it is possible to control the mechanical properties of the final material and to characterize the strain stiffening of an otherwise brittle elastic polymer network.

Results

Composition, homogeneity, elasticity

Multiple networks elastomers have been prepared by sequential swelling and polymerization steps as described in the experimental section and the main synthesis parameters are given in Tables 1 and2. The multiple network elastomers are composed of short and extended chains from the first network and long and nearly Gaussian chains from the second and third networks. All three networks are continuous throughout the sample. The measurement of the extractible shows that the proportion of free chains is negligible, (< 1 wt %) for every double network (DN) and triple network (TN). Different monomers have been used and result in different polymer chains connected together in a network: poly(methyl acrylate) (PMA), poly(ethyl acrylate) (PEA) and poly(butyl acrylate) (PBA). Samples are identified as X y

[Z[Z]],
with 'X' the first network monomer and 'y' its crosslinker concentration, 'Z' the monomer of the second/third network. The question of the miscibility, at the repeat unit level, between these different elastic chains needs to be addressed. Dynamic Mechanical Analyzer measurements of G' and G" at ̇ = 1 s -1 as a function of temperature, show only one T g for DN and TN containing PMA and PEA (Figure 1a), in agreement with a non phase separated system. However, the DN composed of PBA and PMA clearly has two peaks in G" as a function of temperature suggesting phase separation. Those samples stay nevertheless perfectly transparent, indicating that the phase separation may occur at a molecular length scale as in microphase separated block copolymers. [11] At T ~ 40 °C above the highest Tg, every multiple network behaves as a rubbery material in which the modulus increases linearly with the temperature and present a dis sipative behavior characterized by a low value of tanδ as shown in Figure 1b. This remains the case for the microphase separated BA 1 MA.

Equilibrium swelling of the first network

Immersed in a bath of solvent (the monomer methyl acrylate in our case), a polymer network swells to an equilibrium value where the loss in entropy due to chain stretching is compensated by the gain in entropy of mixing with the solvent molecules. [START_REF] Rubinstein | Polymer Physics[END_REF] The swelling ratio is then governed by the crosslinker concentration of the network, its structure and the quality of the solvent. If the methyl acrylate mo nomers that swell the network are polymerized, the chain stretching of the first network occurring during the swelling is then frozen in place by the polymerization of the chains of the second network [12] . If such a DN is immersed again in a monomer bath, it will swell again until the loss in entropy due to the stretching of the 1 st and 2 nd network chains is compensated by the gain in entropy of mixing with the liquid monomer. Such a method gives a versatile way to control the volume fraction of the stiff network chains and has been used for hydrogels as well. [13] However after the 3 rd polymerization, there will be three chain populations, a highly stretched but very diluted population of 1 st network chains, effectively constituting a stiff network, a proportion of lightly stretched 2 nd network chains and a majority of unstretched 3 rd network chains. Figure 2 shows the equilibrium swelling of the PEA first network in the EA y MA double and EA y MAMA triple networks networks after polymerization as a function of the crosslinker concentration in the 1 st network. Each swelling equilibrium defines the degree of isotropic prestretching  0 of the 1st network chains and those quantities for all samples tested in this study are summarized on Table 3.

Mechanical properties in uniaxial deformation

The elastic properties of simple crosslinked networks above their glass transition temperature can be typically tuned by changing the crosslink density. A higher concentration of crosslinker leads to a higher density of elastic chains and therefore to a higher elastic modulus.

The effect of changing the crosslink density on the uniaxial stress-strain curves for three different SN of PEA is reported on Figure 3a. A side effect of increasing the modulus is typically however a strong decrease of the extensibility of the material which is consistent with the Lake-Thomas model [14] .

Compared to these simple polyacrylate networks, our multiple networks present striking differences in mechanical behavior. As shown in Figure 3b for the case of the EA 0.5 family of samples, while the simple network of EA 05 and the second network of polymerized alone MA 2N appear to follow Gaussian elasticity with no dramatic change in curvature, the sequentially polymerized DN EA 05 MA shows a pronounced stiffening at high strain and the TN EA 05 MAMA presents both a stiffening at intermediate strains and then a softening at high strain. Furthermore the initial modulus clearly increases after each polymerization.

For the example of Figure 3b, several mechanical properties are enhanced relative to the parent SN. The initial modulus after each polymerization step increases from E = 0.8 MPa for the SN to 1.5 and 2.4 MPa for the derived DN and TN respectively. Concurrently the elongation at break also increases from λ b = 1.5 to nearly λ b = 2.5 in DN and TN relative to the corresponding SN which is very brittle and poorly extensible (Figure 3a). The nominal stress at break is also highly enhanced by a factor of 8 and 28 relative to the SN for the DN and TN respectively. For the sake of comparison we also show a simple network (MA bulk ) of MA prepared to have a Young's modulus in the range of that of a TN. MA bulk is much more brittle than a TN with the same initial modulus, showing the effectiveness of incorporating prestretched chains in a matrix to combine stiffness and toughness. It should be noted that those values correspond to the highest values obtained from each type of sample.

Figure 3a,c andd shows the effect of the crosslinker concentration of the stiff network in the mechanical properties of single networks and derived multiple networks. As discussed above, while the initial modulus of the SN increases with the amount of crosslinker in the material (Table 4), the crosslinker concentration in the first network has also an important effect on the properties of the DN and TN made from the respective SN's. By adding more crosslinks in the first network, we increase the initial modulus of the DN and TN and more generally the stress level for all λ is very significantly increased. Secondly, we observe clearly that the onset of strain stiffening occurs at a lower extension for the DN and TN made from the more densely crosslinked first networks.

A second parameter that can be changed is the monomer used for the synthesis of the 1 st network. Figure 4 shows the stress-strain curves of three different first networks that have been synthesized from MA, EA and BA with the same weight fraction of crosslinker, and their corresponding double networks, with MA as a second monomer. Although the targeted average number of monomers between crosslinks is different, the targeted average molecular weight between crosslinks, and hence the elastic modulus are the same. The three DN all contain 30 wt % of first network (Table 3). In uniaxial extension, the three materials have indeed a very similar initial modulus in small strain (2 MPa) but their stress-strain curves diverge for λ > 1.4 and interestingly, the weaker first network gives the stronger double network in term of stress at break. The stiffening is observed for lower λ in the BA 1 MA sample than in the MA 1 MA, the stiffening of which is very limited. The nominal stress at break is multiplied by a factor of almost 5 between those two samples. EA 1 MA being between the two extreme.

Finally, multiple networks elastomers of pure PEA have also been prepared starting from a first network of EA 0.5 . Mechanical tests were performed at 20 °C in order to be around 40 °C higher than the T g of the pure EA networks. The uniaxial stress-strain curves for this series are shown in Figure 6 in comparison with the equivalent series but with MA as second and third monomer. The general aspect of the stress-strain curves between the two series is very similar except that the initial modulus and stress level are generally lower for fully EA based networks.

Discussion

While the increase in the mechanical properties of these interpenetrated and very elastic networks is remarkable, it is interesting to discuss in more detail how to control their properties by changing their composition (monomers, crosslinker)

Influence of the choice of monomer

In double network hydrogels, the nature of the two monomers has to be carefully chosen [13] to achieve a high mechanical reinforcement. Typically permanently charged polymers are used in the first network to reach equilibrium swelling ratios of the order of 10 and tough gels can be obtained in only two polymerization steps.

In the case of multiple network elastomers the need for a relatively low T g and an elastic behavior precludes the use of very polar monomers able to hydrogen bond strongly or to form dipole-dipole interactions. Given these restrictions, changing the monomer used to synthesize the stiff network appears to have a limited influence on the mechanical properties. The networks MA 1 , EA 1 and BA 1 have the same average molecular weight between crosslinks M x of 7500 g.mol -1 (from Table 4), but a number of repeat units N x between crosslinks which decreases with the molar mass of the monomer: N x = 86 for MA 1 , 76 for EA 1 and 59 for BA 1 .

Chains between crosslinks have therefore less backbone bonds in BA 1 than in MA 1 and since the maximum extensibility of a chain λ max scales with N x 1/2 one expects λ max to scale with M 0 -1/2 at a fixed value of M x . Because the prestretching of chains λ 0 is identical for the three DN samples of Figure 4b, the stress-strain curves of Figure 4b replotted as a function of should show a strain hardening at the same relative extension. Figure 5 shows that it is not the case for the BA 1 MA which can stretch more relative to its unstretched length than the EA 1 MA and MA 1 MA network. This may be due to the heterogeneous structure of the elastomer resulting from the microphase separation(Table 3). Note that this strain stiffening at higher values of l does appear to modify or reduce the reinforcement mechanism and the stress at break is higher. In elastomers, the multiple network effect is therefore still effective even if the two networks are not fully miscible. The main point is to introduce prestretched chains in the system and a large contrast in crosslinker concentration between the networks.

The comparison between the multiple networks made from an identical 1st network but where the unstretched networks are made from MA or EA is instructive. It shows first that a difference in monomer composition between the stiff and the soft networks is not needed for mechanical reinforcement in multiple network elastomers. However PMA and PEA melts do not have the same average molecular weight between entanglements. Although detailed measurements on PMA are difficult to find there is evidence that EA is less entangled than MA (M e (EA) = 13 kg.mol -1 [15] and M e (MA) = 11 kg.mol -1 [START_REF] Zosel | Advances in Pressure Sensitive Adhesive Technology[END_REF] ) and this may explain the differences in stress level observed. Those simple experiments demonstrate that in multiple network elastomers different monomers can be used to fine tune the properties as long as they can swell the first network sufficiently.

Stretching of chains in the stiff network due to swelling

In the swollen state or in the DN and TN, we assume that, even in the absence of any macroscopic deformation, the chains of the first networks are isotropically stretched by a factor λ 0 relative to the dry state as defined on Equation (1) by the ratio of thickness of the final network ( ) and the thickness of the corresponding first network ( ). For DN, the reported values of λ 0 on Figure 7 stay below 1.7. Given that the 1 st network is synthesized in solvent condition (in the presence of 50% toluene) and then subsequently dried, chains are slightly supercoiled in the preparation condition they are then only slightly extended compared to their Gaussian state once swollen. Once the polymerization of the 2 nd monomer has occurred inside the 1 st network, the entropy of mixing between monomer and network disappears and these DN can again be swollen with monomer. This 2 nd swelling step leads then to a higher stretching of the first network, up to a λ 0 of 2.5 and a volume fraction of 1 st network well below 10%. Such a high level of prestretching is not achievable in one step with neutral polymers.

(

) 1
The degree of prestretching (and the volume fraction) of 1 st network chains can be controlled through the number of swelling/polymerization steps or by the crosslinker concentration in the first network. An increase in crosslinker concentration gives a network with shorter chains between crosslinks and the equilibrium swelling decreases.

It should be noted that Yoo et al. [17] reported no significant mechanical enhancement of the toughness of silicone double networks in a case where the loose network is first prepared and then swollen by short oligomers subsequently crosslinked. Following this path, they did not introduce short prestretched chains in their system and the volume fraction of unstretched short chains was larger than that of the long chains. The prestretching of a fraction of chains in the final multiple networks appears therefore to be important to enhance the mechanical properties.

Origin of the initial modulus

An interesting feature of both DN and TN is the significant increase in modulus which is observed. For every family of multiple networks (DN or TN), the relation between the modulus E and the concentration of chemical crosslinker in the first network seems to be linear (Figure 8). However the absolute values are much higher for the DN series and then the TN series than for the three single networks and the slope is steeper when increasing the number of polymerizations, meaning that the impact of the crosslinker concentration in the first network becomes dominant after three steps of polymerization.

Multiple networks are composed of prestretched chains of the first network at λ 0 , and loosely crosslinked chains on second/third networks which are entangled together and loosely connected. In the TN there are more precisely three populations of chains: extended chains of the first network, lightly stretched entangled chains of the second network (although the very small amount of crosslinker introduced in the second network suggests that the 2 nd network chains can relax during swelling and are only lightly stretched. An estimate of the contribution to the modulus from the prestretched chains of the first network can be made using Equation (2). [28] with the volume fraction of first network, the initial modulus of the first network alone and the isotropic elongation of the first network in the final DN or TN due to swelling. We now propose two ways to take into account the contributions from additional chains of the second and third networks.

If the average molecular weight between entanglements of the second network is significantly lower than the average molecular weight between crosslinks of the first network, the second and third networks should simply contribute to the modulus in proportion to their weight fraction and . In TN, the prestretching of chains of the second network has to be taken into account by introducing an elongation of the second network . The final formula for the estimate of the modulus of DN and TN for this case is given in Equation (3). For every estimate the modulus of the second network alone , mainly due to the entanglement structure, has been taken at 0.9 MPa.

If on the other hand the crosslinks of the first network act as topological constraints and delimit a confining distance between additional entanglements for the second and third network, the expression for the modulus must be modified as presented in Equation (4) for DN. The expression for TN is more delicate to establish and is not treated here, it should

̅ ̅̅̅ (2)
( ) ( ) (3)
include the topological constraints from chains and crosslinks of the second network, as well as its stretching due to swelling.

where is the average molar mass of the gaussian chains of the second network between two stretched strands of 1 st network and can be expressed by Equation (5) in the case of a DN, taking into account the equilibrium degree of swelling and the difference in monomer type.

For that we consider the average weight between crosslinks in the unswollen first network of PEA. In the DN, due to swelling, the mesh is isotropically stretched by a factor λ .

The distance between crosslinks of the first network scale as ( ) λ , leading to the expression of given on Equation (5).

Estimates of the initial modulus obtained from Equations (3) and (4) are reported on Figure 8 with the experimental modulus. The estimate of the modulus from Equation (3) predicts well EA 0.5 MA, but fails for all other samples, showing that the assumption of simple additivity of entanglements from the two networks is incorrect for highly crosslinked first networks.

Equation (4), assumes effectively that the crosslinks of the first network control the entanglement density of the second network in an analogous way to a fixed network of topological obstacles controlling the diffusion of long chains through. [32] The predicted values of E from Equation (4) are in much better agreement with the experimental data than

) (4) ( ) (5)
the previous model but still underestimate the modulus for EA 2 MA. In this last sample the mesh size of the swollen first network corresponds to an equivalent PMA chain of 5500 g.mol -1 , significantly smaller than the average weight between entanglements for pure PMA.

Some additional entanglements between polymer chains of the two networks themselves have to be added for a better description, and not only a contribution from crosslinks of the first network.

From those simple models in DN we can estimate that for loosely crosslinked first networks (i.e.

), the key parameter to explain the modulus is the level of prestretching of the chains of the first network. Then for more crosslinked first networks (i.e.

) the crosslinks and chains of the first network form topological constraints for second network chains and are responsible for the increase in modulus by increasing the density of entangled chains. For TN, the high dilution of stretched chains also dilutes their contribution to the modulus (1 st model ineffective) and the presence of three entangled networks with different degrees of prestretching and some connections between networks, make the second model more difficult to use.

In the supplementary information we discuss the impact of transfer reactions during the polymerization of acrylates in the bulk. As discussed for hydrogels [18] , the remaining unreacted vinyl groups after the polymerization of the 1 st network and the normally occurring transfer reactions to the polymer will create some connecting points between the networks and so participate to the increase in modulus. A direct estimate of the connections between the two networks is difficult after the polymerization of the second and t hird networks. We estimated this contribution to be around 0.25 MPa, it is nevertheless too low to account for the large observed increase in modulus for the DN made from highly crosslinked first networks.

Interpretation of the stiffening at high strain

Double and triple networks show a marked stiffening in large strain in uniaxial extension and a high stress at break. For simple networks, samples break at too low an extension ratio to show stiffening and the second network of PMA alone (MA 2N ) may present stiffening but at a very high elongation which is not accessible with our experimental setup (i.e. λ > 5.5). For the DN, the onset of stiffening, which corresponds to the abrupt change in slope of the stress/strain curve, occurs before an elongation λ of 2, and even at lower values of λ for the TN.

In unfilled rubber-like materials, the phenomenon of strain stiffening is due either to the fact that the chains forming the network are stretched close to their finite extensibility [START_REF] Treloar | The physics of rubber elasticity[END_REF] or to strain induced crystallization [START_REF] Gent | [END_REF] or strain induced clustering [21] . This finite extensibility has been discussed theoretically at the chain level [START_REF] Treloar | The physics of rubber elasticity[END_REF] but although several empirical models exist [22,23] there is currently no molecular model that captures quantitatively the effect of nongaussian elasticity in a network [24,25] . One of the reasons is that this regime is often inaccessible experimentally for a macroscopic material because of premature fracture and models can only progress if more systematic data is available.

Acrylates do not self-associate by hydrogen bonding, and cannot crystallize so the observed strain stiffening in multiple networks has to be due finite extensibility of the shorter and already prestretched 1 st network chains in the system. To confirm this hypothesis we will analyze the data of a family of stress/strain curves of EA 0.5 based multiple networks reported on Figure 9. If we assume that chains of the 1 st network are stretched affinely with the degree of swelling in the final system (Table 5) we can define a degree of stretching at preparation λ 0 and a corrected elongation λ cor defined by Equation (6). In the multiple networks being stretched uniaxially, it corresponds to the real elongation of the chains of the first network in the direction of stretching relative to its end to end distance in the dry first network. Now that the elongation has been rescaled by the level of prestretching  0 (Figure 9), the stress ( cor ) of the DN and TN needs also to be corrected to correspond to the stress of the SN alone at that value of . To do so, we rescaled the reduced stress (defined as  R = ( N +σ C )/(   before the strain stiffening to that of the first network, with σ C a translation constant. We can then fit the master curve obtained by rescaling the stress strain data. with an empirical strain stiffening model inspired by the Arruda and Boyce equation and modified with Mooney parameters [23,26] to incorporate the softening at intermediate strain (Equation (7)).

with the Mooney-Rivlin parameters and λ limit the limiting extensibility of the chains obtained from the fit. Once the master curve is constructed, and are fitted freely on the SN curve, λ limit is adjusted manually to set the stiffening at the right elongation defined b y the rescaled DN and TN curves and an example for the EA 0.5 multiple networks family is shown in Figure 9. The functional form of the stiffening of the double network as a function of the corrected stretch is well described without adjustable parameters by the fit of the corresponding first network to Equation ( 7). This experimental evidence shows that the (6)

( ) (7)
stiffening is indeed controlled by the final extensibility of the minority chains of the first network. The stiffening of the triple network is also well described for λ cor < 4. At higher elongation, the stress of the TN diverges however from the master curve suggesting a damage mechanism due to internal bond breakage [1] . which is not the focus of the present paper. The same master curve construction was also applied to EA 1 and EA 2 based multiple networks with the same success,. Some representative master curves and best fits of Equation ( 7) are shown in Figure 10, and presented in Table 5.

As observed in the EA 0.5 family of samples, the initial modulus of the DN and TN is not well described by the model (Figure 10) confirming that the initial modulus is not purely the result of the prestretching of the chains of the first network. The stiffening however is well described by the model and directly relates to the finite extensibility of the minority chains of the first network. As can be seen in Table 4, the maximum extensibility of the chains of the first network as fitted by Equation ( 7), decreases with increasing crosslinker level.

Maximum extensibility of chains of the first network

It is also in principle possible to estimate the maximum average extensibility of the first network chains from their elastic modulus E. From the number density of elastic chains obtained from the modulus, it is possible to extract the average number of C-C bonds per network chain. Then using Equation ( 8) derived from the Kuhn theory [START_REF] Rubinstein | Polymer Physics[END_REF] , and knowing for PEA, PMA or PBA, the theoretical maximum of extensibility of the chains can be evaluated.

( )

Surprisingly for every family of samples, Figure 11 shows a very good correlation between , estimated from the Young's modulus of the first network, and , estimated from the stiffening parameter of Equation ( 7). To the best of our knowledge this quantitative agreement is remarkable. This result has several interesting implications. First of all this correspondence suggests that the slightly supercoiled chains formed during the first polymerization in the presence of toluene, are nearly unentangled and that the Gaussian chains of the other two networks do not prevent full extension of the chain. In those conditions one can argue that the prestreched chains carry most of the load and that the stiffening measured on the macroscopic sample directly reflects the stiffening of the chains themselves.

More generally, these results suggest that swelling a network in a polymerizable monomer is a good method to characterize the stress-strain curve of the chains composing this network all the way to their fracture. This information can then be used to understand better fracture processes in polymers where stretches at the crack tip greatly exceed the values that can be measured in uniaxial extension. Furthermore the synthesis method presented here, in addition of producing tough materials, is a very good way to study the non-Gaussian elasticity of the polymer chains and could be used for much mo re model first networks as was recently done for hydrogels [27] .

Conclusion

A series of multiple interpenetrating network elastomers have been successfully synthesized with sequential swelling/polymerization steps leading to interpenetrated multiple networks.

This synthesis method leading to elastic tough elastomers can also be used to study the large strain behavior of simple networks that would otherwise break at low strain. We show that the stiffening at high strain in multiple networks can not only be tuned but also be predicted from the crosslinker concentration of the first network and its degree of swelling in the final sample.

One can now furthermore use this simple and robust strategy to design and prepare synthetic rubbers with fine tuned nonlinear elastic properties involving significant strain stiffening. We believe that this strategy is universal and does not depend on polymer type as long as the material is in the rubbery state.

Experimental Section

Materials: The monomers, methyl acrylate (MA), ethyl acrylate (EA) and butyl acrylate (BA) and the crosslinker 1,4-butanediol diacrylate (BDA) were purified over a column of activated alumina to remove the inhibitor. The UV initiator, 2-hydroxy-2-methylpropiophenone (HMP) was used as received. Anhydrous toluene and cyclohexane were used as received as solvents.

All reagents were purchased from Aldrich.

Synthesis: Both double and triple network elastomers were prepared by a multiple step free radical polymerization procedure starting from identical first networks.

Preparation of the single network:

The stiff networks were prepared first by UV free radical polymerization of a solution of monomer (MA, EA or BA), BDA as crosslinker (1.45 -5.81 mol % relative to monomer) and HMP as a UV initiator (1 -1.49 mol % relative to monomer). The reactants were dissolved in anhydrous toluene as solvent at 50 wt %. All the reactants initially deoxygenated and stored in the glove box were mixed for a short time and poured in a 1 mm thick glass mold. The reaction was initiated by UV light (Vilbert Lourmat VL-215.L lamp, 365 nm, 10 µW/cm²) and left to proceed for 90 minutes. The sample was then taken out of the glove box and out of the mold, weighed and immersed in solvent baths for a week to remove every soluble components, unreacted species or free chains. The solve nt mixture was changed every day with a gradual degradation of its quality to induce a slow deswelling of the sample. Samples were finally dried under vacuum at 80 °C overnight and then weighed and stored at room temperature for later use. Those samples will be referred as single networks (SN) and noted X y with 'X' referring to the monomer type (EA, MA, BA)

and 'y' to the concentration of crosslinker.

Preparation of DN and TN:

Starting from a first network (SN), double networks (containing a stiff network embedded in a loose one) were prepared following a swelling and polymerization sequence. A piece of first network was swollen in a bath composed of the second monomer (EA or MA), BDA as crosslinker (0.01 mol %) and HMP as UV initiator (0.01 mol %) as reported on Table 2. The swelling step was performed in a sealed plastic box in the glove box with oxygen-free reactants. Once swollen at equilibrium, after two hours, the sample was carefully extracted from the monomer bath, gently wiped to eliminate the excess of monomer, and placed between siliconized PET sheets and glass plates. The polymerization was initiated and completed via a two hours UV irradiation. The mold was then taken out of the glove box, and the sample was removed, weighed and dried under vacuum at 80 °C over night. It was finally stored at room temperature until later use. Those samples will be referred to as double networks (DN) and noted X y Z with 'X y ' referring to the first network and 'Z' to the second monomer (EA or MA).

To reduce further the volume fraction of the 1st network, the same sequence of swelling/polymerization was repeated starting from double networks, the composition of the swelling bath being kept identical. Once again, after two hours of swelling, the sample was at equilibrium. The swollen double network was sealed between PET sheets and tightened between glass plates. The third network was polymerized inside the first two networks via a two hours exposure to the UV. The sample was then dried under vacuum at 80 °C overnight and stored at room temperature until later use. Those samples will be referred to as triple networks (TN) and noted X y ZZ with 'X y ' referring to the first network and 'Z' to the second and third monomer (EA or MA). It should be note that those network contain a stiff network (crosslinked and swollen), and two quite loose networks (very loosely crosslinked and almost unstretched). Transfer reactions between networks certainly occur during the second and third polymerizations so that these networks are loosely connected by covalent bonds.

Preparation of the networks without solvent: As references, loosely crosslinked networks of EA and MA were prepared alone in solvent free conditions by UV-initiated polymerization (HMP as UV initiator) between glass plates and a subsequent overnight drying step under vacuum at 80 °C to remove unreacted monomers. EA 2N and MA 2N prepared with 0.01 mol % of BDA are exact replicas of the second/third network but prepared alone (compositions in Table 2). MA bulk prepared with 2.5 mol % of BDA has been designed to present an initial modulus comparable to a TN.

Characterizations:

Mechanical tests: Mechanical tests were performed on a standard tensile Instron machine, model 5565, using a 100 N load cell and an environmental chamber to precisely control the temperature (25 °C or 60 °C). A video extensometer gave a local measurement of the stretch λ = l/l 0 where l 0 is the initial gauge length. The relative uncertainty of the measurements given by the load cell and the video extensometer are respectively 0.1 % in the range of 0 to 100 N and 0.11 % at the full scale of 120 mm. Specimens were cut into a dumbbell shape using a normalized cutter (central part: length 20 mm, cross-section 4 mm and thickness 0.6 -2.5 mm depending on the sample). Uniaxial tensile tests from small to large strain were performed at a constant velocity of the crosshead of 500 µm.s -1 and the typical initial strain rate on the central part of the sample was around 0.04 s -1 . Using the nominal stress and strain, the Young's modulus was calculated at a strain of less than 10 %.

For single networks, the classical theory of rubber elasticity was applied to determine the average molecular weight between crosslinks M x.

[3] Table 2: Formulation of the second network swelling bath, (a) [BDA]= n BDA /n ,with n and n BDA the number of moles of monomer and BDA respectively, (b) [HMP]= n HMP /n, with n HMP the number of moles of HMP, (c) M x th the theoretical weight between crosslinks as M x th = (M 0 * n) /(2*n BDA ), with M 0 the molar mass of the monomer.

ToC figure

The best fits of the uniaxial elongation data to Equation S1 give the same contribution to the modulus G x for the two samples. For a sample prepared with no crosslinker, G x should have been equal to zero and the resulting material would have been a polymer melt. This is evidence that there are transfer reactions during the polymerization which create additional crosslinks.

Assuming rubber-like elasticity we estimated the real crosslinks concentration composed of S1). We have assumed here that E x = 3G x from incompressibility.

Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :Figure 6 :

 123456 Figure 1: a) Loss modulus E'' as a function of temperature measured at ̇ = 1 s -1 . Storage

2 Figure 7 :

 27 Figure 7: Swelling elongations or prestretch o f chains of the first network (λ 0 ) versus the

Figure 8 :Figure 9 :Figure 10 :

 8910 Figure 8: Initial modulus at 60°C versus concentration of chemical crosslinker in the first

Figure 11 :

 11 Figure 11: Correlation between λ limit from fitting Equation (7) to the experimental master

  BDA and transfer reaction of concentration respectively [BDA] and [Tr] (Equation S2). The concentration of BDA being set by the starting solution, [Tr ] can be estimated for the two samples around 0.1 mol % of monomer (Table

Table 1 :

 1 Composition of first networks prepared in solvent,(a) [BDA]= n BDA /n monomer ,with n

	and n BDA the number of moles of monomer and BDA respectively, (b) M x	th the theoretical
	weight between crosslinks as M x	th = (M 0 * n)/(2 * n BDA ), with M 0 the molar mass of the
	monomer.
	Sample	Monomer	(a) [BDA]		(b) Mx th	Monomer	Toluene	BDA	HMP
			[mol %]	[g/mol]	[g]	[g]	[µL]	[µL]
	EA 05	EA	1.45		3400	8.6	8.6	235.8	152.5
	EA 1	EA	2.81		1700	8.6	8.6	471.5	152.5
	EA 2	EA	5.81		860	8.6	8.6	943	152.5
	MA 1	MA	2.50		1700	8.6	8.6	471.5	152.5
	BA1	BA	3.72		1700	8.6	8.6	471.5	152.5
	MA bulk	MA	2.5		1700	8.6	0	471.5	152.5

Table 3 :

 3 Composition and thermo-mechanical properties of multiple networks with EA x as first networks, tanδ and E small strain modulus in MPa measured by DMA; *at 60 °C and **

	at 20 °C
	Sample		(°C)
	EA 05MA	20 w t %	13	0.05*	1.5*
	EA 1MA	30 w t %	13	0.03*	2.1*
	EA 2MA	35 w t %	12	0.02*	2.9*
	MA 1MA	30 w t %	22	0.08*	1.7*
	BA1MA	30 w t %	-27 / 15	0.05*	1.9*
	EA 05EA	20 w t %	-15	0.25**	1.1**
	EA 05MAMA	6 w t %	18	0.05*	2.4*
	EA 1MAMA	10 w t %	-	-	-
	EA 05EAEA	6 w t %	-15	0.18**	1.7**

Table 4 :

 4 Elastic modulus E of simple networks measured from uniaxial elongation and experimental average weight between crosslinks, M x exp , derived from the classical rubber elasticity

	Sample	E (MPa)	Mx exp (g/mol)
	EA 05	0.8	10800
	EA 1	1.35	7400
	EA 2	2.3	3800
	MA 1	1.22	7600
	BA1	1.17	7500

Table 5 :

 5 Fit parameters for Equation 7 for the three families of multiple networks, λ limit from fits; λ max from small strain properties of the first network

	1 st Netw ork	DN TN	DN	TN
	EA 0.5	1.7 2.5 0.23 0.65 0.078 0.057 19.5	4.4	4.4
	EA 1	1.6 2.2	0.5	0.9	0.140 0.078 13.2	3.6	3.7
	EA 2	1.3	-	0.25	-	0.15	0.197	6	2.5	2.6

Different brittle and prestretched elastomer networks are embedded at a low volume fraction in a soft extensible matrix.

  The increase in toughness of the final material is directly controlled by the non linear elastic properties of the prestretched network and its volume fraction, providing a general design rule for tough soft materials.

	Keyword
	elastomers, mechanical properties, double network, fracture, non-linear elasticity

Characterizing Large Strain Elasticity of Brittle Elastomeric Networks by Embedding them in a Soft Extensible Matrix

Table S1 :

 S1 Properties of second networks of PMA: MA 2N and MA 2N_NOX at 60 °C, with M mono =86.09 g.mol -1 , ρ=1.12 g.cm-3

		[	] [ ]			(S2)
	Sample	[BDA]		E	E e	E x	[Bda]+[Tr]	[Tr]
	MA 2N	0.01 mol %	0.9 MPa	0.78 MPa	0.25 MPa	0.11 mol %	0.1 mol %
	MA 2N_noX	0 mol %		0.96 MPa	0.85 MPa	0.25 MPa	0.11 mol % 0.11 mol %

Acknowledgements

We gratefully acknowledge the financial support of DSM Royal B.V. and numerous fruitful discussions with Markus Bulters, Paul Steeman and Meredith Wiseman of DSM Ahead in Geleen. We also thank Hugh Brown for his insightful comments and encouragements on this project and Jian Ping Gong for her inspiration.

Weight fraction of first network:

To determine the composition of the final DN, we used the weight and the thickness of the samples. Before swelling, the thickness (h SN ) and the weight (m SN ) of the first network were measured. The same characteristics were collected from the final DN and TN (h and m). The weight fraction ( ) of first network in the final multiple network was determined with Equation ( 9). The swelling ratio of the first network ( ) was estimated from thicknesses (Equation (10)).

(

Extractible part: A solvent extraction of free chains in DN and TN was performed to quantify the extractible part. A piece of material was weighed and then immersed in a large volume of toluene for two weeks. After drying under vacuum, the sample was weighed again.

The weight loss corresponds to the amount of extracted free chains.

Dynamic mechanical analysis (DMA):

DMA experiments were performed in tension mode on a Q800 DMA testing machine (TA instruments) on strips of material with a length close to 15 mm, a width of 4.9 mm and a thickness between 0.6 and 2.5 mm depending on the sample. A tensile preload force of 0.1 N was applied to the sample in order to avoid any buckling during the experiment. From this steady state, small oscillations were applied at 0.1 % of strain. Every test was performed at 1 Hz and 1 °C/min, from -90 °C or -50°C to 80 °C. The Tg of the material is taken at the maximum of the loss modulus.

Supporting Information

Characterizing nearly the same properties as those of MA 2N . Using the Rubinstein and Panyukov equation [25] (Equation S1), we estimated the contribution to the modulus from entanglements and from crosslinks for the two samples (Table S1).


Fabien Campillo
email: fabien.campillo@inria.fr

Nicolas Champagnat
email: nicolas.champagnat@inria.fr

Coralie Fritsch
email: coralie.fritsch@inria.fr

Monday

On the variations of the principal eigenvalue with respect to a parameter in growth-fragmentation models

Keywords: Growth-fragmentation model, eigenproblem, integro-differential equation, invasion fitness, individual-based model, infinite dimensional branching process, piecewise-deterministic Markov process, bacterial population Mathematics Subject Classification (MSC2010): 35Q92, 45C05, 60J80, 60J85, 60J25, 92D25

We study the variations of the principal eigenvalue associated to a growth-fragmentation-death equation with respect to a parameter acting on growth and fragmentation. To this aim, we use the probabilistic individual-based interpretation of the model. We study the variations of the survival probability of the stochastic model, using a generation by generation approach. Then, making use of the link between the survival probability and the principal eigenvalue established in a previous work, we deduce the variations of the eigenvalue with respect to the parameter of the model.

Introduction

In biology, microbiology and medicine, diverse models are used to describe structured populations. For example the growth of a bacterial population or of tumor cells can be represented, in a constant environment, by the following growth-fragmentation-death equation [START_REF] Doumic | Analysis of a population model structured by the cells molecular content[END_REF][START_REF] Doumic Jauffret | Eigenelements of a general aggregationfragmentation model[END_REF][START_REF] Laurençot | Exponential decay for the growthfragmentation/cell-division equations[END_REF][START_REF] Fredrickson | Statistics and dynamics of procaryotic cell populations[END_REF][START_REF] Sinko | A new model for age-size structure of a population[END_REF][START_REF] Bell | Cell growth and division: I. A mathematical model with applications to cell volume distributions in mammalian suspension cultures[END_REF][START_REF] Metz | The dynamics of physiologically structured populations[END_REF])

∂ ∂t m t (x) + ∂ ∂x g(x) m t (x) + b(x) + D m t (x) = 2 M x b(z) z q z, x z m t (z) dz , 1
which describes the time evolution of the mass density m t of the population of cells which is subject to growth at speed g, cell division at rate b, with daughter cells generated by a division kernel q and death at rate D. In order to study the asymptotic growth of the population, the eigenproblem associated to this equation is generally considered. The eigenvalue, also called Malthus parameter in this context, gives the asymptotic global growth rate of the population and allows to determine if the environment favors the development of the population. Biologically, it is interesting to study the variation of this growth rate when its environment is changed (either by the action of an experimentalist or due to fluctuations of external conditions). In this article, we consider the model described previously, in which the growth function and the division rate depend on an environmental parameter S describing the constant environment. The death rate is assumed independent of S since we have in mind chemostat in which death is due to dilution at fixed rate. This parameter can, for example, represent an external resource or the influence of other populations supposed to be at equilibrium. The study of the influence of this parameter on the growth of the population is a question of biological interest for a better understanding of the model, but also of numerical interest, for example, for the study of mutant invasions in adaptive dynamics problems [START_REF] Fritsch | A numerical approach to determine mutant invasion fitness and evolutionary singular strategies[END_REF].

This new question seems to be difficult to approach with standard deterministic mathematical tools where, up to our knowledge, no result is available except a study of the influence of asymmetric division by [START_REF] Michel | Optimal proliferation rate in a cell division model[END_REF][START_REF] Michel | Fitness optimization in a cell division model[END_REF] and an asymptotical study of the influence of the parameters by [START_REF] Calvez | Self-similarity in a general aggregationfragmentation problem ; application to fitness analysis[END_REF]. See also the work of [START_REF] Olivier | How does variability in cells' aging and growth rates influence the Malthus parameter?[END_REF] for a study of the impact of the variability in cells' aging and growth rates as well as the one of [START_REF] Clairambault | Circadian rhythm and tumour growth[END_REF] for comparison of Perron eigenvalue (for constant in time birth and death rates) and Floquet eigenvalue (for periodic birth and death rates).

The approach that we propose in this article uses the probabilistic interpretation of the growth-fragmentation-death equation under the form of a discrete stochastic individualbased model. This class of piecewise deterministic Markov processes is studied a lot, with a particular recent interest to the estimation of the parameters of the model [START_REF] Doumic | Statistical estimation of a growth-fragmentation model observed on a genealogical tree[END_REF][START_REF] Hoang | Estimating the division kernel of a size-structured population[END_REF][START_REF] Hoffmann | Nonparametric estimation of the division rate of an age dependent branching process[END_REF]. In this individual-based model, the growth of the population is determined by its growth rate, but also by its survival probability in some constant environment. The link between the eigenvalue of the deterministic model and the survival probability of the stochastic model, which correspond to two different definitions of the biological concept of invasion fitness [START_REF] Metz | How should we define 'fitness' for general ecological scenarios?[END_REF][START_REF] Metz | Fitness[END_REF], was established by [START_REF] Campillo | Links between deterministic and stochastic approaches for invasion in growth-fragmentation-death models[END_REF]. Our goal is to use this link to deduce variation properties of the eigenvalue with respect to the environmental parameter S from the variations on the survival probability. The probabilistic invasion fitness allows to use a generation by generation approach, which is more difficult to apply to the eigenproblem since generations overlap. Using this approach, the variations of the survival probability can be obtained by applying a coupling technique to the random process.

In an adaptive dynamics context, the variation of both invasion fitnesses are numerically very useful. For instance, considering the time evolution of a bacterial population in a chemostat, the invasion fitness determines if some mutant population can invade a resident one when a mutation occurs [START_REF] Metz | Adaptive dynamics: A geometric study of the consequences of nearly faithful reproduction[END_REF]. This invasion fitness is the one of the mutant population in the environment at the equilibrium determined by the resident one. In this example, the environmental parameter S represents the substrate concentration at the equilibrium of the resident population. When the mutant population appears in the chemostat it appears in small size, hence its influence on the resident population and on the resource concentration can be neglected, which allows to assume the substrate concentration S to be constant as long as the mutant population is small. Moreover, due to the small number of mutant individuals, it is essential to use a stochastic model [START_REF] Fritsch | A modeling approach of the chemostat[END_REF][START_REF] Campillo | Weak convergence of a mass-structured individualbased model[END_REF]. However, the stochastic invasion fitness is numerically less straightforward to compute than the deterministic one. The mutual variations of both invasion fitnesses established in this article allow to considerably simplify the numerical analysis of a mutant invasion since the problem is reduced to the computation of a single eigenvalue in order to characterize the possibility of invasion of the mutant population [START_REF] Fritsch | A numerical approach to determine mutant invasion fitness and evolutionary singular strategies[END_REF].

In Section 2, we present the deterministic and the stochastic versions of our growthfragmentation-death model. We give the definitions of invasion fitness in both cases : for the stochastic one, it is defined as the survival probability and for the deterministic one, it corresponds to the eigenvalue of an eigenproblem. We extend some results from [START_REF] Campillo | Links between deterministic and stochastic approaches for invasion in growth-fragmentation-death models[END_REF], in particular Theorem 2.4 linking these two invasion fitnesses, to our more general context. Section 3.1 is devoted to the monotonicity properties of the survival probability of the stochastic model with respect to the initial mass and the death rate. In Section 3.2 we prove, under suitable assumptions, the monotonicity of the survival probability with respect to the environmental parameter S. In Section 3.3, we deduce from the previous results and from the link between the two invasion fitnesses, the monotonicity of the eigenvalue with respect to S. Our assumptions are based on the realistic biological idea that the larger a bacterium is, the faster it divides and the larger the parameter S is, the faster a bacterium grows. This is biologically consistent in the case where S represents the substrate concentration. The monotonicity of fitnesses is obtained under additional assumptions which are detailed in the following sections. We extend this result assuming a particular form of the growth rate g and give a more general approach in Section 3.4.

Models description

In this Section we present two descriptions of the growth-fragmentation-death model. This model is the one studied by [START_REF] Campillo | Links between deterministic and stochastic approaches for invasion in growth-fragmentation-death models[END_REF], in which we add a dependence in a onedimensional environmental parameter S, which is supposed to be fixed in time. In Section 3, we study the variation of the invasion possibility of the population (whose definition depends on the considered description) with respect to S for both descriptions.

Basic mechanisms

We consider models in which each individual is characterized by its mass x ∈ [0, M ], where M is the maximal mass of individuals, and is affected by the following mechanisms:

1. Division: each individual of mass x divides at rate b(S, x), into two individuals with masses α x and (1α) x, where the proportion α is distributed according to the probability distribution Q(x, dα) = q(x, α) dα on [0, 1].

x ↵ x

(1 ↵) x taux b(s, x)
2. Death: each individual dies at rate D.

3. Growth: between division and death times, the mass of an individual grows at speed g : R + × [0, M ] → R + depending on an environmental parameter S, i.e.

d dt x t = g(S, x t ) . (1)
In this model, individuals do not interact between themselves and the environmental parameter S is fixed in time. This means that the resource S is not limiting for the growth of the population, this is for example the case if the resource is continuously kept at the same level or the consumption of the resource is negligible with respect to the resource quantity. This model is relevant for a population with few individuals in a given environment such that the resource consumption is low.

For any S > 0, let A S t be the flow associated to an individual's mass growth in the environment S, i.e. for any x ∈ (0, M ) and t ≥ 0,

A S t (x) = x + t 0 g(S, A S u (x)) du . (2)
Throughout this paper we assume the following set of assumptions.

Assumptions 2.1. 1. For any x ∈ [0, M ], the kernel q(x, .) is symmetric with respect to 1/2:

q(x, α) = q(x, 1 -α), ∀α ∈ [0, 1] such that 1 0 q(x, α) dα = 1. 2. For any α ∈ [0, 1], the function x → q(x, α) is continuous on [0, M ].
3. There exists a function q : [0, 1] → R + such that q(x, α) ≤ q(α) for any x ∈ (0, M ) and 1 0 q(α) dα < +∞. 4. g(S, 0) = g(S, M ) = 0 and g(S, x) > 0 for any x ∈ (0, M ) and S > 0.

5. g(S, .) ∈ C[0, M ] ∩ C 1 (0, M ) , where C[0, M ] and C 1 (0, M ) respectively represent
sets of continuous functions on [0, M ] and continuously differentiable functions on (0, M ).

6. b(S, .) ∈ C[0, M ] and there exists

m div ∈ [0, M ) and b > 0 such that b(S, x) = 0 if x ≤ m div , 0 < b(S, x) ≤ b if x ∈ (m div , M ) .
Assumptions 2.1-5 and 2.1-4 ensure existence and uniqueness of the growth flow defined by (2) for x ∈ (0, M ) until the exit time (Demazure, 2000, Th. 6.8.1). We define this flow as constant when it starts from M . Note that the exit time T exit (x) is infinite if the convergence lim x→M g(S, x) = 0 is sufficiently fast (see for example (Campillo et al., 2016, Assumption 3.) for more details). Assumption 4 means that the maximal biomass of an individual is the same for any concentration of resources. This may not be true in general, but we can always change the scale of biomass for each value of S so that the maximal value of x is always M and modify the growth and birth parameters accordingly. This is what we shall assume in the sequel.

T exit (x) := inf{t > 0 | A S t (x) ≥ M } of (0, M ) and that A S ∈ C 1 (D) with D = {(t, x), t < T exit (x)}

Growth-fragmentation-death integro-differential model

The deterministic model associated to the previous mechanisms is given by the integrodifferential equation

∂ ∂t m S t (x) + ∂ ∂x g(S, x) m S t (x) + b(S, x) + D m S t (x) = 2 M x b(S, z) z q z, x z m S t (z) dz , (3)
where m S t (x) represents the density of individuals with mass x at time t evolving in the environment determined by S, with a given initial condition m S 0 . Let G S be the non local transport operator such that

∂ t m S t (x) = G S m S t (x): for any f ∈ C 1 (0, M ), x ∈ (0, M ), G S f (x) def = -∂ x (g(S, x) f (x)) -(D + b(S, x)) f (x) + 2 M x b(S, z) z q z, x z f (z) dz , (4)
and G * S its adjoint operator defined for any f ∈ C 1 (0, M ), x ∈ (0, M ) by

G * S f (x) def = -(D + b(S, x)) f (x) + g(S, x) ∂ x f (x) + 2 b(S, x) 1 0 q(x, α) f (α x) dα . (5)
We consider the eigenproblem

G S ûS (x) = Λ S ûS (x) , (6a) lim x→0 g(S, x) ûS (x) = 0 , D + Λ S > 0 , ûS (x) ≥ 0 , M 0 ûS (x) dx = 1 (6b)
and the adjoint problem

G * S vS (x) = Λ S vS (x) , vS (x) ≥ 0 , M 0 vS (x) ûS (x) dx = 1 . (7)
The eigenvalue Λ S is then interpreted as the exponential growth rate (or decay rate if it is negative) of the population.

In the rest of the paper, we will assume that the following assumption is satisfied. [START_REF] Campillo | Links between deterministic and stochastic approaches for invasion in growth-fragmentation-death models[END_REF] have given some conditions under which this assumption holds (see also [START_REF] Doumic | Analysis of a population model structured by the cells molecular content[END_REF][START_REF] Doumic Jauffret | Eigenelements of a general aggregationfragmentation model[END_REF] for sligthly different models and [START_REF] Perthame | Exponential decay for the fragmentation or celldivision equation[END_REF][START_REF] Laurençot | Exponential decay for the growthfragmentation/cell-division equations[END_REF][START_REF] Mischler | Spectral analysis of semigroups and growthfragmentation equations[END_REF] for exponential stability of the eigenfunctions).

Assumption 2.2. For any S > 0, the system (6)-( 7) admits a solution (û S , vS , Λ S ) such that ûS ∈ C 1 (0, M ) and vS ∈ C[0, M ] ∩ C 1 (0, M ).

Growth-fragmentation-death individual-based model

The mechanisms described in Section 2.1 can also be represented by a stochastic individualbased model, where the population at time t is represented by the counting measure

η S t (dx) def = Nt i=1 δ X i t (dx) , (8)
where

N t = M 0 η S t (dx)
is the number of individuals in the population at time t and (X i t , i = 1, . . . , N t ) are the masses of the N t individuals (arbitrarily ordered).

The stochastic individual-based model is relevant for small population whereas the deterministic one is relevant for large population [START_REF] Campillo | Weak convergence of a mass-structured individualbased model[END_REF].

The process (η S t ) t≥0 is defined by

η S t = N 0 j=1 δ A S t (X j 0 ) + [0,t]×N * ×[0,1] 3 1 {j≤Nu-} 1 {θ 1 ≤b(S,X j u -)/ b} 1 {θ 2 ≤q(X j u -,α)/q(α)} -δ A S t-u (X j u -) + δ A S t-u (α X j u -) + δ A S t-u ((1-α) X j u -) N 1 (du, dj, dα, dθ 1 , dθ 2 ) - [0,t]×N * 1 {j≤Nu-} δ A S t-u (X j u -) N 2 (du, dj) (9)
where N 1 (du, dj, dα, dθ 1 , θ 2 ) and N 2 (du, dj) are two independent Poisson random measures defined on

R + × N * × [0, 1] × [0, 1] × [0, 1] and R + × N * , corresponding respectively
to the division and death mechanisms, with respective intensity measures

n 1 (du, dj, dα, dθ) = b du ≥1 δ (dj) q(α) dα dθ 1 dθ 2 , ( 10
)
n 2 (du, dj) = D du ≥1 δ (dj) , (11)
(see [START_REF] Campillo | Weak convergence of a mass-structured individualbased model[END_REF] and [START_REF] Campillo | Links between deterministic and stochastic approaches for invasion in growth-fragmentation-death models[END_REF] for more details). This population process can be seen as a multitype branching process with a continuum of types. We are interested in its survival probability. We suppose that, at time t = 0, there is only one individual, with mass x 0 , in the population, i.e.

η S 0 (dx) = δ x 0 (dx) .

The extinction probability of the population with initial mass x 0 is

p S (x 0 ) def = P S δx 0 (∃t > 0, N t = 0) , where P S δx 0
is the law of the process (η S t ) t≥0 under the initial condition η S 0 = δ x 0 . The survival probability is then given by P S δx 0 (survival) = 1p S (x 0 ). We define the n-th generation as the set of individuals descended from a division of one individual of the (n -1)-th generation. The generation 0 corresponds to the initial population. We denote by Z n the number of individuals at the n-th generation and we define the extinction probability before the n-th generation as

p S n (x 0 ) def = P S δx 0 (Z n = 0) , n ∈ N .

It is obvious that lim

n→∞ p S n (x 0 ) = p S (x 0 ) .
Let τ be the stopping time of the first event (division or death). Then at time τ the population is given by

η S τ def = 0 if death , δ X 1 + δ X 2 if division , (12)
with X 1 = α A S τ (x 0 ) and X 2 = (1α) A S τ (x 0 ) where the proportion α is distributed according to the kernel q(A S τ (x 0 ), α) dα.

Applying the Markov property at time τ and using the independence of particles, it is easy to prove (see [START_REF] Campillo | Links between deterministic and stochastic approaches for invasion in growth-fragmentation-death models[END_REF]) that for any x ∈ [0, M ] and

n ∈ N * p S n (x) = D ∞ 0 e -D t e -t 0 b(S,A S u (x)) du dt + ∞ 0 b(S, A S t (x)) e -t 0 b(S,A S u (x)) du-D t 1 0 q(A S t (x), α) p S n-1 α A S t (x) p S n-1 (1 -α) A S t (x) dα dt . ( 13
)
with p S 0 (x) = 0. It can then be deduced (Campillo et al., 2016, Proposition 3) that p S is the minimal non negative solution of

p S (x) = ∞ 0 D e -D t e -t 0 b(S,A S u (x)) du dt + ∞ 0 b(S, A S t (x)) e -t 0 b(S,A S u (x)) du-D t 1 0 q(A S t (x), α) p S α A S t (x) p S (1 -α) A S t (x) dα dt , (14)
in the sense that for any non negative solution p we have p ≥ p S . q(y, α) p S α y p S (1α) y dα dy .

Therefore, the extinction probability is solution of

g(S, x)∂ x p S (x)+D (1-p S (x))+b(S, x) 1 0 q(x, α) p S (α x) p S ((1-α) x) dα-p S (x) = 0 .
For any x ∈]0, M [ and y > 0 such that x ≤ y, let t S (x, y) be the first hitting time of y by the flow A S t (x), i.e.

t S (x, y) def = inf{t ≥ 0, A S t (x) = y} = A -1 S,x (y) , if x ≤ y < M , +∞ , if y ≥ M , (15)
where A -1 S,x is the inverse function of the C 1 -diffeomorphism t → A S t (x). [START_REF] Campillo | Links between deterministic and stochastic approaches for invasion in growth-fragmentation-death models[END_REF] we have made the link between the survival probability of the stochastic process and the eigenvalue of the deterministic model, given by the theorem below. This result was proved for a kernel q(x, .) which does not depend on x ∈ (0, M ), but it can easily be extended to our case where q(x, .) depends on the mass x at the division time as explained below.

Theorem 2.4 (Campillo, Champagnat, Fritsch (2015)). Under Assumptions 2.1 and 2.2, we have the following relation between the two invasion criteria Λ S > 0 ⇐⇒ P S δx (survival) > 0 , ∀x ∈ (0, M ) . Note that, contrary to the works of [START_REF] Perthame | Exponential decay for the fragmentation or celldivision equation[END_REF]; [START_REF] Doumic | Analysis of a population model structured by the cells molecular content[END_REF]; [START_REF] Laurençot | Exponential decay for the growthfragmentation/cell-division equations[END_REF]; Doumic Jauffret and Gabriel (2010); [START_REF] Mischler | Spectral analysis of semigroups and growthfragmentation equations[END_REF], we assume here a compact set [0, M ] of biomasses to keep things simple in the sequel. The extension of our approach to a non-compact case would require to identify the good assumptions at infinity for the last result to hold (the rest of our arguments should work similarly). The last problem is not so easy because it strongly depends on the growth at infinity of the eigenfunctions û and v of Assumption 2.2. Note in addition that the problem of existence of these eigenfunctions also requires a careful study at infinity (see Doumic Jauffret and Gabriel ( 2010)).

Proof. The key argument of the proof is that the process (e -Λ S t Nt i=1 vS (X i t )) t≥0 is a P S δx -martingale such that

e -Λs t Nt i=1 vS (X i t ) ---→ t→∞ Z P S δx -a.s.
where Z is an integrable random variable (see [START_REF] Campillo | Links between deterministic and stochastic approaches for invasion in growth-fragmentation-death models[END_REF], Theorem 2 and Lemma 3)). The arguments of [START_REF] Campillo | Links between deterministic and stochastic approaches for invasion in growth-fragmentation-death models[END_REF] to prove that 1. if Λ S > 0 then P S δx (survival) > 0 for any x ∈ (0, M ) 2. if Λ S < 0 then P S δx (survival) = 0 for any x ∈ [0, M ] can be directly applied for a kernel q(x, .) depending on the mother mass x. The first statement is proved using that if Λ S > 0 then Z is bounded in L 2 whereas the second one comes from the inequality

E δx (N t ) ≤ C x e Λ S t , ∀t ≥ 0
where C x > 0 is a constant depending on the initial mass x ∈ (0, M ). Its proof by Campillo et al. ( 2016) is technical, but the extension to kernels q depending on x is straightforward. The only difficulty concerns the third point of the proof of (Campillo et al., 2016, Theorem 2) in which we prove that if Λ S = 0 then M ε t → 0 a.s. with M ε t the number of individuals with mass in [ε, Mε] at time t for 0 < ε < M 2 . Then, the fourth point of the proof, stating that M ε t → 0 implies extinction, follows similarly. The main idea of this fourth step is that the number of individuals M ε t cannot indefinitely stay in a compact subset {1, 2, . . . , c} of N. Then either lim sup t→∞ M ε t = ∞ or M ε t → 0 a.s. However, lim sup t→∞ M ε t = ∞ contradicts the fact that Z is integrable if Λ S = 0, hence M ε t → 0 a.s.

For the proof of the third point (see details in [START_REF] Campillo | Links between deterministic and stochastic approaches for invasion in growth-fragmentation-death models[END_REF])), it is sufficient that for c > 0, there exists t 0 such that

1 > γ := inf ε≤x≤M -ε P S δx M ε t 0 ≥ c > 0 . (16)
If q(x, .) = q(.) is independent of the mother mass, it is sufficient to take ε > 0 such that x ∈ (ε, Mε) and q ([ε/(M -2 ε), 1/2]) > 0 to obtain (16) for one t 0 . In our case, the last condition must be replaced by inf ε≤x≤M -ε q(x, [ε/(M -2 ε), 1/2]) > 0 for some ε > 0. Note that the infimum above is reached at some x 0 (ε) ∈ [ε, Mε], by Assumptions 2.1-2 and 3. Therefore, we proceed by contradiction and assume that for all ε > 0, there exists

x 0 (ε) ∈ [ε, M -ε] such that q(x 0 (ε), α) = 0 for almost all α ∈ ε M -2 ε ; 1 2 .
Then, from the sequence x 0 1 n n , we can extract a subsequence which converges towards x * 0 . By continuity of x → q(x, α), we then get q(x * 0 , α) = 0 for almost all α ∈ (0, 1). Hence 1 0 q(x * 0 , α) dα = 0, which contradicts Assumption 2.1-1.

Variations of the invasion fitnesses with respect to the environmental variable

Our goal is to study the variation of Λ S w.r.t S. For this, we start by studying the monotonicity properties of the survival probability in the stochastic model.

Monotonicity properties w.r.t. the initial mass and the death rate on the stochastic model

From a biological point of view, little is known about the dependence of the division kernel q w.r.t. x [START_REF] Osella | Step by step, cell by cell: quantification of the bacterial cell cycle[END_REF]. Most often, it is assumed independent of x in applications.

In order to obtain the most general result, we assume that q depends on x, and we need to state assumptions about this parameter. Note however that the self-similar fragmentation is included in our assumptions. Moreover, although q(x, α) is assumed to be regular w.r.t. α, more general kernels can be consider, in particular the following results should hold for self-similar equal mitosis.

For any x ∈ (0, 1), let F x : [0, 1] → [0, 1] be the cumulative distribution function associated to the law q(x, α) dα, that is for any u ∈ [0, 1]

F x (u) = u 0 q(x, α) dα and let F -1
x be its inverse function defined by

F -1 x (v) = inf u∈[0,1] {F x (u) ≥ v} .
Assumption 3.1. The cumulative distribution function F x satisfies, for any u ∈ (0, 1) and any x ≤ y,

x F -1 x (u) ≤ y F -1 y (u) and (1 -x) F -1 x (u) ≤ (1 -y) F -1 y (u) .
As we will see in Lemma 3.4 below, this assumption corresponds to a coupling condition on the mass of offspring born from individuals of different sizes. We need this condition because our method can be seen as a construction of a coupling of the masses of individuals at each generation in two stochastic processes starting from different initial masses (see our comments below, particularly Remark 3.7).

Remark 3.2. If F -1

x is such that for any u ∈ (0, 1), x → F -1

x (u) ∈ C 1 ([0, M ]) and satisfies for any x ∈ (0, M ),

x ∂ x F -1 x (u) ∈ [-F -1 x (u), 1 -F -1 x (u)] , then ∂ x (x F -1 x (u)) = F -1 x (u) + x ∂ x F -1 x (u) ≥ 0 . Hence x F -1
x (u) is non decreasing. In the same way, (1x) F -1 x (u) is non decreasing too. Therefore, Assumption 3.1 holds.

Examples 3.3. We give some examples which satisfy Assumption 3.1.

1. We consider the following division kernel,

q(x, α) = 1 {l(x)≤α≤1-l(x)} 1 -2 l(x) .
where l ∈ C 1 ([0, M ], (0, 1/2)). Then for u ∈ (0, 1),

F -1 x (u) = (1 -2 u) l(x) + u
and, by Remarks 3.2, Assumption 3.1 holds if for any x, 0 ≤ x l (x) + l(x) ≤ 1.

2. We can extend the previous example considering the following function q, q(x, α) = (αl(x)) β(x) C(x)

1 {l(x)≤α≤1/2} + (1 -α -l(x)) β(x) C(x) 1 {1/2≤α≤1-l(x)} (17)
where C(x) = 2 (1/2l(x)) β(x)+1 /(β(x) + 1) is a normalizing constant. The previous example corresponds to β(x) = 0 for any x ∈ [0, M ]. Then

F x (u) = 1 2 u -l(x) 1 2 -l(x) β(x)+1 1 {l(x)≤α≤1/2} +   1 - 1 2 1 -u -l(x) 1 2 -l(x) β(x)+1   1 {1/2<α≤1-l(x)} + 1 {1-l(x)<α} (18)
and for any u ∈ (0, 1)

F -1 x (u) = 1 2 -l(x) (2 u) 1/(β(x)+1) + l(x) 1 {0<u≤1/2} + 1 -l(x) - 1 2 -l(x) (2 (1 -u)) 1/(β(x)+1) 1 {1/2<u<1} . (19)
An example of such functions is given in Figure 1.

For u ∈ (0, 1/2],

∂ x F -1 x (u) = -l (x) - 1 2 -l(x) β (x) (β(x) + 1) 2 ln(2 u) (2 u) 1/(β(x)+1) + l (x)
and for u ∈ [1/2, 1),

∂ x F -1 x (u) = l (x) + 1 2 -l(x) β (x) (β(x) + 1) 2 ln(2 (1 -u)) (2 (1 -u)) 1/(β(x)+1) -l (x) Assumption 3.1 holds if 0 ≤ x ∂ x F -1
x (u)+F -1 x (u) ≤ 1 for any u ∈ (0, 1), for example if β is a constant function and if 0 ≤ l(x) + x l (x) ≤ 1 for any x ∈ (0, M ).

Lemma 3.4. Let f be a non-increasing function on [0, M ]. Then, under Assumption 3.1, the function

x → 1 0 q(x, α) f (α x) f ((1 -α) x) dα is non-increasing.
Proof. For any x ∈ (0, 1), let θ x defined by θ x = F -1

x (U ) where U is uniformly distributed on [0, 1]. Therefore the law of the variable θ x is q(x, α) dα. By Assumption 3.1, x (u)

∂ x (x θ x ) = θ x + x ∂ x F -1 x (U ) ≥ θ x -F -1 x (U ) = 0 a.s.
Figure 1: Representation of the function q (left), F x (center) and F -1 x (right) respectively defined by Equations ( 17), ( 18) and ( 19) with l(x) = 0.25 and β(x) = 5. and

∂ x (x (1 -θ x )) = 1 -θ x -x ∂ x F -1 x (U ) ≥ 1 -θ x -(1 -F -1
x (U )) = 0 . Therefore, for any x < y we have x θ x ≤ y θ y a.s. and x (1θ x ) ≤ y (1θ y ) a.s. Hence,

1 0 q(x, α) f (α x) f ((1 -α) x) dα = E (f (θ x x) f ((1 -θ x ) x)) ≤ E (f (θ y y) f ((1 -θ y ) y)) = 1 0 q(y, α) f (α y) f ((1 -α) y) dα .
Remark 3.5. Note that the last proof makes use of a probabilistic coupling argument, since we actually prove and use the following property: the pair of random variables (xθ x , x(1θ x )), where θ x is distributed as q(x, α)dα, is stochastically increasing w.r.t. x. This means that, for all x ≤ y, there exists a coupling of the random variables θ x and θ y , i.e. two random variables θ x and θ y with the same laws as θ x and θ y can be constructed on the same probability space, such that xθ x ≤ yθ y and x(1-θ x ) ≤ y(1-θ y ). Therefore, Assumption 3.1 means that the offspring masses of two individuals reproducing at respective masses x and y can be coupled so that the masses of the offspring are in the same order as those of the parents.

Proposition 3.6. Under Assumption 3.1, if the division rate b(S, .) is non decreasing then the extinction probability p S : x → p S (x) is non increasing.

The assumption that b increases with the mass x of an individual is biologically natural, since a bigger total biomass usually means a bigger fraction of biomass devoted to the biomolecular mechanisms involved in cellular division. We give below an analytical proof of this proposition, but it can also be proved using probabilistic arguments, as explained in Remark 3.7 below.

In other words, for the chemostat model, under the assumptions of the previous theorem, the higher the substrate concentration in the chemostat at the mutation time is, the higher the survival probability is.

Remark 3.11. Following Remark 3.7, the last result could be also proved by probabilistic coupling arguments. These arguments would actually only require to assume that, for all x ≤ y and S 1 ≤ S 2 , there exists a coupling between the sets M x,S 1 1 and M y,S 2 1 of biomasses at birth of the individuals of the first generation in the branching processes (η S 1 t , t ≥ 0) such that η S 1 0 = δ x and (η S 2 t , t ≥ 0) such that η S 2 0 = δ y respectively, satisfying the following property: |M y,S 2 1 | = 0 implies |M x,S 1 1 | = 0 and when both have cardinal 2, then M x,S 1 1 = {x 1 , x 2 } and M y,S 2 1 = {y 1 , y 2 } with x 1 ≤ y 1 and x 2 ≤ y 2 . The proof of Theorem 3.10 given below actually consists in checking that the coupling assumption above is implied by Assumptions 3.1 and 3.9. However, this coupling assumption is hard to check in practice and this is why we chose to give an analytical proof based on the Assumptions 3.1 and 3.9, which are stronger, but easier to check.

Of course, Theorem 3.10 is certainly valid under weaker assumptions, for example if b or g are not monotonic w.r.t. S, but our probabilistic approach requires coupling assumptions like the one stated in this remark, so the method would not extend easily to such cases.

Proof. For any y ∈ (0, M ) the function S → g(S, y) is non decreasing then A S 1 u (x) ≤ A S 2 u (x) for any u ≥ 0. Moreover the function (S, x) → b(S, x) is non decreasing in the two variables S and x, then we have

p S 1 1 (x) -p S 2 1 (x) = D ∞ 0 e -D t e -t 0 b(S 1 ,A S 1 u (x)) du -e -t 0 b(S 2 ,A S 2 u (x)) du dt ≥ 0 .
The function S → p S 1 (x) is then non increasing for any x ∈ (0, M ). Let n ∈ N * , we assume that the function S → p S n (x) is non increasing for any x ∈ (0, M ). The function t

→ t 0 (b(S, A S u (x)) + D) du is a bijection from [0, ∞[ to [0, ∞[. Hence, for X ≥ 0, there exists a unique T S x (X) such that X = T S x (X) 0 (b(S, A S u (x)) + D) du. By the change of variable X = t 0 (b(S, A S u (x)) + D) du in (13), we obtain p S n+1 (x) = ∞ 0 D b S, A S T S x (X) (x) + D + b S, A S T S x (X) (x) b S, A S T S x (X) (x) + D Ψ S,x n (X) e -X dX with Ψ S,x n (X) = 1 0 q A S T S x (X) (x), α p S n α A S T S x (X) (x) p S n (1 -α) A S T S x (X) (x) dα . (22)
Moreover, for all X ≥ 0, for S 1 ≤ S 2 , by the changes of variable A S i u (x) = y for i = 1, 2 and by Assumption 3.9-3, we have x). We deduce from Lemma 3.4 and Proposition 3.6 that

A S 2 T S 2 x (X) (x) x b(S 2 , y) + D g(S 2 , y) dy = X = A S 1 T S 1 x (X) (x) x b(S 1 , y) + D g(S 1 , y) dy ≥ A S 1 T S 1 x (X) (x) x b(S 2 , y) + D g(S 2 , y) dy therefore A S 2 T S 2 x (X) (x) ≥ A S 1 T S 1 x (X) (
Ψ S 1 ,x n (X) ≥ Ψ S 2 ,x n (X). Hence, using Ψ S 1 ,x n (X) ≥ Ψ S 2 ,
x n (X) in the expression of p S 1 n+1 , subtracting the expression of p S 2 n+1 and factorizing the terms, we obtain

p S 1 n+1 (x) -p S 2 n+1 (x) ≥ ∞ 0 D b S 1 , A S 1 T S 1 x (X) (x) + D - D b S 2 , A S 2 T S 2 x (X) (x) + D (1 -Ψ S 2 ,x n (X)) e -X dX
and as Ψ S 2 ,x n (X) ≤ 1, by Assumptions 3.9-1, p S 1 n+1 (x) ≥ p S 2 n+1 (x). Finally, passing to the limit, we get

p S 1 (x) -p S 2 (x) = lim n→∞ (p S 1 n (x) -p S 2 n (x)) ≥ 0 .

Properties on the variations of the eigenvalue

Until now, we only studied the probability of survival of the branching process η S . The underlying coupling arguments require to consider the population state at each generation in a process where generations actually overlap. This is why such an approach is hard to apply directly to the integro-differential eigenvalue problem, where the notion of generations is difficult to define. However, the link between the stochastic and deterministic problems stated in Theorem 2.4 allows to extend the monotonicity properties of Theorem 3.10 to the eigenvalue Λ S , as proved below. The next corollary is a direct consequence of Theorems 2.4 and 3.10.

Corollary 3.12. Under Assumptions 3.1 and 3.9, 1. if there exists S 1 > 0 such that Λ S 1 > 0, then Λ S 2 > 0 for any S 2 > S 1 ; 2. if there exists S 1 > 0 such that Λ S 1 ≤ 0, then Λ S 2 ≤ 0 for any S 2 < S 1 .

This Corollary allows to deduce the following result about variation of the eigenvalue with respect to S. Corollary 3.13. Under Assumptions 3.9, the function S → Λ S is non decreasing.

The monotonicity of b is important to obtain the monotonicity of the eigenvalue (and of the survival probability). For example, one can imagine cases where a fast growth rate g transports individuals to big masses and if the division rate is low for high values, the monotonicity of the eigenvalue does not hold (see [START_REF] Calvez | Self-similarity in a general aggregationfragmentation problem ; application to fitness analysis[END_REF] for non monotonic examples).

Proof. Let S * > 0 be fixed. We set D = D + Λ S * > 0. Let Λ S be the eigenvalue of the following eigenproblem: Hence Λ S ≤ Λ S * .

Extensions and concluding remarks

The previous method can be applied for more general g, for which the growth in one environment is larger than the growth in the other one for all masses. A particular case is given in the following corollary.

Corollary 3.14. We assume that the division rate function b does not depend on the variable S and is non decreasing in the variable x and that the growth speed g is of the form g(S, x) = µ(S) g(x) , where g(S, x) > 0 for any x ∈ (0, M ) and g ∈ C[0, M ]∩C 1 (0, M ) is such that g(0) = g(M ) = 0. Then, we have P S 1 δx (survival) ≤ P S 2 δx (survival) ⇐⇒ µ(S 1 ) ≤ µ(S 2 ) and Λ S 1 ≤ Λ S 2 ⇐⇒ µ(S 1 ) ≤ µ(S 2 ) .

More generally, the following result states the link between the comparison of the survival probability and the comparison of the eigenvalue.

We extend the notations of the survival probability P S,D δx (survival) and the eigenvalue Λ D S with a dependence to the death rate D.

Proposition 3.15. Let S 1 , S 2 > 0. If for any x ∈ [0, M ] and for any D > 0, we have P S 1 ,D δx (survival) ≥ P S 2 ,D δx (survival), then

∀D > 0 , Λ D S 1 ≥ Λ D S 2 .
The condition on the survival probability stated in the previous theorem could be of course obtained under the appropriate coupling assumptions (as in Remark 3.7), but it seems hard to find general practical conditions on the parameters of the model ensuring such a property. Note also that this coupling method could be applied for example to the case where the division distribution q also depends on the variable S. The results of Section 3.1 would remain true as, for this section, the substrate concentration is fixed. The difficulties are in the control of the variation in S of Ψ S,x n defined by ( 22).

Proof. Let S 1 > 0. We set D = D + Λ D S 1 > 0. Let Λ S be the eigenvalue associated to the eigenproblem

∂

  x (g(S, x) û S (x)) + (b(S, x) + D + Λ S )û S (x) = 2 S * , we have Λ S * = 0, then from Corollary 3.12, for any S ≤ S * , Λ S ≤ 0. MoreoverΛ S = Λ S + D -D = Λ S -Λ S * .

∂

  x (g(S, x) û S (x)) + (b(S, x) + D + Λ S )û S (x) = 2 z) dz .For S = S 1 , Λ S 1 = 0, we then deduce, from Theorem 2.4, that P S 1 ,D δx (survival) = 0 and then, by assumption, that P S 2 ,D δx(survival) = 0. From Theorem 2.4 Λ S 2 ≤ 0. Moreover Λ S 2 = Λ D S 2 + D -D = Λ D S 2 -

Proof. We prove by induction that the function p S n is non increasing for any n ∈ N * , where p S n is given by ( 13). Let 0 < x < y < M . As A S u (x) < A S u (y), for any u ≥ 0, Then the function p S 1 is non increasing. Let n ∈ N * , we assume that the function p S n is non increasing.

We can write p S n+1 (x) as p S n+1 (x) = p S 1 (x) + P S δx {extinction before the (n + 1)-th generation} ∩ η S τ = 0 , with P S δx {extinction before the (n + 1)-th generation}

The following relation holds

with p S n+1 (x | η S τ = 0) = P S δx (extinction before the (n + 1)-th generation | η S τ = 0) .

Since for any t ≥ 0, A S t(x,y)+t (x) = A S t (y), then, by a change of variable,

For any t ∈ [0, t S (x, y)] we have A S t (x) ≤ y. Since we assume that the function p S n is non increasing, from Lemma 3.4, we then get

Still because the function p S n is non increasing and from Lemma 3.4,

Adding ( 20) and ( 21), and using the last inequality, we then get

Hence,

This ends the induction. Passing to the limit, we finally get

Remark 3.7. The last result can also be proved by a probabilistic coupling argument as follows. First, for all x ∈ (0, M ), the time of death or division of an individual of mass x can be constructed from an exponential random variable E with parameter 1 as

Ty (y). Second, we observe that the probability of death given death or division occurs for an individual of mass x, D/(D + b(S, x)), is non-increasing as a function of x. Hence, using Remark 3.5, given x ≤ y, we can construct a coupling between the branching processes (η S t , t ≥ 0) with η S 0 = δ x and (η S t , t ≥ 0) with ηS 0 = δ y such that the random sets M 1 and M1 of masses at birth of the individuals of the first generation satisfy the following property: the cardinals

It then follows by induction that the processes (η S t , t ≥ 0) and (η S t , t ≥ 0) can be coupled so that, for all n ≥ 0, the masses at birth of all the individuals of the n-th generation can be ordered into two vectors V n = (x n 1 , . . . , x n Gn ) and V n = (x n 1 , . . . , xn

Ĝn

), where G n and Ĝn are the random sizes of generation n in η S and ηS respectively, satisfying the following property: for all n, G n ≤ Ĝn and for all 1 ≤ i ≤ G n , x n i ≤ xn i . This implies Proposition 3.6 since survival of η S means that G n ≥ 1 for all n and this implies that ηS also survives. Hence p S (x) ≥ p S (y).

We now extend the notation of the extinction probability with a dependence in D : let p S,D (x) be the extinction probability of the population evolving in the environment determined by S, with a death rate D and a initial individual with mass x.

Monotonicity properties w.r.t. S on the stochastic model

We now study the variations of the survival probability w.r.t. the environmental parameter S. We need additional assumptions.

Assumptions 3.9.

1. The division rate function b is non decreasing in the two variables S and x.

2. The growth speed g in non decreasing in S: g(S 1 , x) ≤ g(S 2 , x) , ∀x ∈ [0, M ], 0 < S 1 < S 2 .

3. For any x ∈ (0, M ), the function S → b(S,x) g(S,x) is non increasing.

Assumptions 1 and 2 above are natural from the biological point of view since a bigger total biomass means a bigger fraction of biomass devoted to division and a larger amount of resources means a more efficient growth and division of cells. Assumption 3 means that the growth rate increases faster in S that the division rate. This excludes that, increasing S, a faster division produces too small individuals to grow and reproduce. Note that these assumptions are satisfied for instance if b does not depends on the variable S and if g is of the form g(S, x) = µ(S) g(x), where µ is an non decreasing function, for example a Monod kinetics [START_REF] Monod | The Growth of Bacterial Cultures[END_REF] µ(S) = µ max S K+S where µ max and K are constants. The form g(S, x) = µ(S) g(x) means that the resource concentration S influences the speed of growth of bacteria independently of the way x influences growth. In other words, the flow t → A S t (x) is just a proportional time change of t → A S t (x) for all S, S .

Theorem 3.10. Under Assumptions 3.1 and 3.9, we have for any x ∈ (0, M ) P S 1 δx (survival) ≤ P S 2 δx (survival) , ∀ 0 < S 1 ≤ S 2 .