Dr Erwan Prignot

Chanjuan Zhang

Dr Olivier Jeannin

Zakaria Halime
email: zakaria.halime@universite-paris-saclay.fr

Dr Rafael Gramage-Doria
email: rafael.gramage-doria@univ-rennes1.fr

Franck Camerel
email: franck.camerel@univ-rennes1.fr

Dr C Zhuang

Divergent behavior in the chemistry of metal-bis(dithiolene) complexes appended with peripheral aliphatic butyl chains

Keywords: Metal-bis(dithiolene), Electronic properties, Crystallography, Ligand binding, CO2 reduction

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

Introduction

Square-shaped metal-bis(dithiolene) complexes, combining a metal center and two non-innocent redox ligands, exhibit remarkable optical and electrochemical properties and have been intensively investigated in materials science, especially as precursors from conducting and magnetic crystals, [1][2] as NIR dyes for lasers [3][START_REF] Mueller-Westerhoff | Compr. Coord. Chem[END_REF] and as photothermal agents for photothermal therapy [START_REF] Mebrouk | [END_REF][6] or to develop photoresponsive polymers. [7][8] The electrochemistry associated with square-shaped metalbis(dithiolene) complexes have been also largely investigated and applied in different fields. [9] For instance, nickel-bis(dithiolene) complexes were used in the selective separation of olefins from hydrocarbon mixture through reversible electrochemically controlled olefins binding. [10] This work was extended to the study of the reversible electrochemically controlled binding of phosphine on iron and cobalt bis(dithiolene) complexes. [11] Metalbis(dithiolene) complexes were also used as catalysts for the photocatalytic and electrocatalytic reduction of protons. The first experiments on the homogeneous photoreduction of water with [Ni(S2C2Ph2)2] complex as photosensitizer in a THF/water solvent mixture under light irradiation above 290 nm were reported in 1980. [12] Since then, a plethora of symmetric, homoleptic metal-(bis)dithiolene complexes were examined for photogeneration of hydrogen under light irradiation. [13] However, the energy storage efficiencies remained modest and dithiolene complexes with symmetrical substituents or electron withdrawing groups on the aromatic rings were found unstable. Recently, several monomeric and dimeric anionic cobalt bis(dithiolene) complexes were found to be stable and highly active photocatalysts for the reduction of protons in aqueous solvents mixtures when combined with [Ru(bpy)3] 2+ as photosensitizer and electron donors. [START_REF] Mcnamara | Proc. Natl. Acad. Sci[END_REF][START_REF] Letko | [END_REF][16] The proposed elementary steps in the catalytic process include reduction of the monoanionic catalyst followed by a protonation, either on the cobalt or sulfur atom. The same complexes also act as electrocatalysts for proton reduction in aqueous solvent mixtures in the presence of a weak acid.

Metal-bis(dithiolene) complexes with a low reductive potential are also good candidate for the electro-or photocatalytic reduction of CO2. Fontecave et al. developed several bis(dithiolene) complexes of molybdenum and nickel mimicking molybdopterin, producing mainly formate, with minor amounts of carbon monoxide and hydrogen. [17][18][19] A molybdenum-containing Desulfovibriodesulfuricans formate dehydrogenase (Dd FDH) was also reported to catalyze the reduction of CO2 to formate. [20] Herein, we report on a complete series of metal-bis(dithiolene) complexes bearing butyl aliphatic chains at the periphery with Ni(II), Co(II), Pd(II), Pt(II) and Au(III) metal centers. The coordination behavior and the electronic properties associated with this series of structurally similar metal-based dithiolene complexes were investigated by various characterization techniques including NMR, UV-vis, IR, HRMS, CV, and X-ray diffraction. Whereas the square planar species derived from Ni(II), Pd(II), Pt(II) and Au(III) are monomeric in nature, the Co(II) ones are prone to form dimeric species. The binding of pyridine derivatives to this class of complexes was also investigated.

Results and Discussion

The metal-based dithiolene complexes of this study comprise peripheral butyl aliphatic chains as shown in Scheme 1. Their synthesis was adapted from a classical and well-established three-step protocol reported in the literature [21][22][23] (see Experimental Section). It begins with the virtually quantitative deprotection (95% yield) of the methoxy groups of 4,4'dimethoxybenzil 1'' under a concentrated mixture of HBr and acetic acid at 140 o C. The 1 H NMR spectrum of the crude reaction mixture revealed the disappearance of the signal corresponding to the methoxy groups at 3.88 ppm. We noted that when the reaction was conducted at lower temperatures (110 o C), the deprotection was not quantitative and the resulting mixture of starting material, product and mono-deprotected compound were not trivial to separate by column chromatography. The second step is the O-alkylation of 4,4'-dihydroxybenzyl 1' with 1bromobutane in the presence of potassium carbonate. The resulting bis-alkylated product 1 was obtained in 69% yield after purification by column chromatography followed by recrystallization. The metal-bis(dithiolene) complexes 1-M (M = Ni, Pd, Pt, Co, Au) were obtained via a two-step sequence by an initial treatment of 4,4'-dibutyloxybenzyl with P4S10 that were further reacted with 0.5 equivalents of a metallic chloride salt to yield the corresponding metal-bis(dithiolene) species in yields varying from 10 to 54 % after purification by column chromatography followed by recrystallization. It is relevant to note that the last step was also reported with 1,3-dimethyl-2imidazolidinone (DMI) as the solvent, typically leading to higher yields, nevertheless, we did not verify this protocol. [21,22] Unlike the nickel (1-Ni) and (1-Au) gold bis-(dithiolene) complexes and that were already reported, [21][22][23] palladium (1-Pd), platinum (1-Pt) and cobalt (1-Co) complexes are reported here for the first time. The spectroscopic characterization of the previously reported 1-Ni and 1-Au complexes were identical to those reported in the literature. [21][22][23] The new metal-bis(dithiolene) complexes 1-Pd, 1-Pt, and 1-Co were fully characterized by 1 H and 13 C NMR spectroscopy as well as IR, UV-vis and MS (MALDI-TOF). Whereas the 1-Ni, 1-Pd, 1-Pt, and 1-Co, with a metal ions formally in the +2 oxidation state, are isolated as neutral diamagnetic complexes, the gold complex 1-Au, with a metal ions formally in the +3 oxidation state, is isolated as a neutral radical complex. [22][23] The 1 H NMR fingerprint of these diamagnetic complexes are rather similar. For instance, a classical AX spin system at ca. 7 ppm was observed for the aromatic protons. Those peaks in the free ligand are down-field shifted to ca. 8 ppm. The methoxy CH2 protons display a signal at ca. 4 ppm as a welldefined triplet. Besides the other aliphatic CH2 protons, the CH3 methyl group have a well-defined triplet signal at ca. 1 ppm. The 13 C NMR fingerprint of these diamagnetic complexes are rather similar too with the unexpected exception of the cobalt complex 1-Co. In fact, the 13 C NMR of the cobalt complex 1-Co differs by the absence of a peak corresponding to the quaternary carbon atom linked to the cobalt-coordinating sulfur (Figure 1). In the case of 1-Ni, 1-Pd and 1-Pt complexes, the signal associated with this carbon atom displays a  = 180 ppm chemical shift. For comparison purposes, a cobalt complex similar to 1-Co, but comprising a longer aliphatic hexyl chain (2-Co) was synthesized (Scheme 2) and, like for 1-Co complex, the same carbon signal is absent in its 13 C NMR spectrum (Figure 1). This unprecedented quaternary carbon quenching in 1-Co NMR is probably attributed to the closer proximity of these carbon atoms to the paramagnetic Co(II) center, although the overall dimeric complex is diamagnetic (vide infra). Although cobalt-dithiolene complexes with different functional groups on the peripheral positions have been synthesized and studied in diverse areas of research, to the best of our knowledge, these observations associated with 13 C NMR spectroscopy have not been reported so far. Note that DOSY NMR studies performed on 1-Ni and 1-Co displayed almost identical diffusion values (D = 1.4 x 10 -5 cm 2 .s -1 ). According to precedents in the literature, [21][22][23] we noted that the IR spectra of all 1-M complexes displayed similar profiles. Elemental analysis also confirmed the homogeneity of the different metal complexes. Such divergent behaviour encountered in solution for the cobalt complexes 1-Co and 2-Co with respect to the other metalbis(dithiolene) complexes was also evidenced by UV-vis spectroscopy studies with the maximal absorbance wavelength strongly depending on the metal ion (Figure 2). For instance, 1-Ni, 1-Pd and 1-Pt complexes displayed respectively maximum absorbance at 933 (ε = 34791 L.mol -1 .cm -1 ), 956 (ε = 46862 L.mol - 1 .cm -1 ) and 871 nm (ε = 41497 L.mol -1 .cm -1 ). The gold complex 1-Au displayed a maxima ca. 1566 nm (ε = 28018 L.mol -1 .cm -1 ), which was ascribed to the classical relativistic effects this metal displays. [22] Unexpectedly, the cobalt complexes 1-Co and 2-Co featured a maximum absorbance at a significantly lower wavelengths of 742 nm (ε = 9200 L.mol -1 .cm -1 for 1-Co and ε = 15300 L.mol -1 .cm -1 for 2-Co) compared to the other similarly oxidized metals (nickel, palladium, platinum). To complement the characterization studies, we noted that the cobalt complex 1-Co displayed m/z peaks corresponding to both monomeric (m/z = 832) and dimeric (m/z = 1662) species by MS analysis. These features were not observed in the other metal complexes, which only showed peaks corresponding to the monomeric species. The above-described combination of data suggests that the cobaltdithiolene complexes exist in a dimeric form in solution [49] and that the dimer is partially dissociated during the MS analysis. Further evidences for the dimeric form of the cobaltdithiolene complexes in solution was inferred by cyclic voltammetry (CV) studies (Table 1, Figures S19-S22). [50] Whereas the 1-Ni, 1-Pd, 1-Pt and 1-Au displayed characteristic two reversible one electron reduction waves corresponding respectively to LM 0/-1 and LM -1/-2 redox couples, the cobalt complexes 1-Co and 2-Co displayed an additional irreversible reduction wave. As it was reported elsewhere for other dinuclear cobalt complexes. [START_REF] Mueller-Westerhoff | Compr. Coord. Chem[END_REF]31] In this case, due to the dimeric nature of theses complexes, the first reversible reduction wave is one electron process corresponding to (LM-LM) 0/-1 redox couple. The second one electron reduction process yield (LM-LM) 2-that dissociate immediately in solution to form two monomer as indicated by the irreversibility of the reduction wave. The current intensity of the last reversible wave corresponding to LM -1/-2 is two times higher than that of the two previous waves because the concentration of the monomer is twice that of the initial dimer, confirming once again the dimeric nature of the cobalt complexes in solution. In addition, we were able to obtain single crystals suitable for X-ray diffraction studies for complexes 1-Pd, 1-Pt and 2-Co (Figure 3, Table 2 and3). In the solid state, 1-Pd and 1-Pt species are monomeric with an almost perfect square planar geometry around the Pd 2+ and Pt 2+ metal centers (Figure 3, top), thus being isostructural with respect to the known 1-Ni and 1-Au species. [21][22][23] On the other hand, 2-Co crystallized as a welldefined dimeric structure in which the Co 2+ center adopt a slightly distorted square planar geometry around the sulfur coordinating atoms, with an additional apical binding to a Co 2+ center and sulfur atoms from another 2-Co molecule. The intermetallic Co…Co distance was found to be 2.774 Å and the intermolecular Co…S distance was 2.364 Å, thereby leading to a dinuclear cluster-like structure (Figure 3, bottom). Additional single crystal information are presented in SI. The central C-C bond and the C-S bonds on the dithiolene ligands are close (see Table 3), indicating that all ligands are in the same oxidation state. These distances, which are intermediate to those observed in the ene-1,2-dithiolate (C-S = 1.75 and C-C = 1.36 Å) and -dithione forms (C-S = 1.66 and C-C = 1.44 Å), confirm the presence of monoanionic -radical ligands on all the complexes. [51] The bond analysis confirms that the metal center in the Pd, Pt, Ni and Co complexes are in a +II state. The formation of the diamagnetic dimeric cobalt complex can be viewed as two antiferromagnetically coupled paramagnetic Co(II) complexes each having two monoanionic radical ligands. Comparison of the bond distances between 2-Co and a parent structure with methoxy groups (Table 3, Co-OMe, Co-Co: 2.7703(8) Å) [25] shows that there is no impact of the chain length on the central metal-containing core. The S deviation from the square planar D4h symmetry calculated using the Continuous Shape Measures (CShM's) algorithm implemented in the SHAPE 2.1 software [52] for 1-Pd, 1-Pt and 1-Ni are very low (0.112, 0.154 and 0.074, respectively). When the square planar geometry is considered for 2-Co, the S value significantly increase (S = 0.849) compared to the "true square planar" while a much lower S value of 0.184 is obtained when considering a ML5 coordination in a C4v square pyramid geometry.

Compound d C-C (Å) d C-S (Å) d M-S (Å) 1-Pd C1-C2 1.392(5) C3-C4 1.391(6)
The overall combination of data supports that the metal dithiolene complexes 1-Co and 2-Co species exist in a dimeric form (Scheme 3). Alternatively, the other metal dithiolene complexes derived from nickel, palladium, platinum and gold are monomers in both solution and solid state. Next, we explored the potential cleavage of the dimeric structure 1-Co using different Lewis base adducts derived from pyridine in solution. [50] However, no changes were observed in the 1 HNMR spectrum of 1-Co upon the addition of equimolar amounts of the pyridine derivatives. UV-vis studies also showed no evidence for an interaction between the pyridine and the metaldithiolene complexes. This was further supported by X-ray diffraction studies performed on single crystals derived from 1-Ni grown in the presence of an excess of pyridine (Figure 4 and Table 2). In the solid state, the nickel center from 1-Ni adopts an almost perfect square planar geometry around the dithiolene ligands with the pyridine molecule located in the unit cell far away from the nickel center (dNi…N = 10.080 Å). This observation strikingly contrasts with other nickel-dithiolene complexes that do exhibit coordination with pyridine derivatives leading to octahedral nickel complexes comprising two-fold pyridine binding in trans,trans-or cis,cis-configuration. [53][54] Attempts of growing crystals suitable for X-ray diffraction studies of 1-Ni in the presence of an even stronger coordinating ligand such as 4dimethylaminopyridine were also unsuccessful. 2,3,5,6tetrafluoropyridine was also evaluated as a potential ligand to coordinate 1-Ni in solution as both 1 H and 19 F could be used as a probe for NMR spectroscopy studies. We did only observe a limited chemical shifts associated with the proton atom and both fluorine atoms upon titration of 1-Ni with increasing quantities of 2,3,5,6-tetrafluoropyridine (Figure S27-S32). [50] Although the titration suggest a 1:1 assembly, the curve fitting failed to establish a reliable association constant due to the very minor variation of the chemical shifts, thereby suggesting a negligible interaction taking place in this system. In solution, triphenylphosphine ligand did not coordinate to the 1-Ni complex according to 1 H and 31 P NMR spectroscopy studies neither. [28] Therefore, it can be concluded that this class of metalbis(ditiolene) complexes with phenyl substituents is too electronrich to accept the lone pair of the pyridine and triphenylphosphine ligand on the metal center.

Conclusion

In summary, we have disclosed the difference of coordination behavior between monomeric and dimeric structures in a series of structurally related metal-bis(dithiolene) complexes comprising peripheral butyl aliphatic chains. In contrast to the nickel, palladium, platinum and gold derivatives, cobalt complexes have a strong preference for rearranging as dimeric species according to MS, 13 C NMR and CV spectroscopy as well as X-ray diffraction studies, in particular. It is remarkable that such long and bulky aliphatic substituents do not forbid dimeric formation for the case of 1-Co and 2-Co. All the reported metal complexes are chemically robust and do not bind ligands such as pyridine and triphenylphosphine neither in solution nor in the solid state. Overall, the present contribution highlights that even well-known and exploited metal-bis(dithiolene) complexes can still lead to interesting finding as regards of the coordination chemistry and electrochemistry herein discussed.

Experimental Section

General Methods. Solvents were purified with an MB SPS-800 purification system. Toluene was dried with CaH2 and distilled prior to use. CDCl3 was filtered through alumina and stored under argon over molecular sieves. All employed chemicals were purchased from commercial sources and used as received. Unless otherwise specified, reactions were carried out under nitrogen atmosphere by employing standard Schlenk and vacuum-line techniques. 1 H and 13 C NMR spectra were recorded with a Bruker (400 MHz and 300 MHz) spectrometer. 1 H NMR spectra were referenced to residual protonated solvent (δ = 7.26 ppm for CDCl3 and δ = 2.17 ppm for acetone-d6). 13 C NMR spectra were referenced to CDCl3 δ = 77.16 ppm). Abbreviations for signal couplings are: br, broad; s, singlet; d, doublet; t, triplet; q, quadruplet; p, quintuplet; hept, heptuplet; m, multiplet; dd, doublet of doublets; dt, triplet of doublets; td, doublet of triplets; tt, triplet of triplets; tdd, doublet of doublet of triplets. Coupling constants, J, were reported in hertz unit (Hz). UV/Vis absorption spectra were recorded with a Shimadzu 3600 plus 205 UV/Vis/NIR spectrophotometer and quartz cuvettes of 1 cm path length. Mass spectroscopy were recorded with a MALDI-TOF (less resolution). FT-IR spectra were recorded using a Bruker VERTEX 70 spectrometer equipped with an ATR apparatus. All electrochemical experiments were performed on a Metrohm Autolab 302N potentiostat. Homogeneous cyclic voltammograms were obtained in a three-electrode electrochemical cell, with a glassy carbon (3 mm diameter) working electrode (GCE), saturated calomel reference electrode (SCE), and a platinum wire counter electrode. [Bu4N]PF6 in DMF was used as the electrolyte. When the electrolyte was saturated with CO2, mQ H2O was added as the proton source for possible CO2 electro-reduction. Potentials were converted to the normal hydrogen electrode (NHE) as reference scale in homogeneous electrochemical study by the following formula: E (V vs NHE) = Applied potential (V vs SCE) + 0.244 V.

Synthesis and characterization of of 4,4'-dihydroxybenzil (1'). 4,4'dimethoxybenzil 1'' (2.37 g, 8.8 mmol) was added in a mixture of glacial acetic acid (63 mL) and hydrobromic acid 38 % wt (63 mL). This mixture was stirred for 18 h at 140 °C. After the reaction, water (300 mL) was added to the mixture and the compound was extracted with ethyl acetate (3 x 250 mL). The organic phase was neutralized with sodium hydroxide to pH = 7 and washed with water (3 x 250 mL). The organic phase was dried over MgSO4 and the ethyl acetate was removed. The product 1' was isolated as an orange powder (2.03 g, 95% yield). 1 H NMR (400 MHz, acetone-d6): δ = 9.75 (s, 2H), 7.87 (dt, J = 8.7, 1.8, 4H), 7.04 (dt, J = 8.8, 2.0, 4H) ppm. The analytical match those previously reported. [START_REF] Perochon | ) complexes bearing four butyl and hexyl aliphatic chains at the periphery with Ni(II), Co(II), Pd(II), Pt(II) and Au(III) have been investigated by a variety of characterization techniques including NMR, UV-vis, IR, MS, CV[END_REF] Synthesis and characterization of 4,4'-dibutyloxybenzil (1): A mixture of 4,4'-dihydroxybenzil 1' (0.43 g, 1.8 mmol), bromobutane (0.50 g, 3.6 mmol) and potassium carbonate (0.51 g, 3.7 mmol) was stirred in DMF (50 mL) under N2 pressure for 48 h at 110°C. The DMF was evaporated under a vacuum. The residual was extracted with CH2Cl2 (3 x 40 mL). The organic phases were washed with water (3 x 40 mL), dried over MgSO4 and the solvent was evaporated to dryness. The product was purified with chromatography on silica gel, eluting with dichloromethane/petroleum ether (7:3). Needle yellow crystal was obtained by slow diffusion of methanol on a solution of the product in dichloromethane (0.41 g, 65%). The product 1 was isolated as an orange powder (2.03 g, 95% yield). 1 H NMR (300 MHz, CDCl3): δ = 7.94 (dt, J = 9.1, 2.1, 4H), 7.14 (dt, J = 8.8, 2.0, 4H), 4.18 (t, J = 6.5, 4H), 1.88-1.77 (m, 4H), 1.61-1.47 (m, 4H), 1.01 (t, J = 7.3, 6H) ppm. 13 C{ 1 H} NMR (75 MHz, CDCl3): δ = 195.25, 166.30, 133.64 (CH), 127.75, 116.60 (CH), 69.76 (OCH2), 32.57 (CH2), 20.55 (CH2), 14.78 (CH2) ppm. The analytical match those previously reported. [START_REF] Perochon | ) complexes bearing four butyl and hexyl aliphatic chains at the periphery with Ni(II), Co(II), Pd(II), Pt(II) and Au(III) have been investigated by a variety of characterization techniques including NMR, UV-vis, IR, MS, CV[END_REF] Synthesis and characterization of nickel-dithiolene complex 1-Ni: A mixture of 4,4'-dibutyloxy-benzil 1 (0.39 g, 1.1 mmol) and phosphorous pentasulfide (0.99 g, 2.2 mmol) was stirred in 1,4-dioxane (5 mL) under N2 pressure for 2 h at 110 °C. The mixture was cooled, the precipitated was filtered and washed with 1,4-dioxane. A solution of nickel chloride hexahydrate (II) (0.13 g, 0.6 mmol) in water (1 mL) was introduced at filtrate. The mixture was stirred for 2 h at 110 °C. Methanol was added to the mixture and the precipitate was filtered, and washed with methanol. The purification of the product was carried out by chromatography on silica gel, eluting with petroleum ether/dichloromethane (1:1). The resulting

Scheme 1 .

 1 Scheme 1. Synthesis of metal-bis(dithiolene) complexes 1-M containing butyl aliphatic chains at the periphery. Reaction conditions (i) AcOH/HBr, 140 o C, 18 h, air; (ii) bromobutane (2.1 eq), K2CO3 (2.1 eq), DMF, 110 o C, 65 h, N2; (iii) P4S10 (2.1 eq), 1,4-dioxane, 110 o C, 2 h, N2.

Figure 1 .

 1 Figure 1. 13 C 1 H NMR (101 MHz, CDCl3) spectra of 1-Ni, 1-Pd, 1-Pt, 1-Co and 2-Co highlighting the aromatic area with the missing signal at ca. 180 ppm for 1-Co and 2-Co.

Figure 2 .

 2 Figure 2. UV-visible spectra (in CH2Cl2) of metal-bis(dithiolene) complexes 1-Ni (blue), 1-Pd (orange), 1-Pt (red), 1-Au (yellow), 1-Co (light green) and 2-Co (dark green). Concentration ≈ 2.10 -5 mol.L -1 .

Figure 3 .

 3 Figure 3. Single crystal X-ray diffraction studies of 1-Pd (top, left), 1-Pt (top, right) and 2-Co (bottom). Hydrogen atoms and aliphatic chains are not represented for the sake of clarity. Color code: carbon (grey), oxygen (red), sulfur (yellow), palladium (dark green), platinum (pink) and cobalt (dark blue).

3 .

 3 Scheme 3. Self-assembly formation of a dimeric structure involving the cobalt dithiolene complexes 1-Co and 2-Co. R = butyl or hexyl.

Figure 4 .

 4 Figure 4. Single crystal X-ray diffraction studies of 1-Ni with pyridine. Hydrogen atoms and aliphatic chains are not represented for the sake of clarity, and the pyridine is represented in spacefill style. Color code: carbon (grey), oxygen (red), sulfur (yellow), nitrogen (blue) and nickel (light green).

Finally, very

  preliminary experiments seems to indicate that the dinuclear cobalt complexes 1-Co and 2-Co display some electrocatalytic activity for CO2 reduction in DMF (see FigureS22in the Supporting Information).

Table 1 .

 1 Measured electrochemical data (in V vs. NHE, 0.1 M [Bu4N]PF6 ,100 mV s -1 ).

	Complex	E 1	E 2	E 3	Solvent
	1-Ni	0.202	-0.649	-	THF
	1-Pd	1.139	0.181	-0.436	DCM
	1-Pt	1.225	-0.110	-0.614	DCM
	1-Au	1.616	1.022	0.480	DCM
	1-Co	0.183	-0.282	-0.781	DMF
	2-Co	0.183	-0.278	-0.776	DMF

Table 2 .

 2 Selected crystallographic data.

		1-Pd	1-Pt	2-Co	1-Ni(pyr)
	system	triclinic	triclinic	triclinic	triclinic
	space group	P-1	P-1	P-1	P-1
	a (Å)	9.3759(14)	9.3337(9)	9.3034(14)	9.501(2)
	b (Å)	13.7748(19)	13.8450(9)	16.496(2)	14.728(3)
	c (Å)	18.341(3)	18.3912(12)	18.089(3)	17.058(4)
	 (deg)	104.658(3)	104.680(2)	97.962(4)	105.343(5)
	(deg)	100.277(4)	100.244(2)	93.160(4)	95.001(5)
	(deg)	107.115(3)	107.270(2)	105.831(4)	93.403(5)
	V (Å 3 )	2106.7(3)	2111.4(2)	2632.4(7)	2284.7(9)

Table 3 .

 3 Selected crystallographic bond distances.

Acknowledgements E.P., O.J., R.G.-D. and F.C. acknowledge the CNRS, Université de Rennes 1, Rennes Métropole and Institut des Sciences Chimiques de Rennes ("Projet inter-Equipe") for financial support. C.Z. and Z.H. acknowledge CNRS, CEA Saclay, ICMMO and University Paris-Saclay for the financial support. We thank the

China Scholarship Council for supporting C. Zhang (CSC student number 201904910525).

product 1-Ni was isolated as black crystalline powder (0.25 g, 54% yield). 1 H NMR (400 MHz, CDCl3): δ = 7.32 (dt, J = 8.9, 2.0, 8H), 6.79 (dt, J = 8.8, 1.7, 8H), 3.97 (t, J = 6.5, 8H), 1.82-1.73 (m, 8H), 1.55-1.45 (m, 8H), 0.99 (t, J = 7.4, 12H) ppm. 13 Synthesis and characterization of palladium-dithiolene complex 1-Pd: A mixture of 4,4'-dibutyloxy-benzil 1 (0.20 g, 0.6 mmol) and phosphorous pentasulfide (0.51 g, 1.1 mmol) was stirred in 1,4-dioxane (7 mL) under N2 pressure for 2 h at 110 °C. The mixture was cooled, the precipitated was filtered and washed with 1,4-dioxane. A solution of potassium tetrachloropalladate (II) (0.084 g, 0.25 mmol) in water (1 mL) was introduced at filtrate. The mixture was stirred for 2 h at 110 °C. Methanol was added to the mixture and the precipitate was filtered, and washed with methanol. The purification of the product was carried out by chromatography on silica gel, eluting with petroleum ether/dichloromethane (1:1). The resulting product 1-Pd was isolated as black crystalline needles (0.096 g, 42% yield). 1

Synthesis and characterization of platinum-dithiolene complex 1-Pt:

A mixture of 4,4'-dibutyloxy-benzil 1 (0.19 g, 0.53 mmol) and phosphorous pentasulfide (0.51 g, 1.1 mmol) was stirred in 1,4-dioxane (6 mL) under N2 pressure for 2 h at 110 °C. The mixture was cooled, the precipitated was filtered and washed with 1,4-dioxane. A solution of potassium tetrachloroplatinate (II) (0.113 g, 0.3 mmol) in water (1 mL) was introduced at filtrate. The mixture was stirred for 2 h at 110 °C. Methanol was added to the mixture and the precipitate was filtered, and washed with methanol. The purification of the product was carried out by chromatography on silica gel, eluting with petroleum ether/dichloromethane (1:1). The resulting product 1-Pt was isolated as black crystalline needles (0.094 g, 35% yield). 1 H NMR (400 MHz, CDCl3): δ = 7.24 (dt, J = 8.9, 1.7, 8H), 6.80 (dt, J = 8.7, 2.2, 8H), 3.97 (t, J = 6.5, 8H), 1.83-1.71 (m, 8H), 1.56-1.44 (m, 8H), 0.98 (t, J = 7.4, 12H) ppm. 13 Synthesis and characterization of gold-dithiolene complex 1-Au: A mixture of 4,4'-dibutyloxy-benzil 1 (0.19 g, 0.53 mmol) and phosphorous pentasulfide (0.51 g, 1.1 mmol) was stirred in 1,4-dioxane (6 mL) under N2 pressure for 2 h at 110 °C. The mixture was cooled, the precipitated was filtered and washed with 1,4-dioxane. A solution of potassium tetrachloroaurate (III) (0.113 g, 0.3 mmol) in water (1 mL) was introduced at filtrate. The mixture was stirred for 2 h at 110 °C. Methanol was added to the mixture and the precipitate was filtered, and washed with methanol. The purification of the product was carried out by chromatography on silica gel, eluting with petroleum ether/dichloromethane (1:1). The resulting product 1-Au was isolated as black crystalline needles (0.027 g, 10% yield). MS (Maldi-TOF): m/z calcd. for AuS4O4C44H52 [M] + : 969.24 , found: 968.9. Elemental analysis (%) calcd. for AuS4O4C44H52: C 54.48, H 5.40, N 0.00, S 13.22; found: C 54.95, H 5.44, N 0.00, S 13.14.

Synthesis and characterization of cobalt-dithiolene complex 1-Co: A mixture of 4,4'-dibutyloxy-benzil 1 (0.20 g, 0.6 mmol) and phosphorous pentasulfide (0.50 g, 1.1 mmol) was stirred in 1,4-dioxane (5 mL) under N2 pressure for 2 h at 110 °C. The mixture was cooled, the precipitated was filtered and washed with 1,4-dioxane. A solution of cobalt chloride (II) (0.037 g, 0.3 mmol) in water (1 mL) was introduced at filtrate. The mixture was stirred for 2 h at 110 °C. Methanol was added to the mixture and the formed precipitate was filtered, and washed with methanol. The purification of the product was carried out by chromatography on silica gel, eluting with petroleum ether/dichloromethane (1:1). The resulting product 1-Co was isolated as black crystalline needles (0.10 g, 37% yield). 1 H NMR (300 MHz, CDCl3): δ = 7.16 (d, 8H), 6.66 (d, 8H), 3.93 (t, J = 6.5, 8H), 1.85-1.69 (m, 8H), 1.57-1.42 (m, 8H), 0.99 (t, J = 7.5, 12H) ppm. 13 Synthesis and characterization of 4,4'-dihexyloxybenzil (2): A mixture of 4,4'-dihydroxybenzil 1' (1.00 g, 4.1 mmol), bromohexane (1.36 g, 8.3 mmol) and potassium carbonate (1.16 g, 8.4 mmol) was stirred in DMF (25 mL) under N2 pressure for 65 h at 110°C. The DMF was evaporated under a vacuum. The residual was extracted with CH2Cl2 (3 x 30 mL). The organic phases were washed with water (3 x 30 mL), dried with MgSO4 and the solvent was evaporated to dryness. The product 2 was purified with chromatography on silica gel, eluting with dichloromethane/petroleum ether (7:3). Needle yellow crystal was obtained by slow diffusion of methanol on a solution of the product in dichloromethane (1.03 g, 70% yield). 1 H NMR (400 MHz, acetone-d6): δ = 7.94 (dt, J = 9.0, 2.0, 4H), 7.15 (dt, J = 9.0, 2.2, 4H), 4.18 (t, J = 6.5, 4H), 1.89-1.81 (m, 4H), 1.57-1.48 (m, 4H), 1.43-1.33 (m, 8H), 0.93 (t, J = 6.9, 6H) ppm. The analytical match those previously reported. [START_REF] Perochon | ) complexes bearing four butyl and hexyl aliphatic chains at the periphery with Ni(II), Co(II), Pd(II), Pt(II) and Au(III) have been investigated by a variety of characterization techniques including NMR, UV-vis, IR, MS, CV[END_REF] Synthesis and characterization of of cobalt-dithiolene complex 2-Co: A mixture of 4,4'-dihexyloxybenzil 2 (0.21 g, 0.5 mmol) and phosphorous pentasulfide (0.45 g, 1.0 mmol) was stirred in 1,4-dioxane (5 mL) under N2 pressure for 2 h at 110 °C. The mixture was cooled, the precipitated was filtered, and washed with 1,4-dioxane. A solution of cobalt chloride (II) (0.033 g, 0.25 mmol) in water (1 mL) was introduced at filtrate. The mixture was stirred for 2 h at 110 °C. Methanol was added to the mixture and the precipitate was filtered, washed with methanol. The purification of the product was carried out by chromatography on silica gel, eluting with petroleum ether/dichloromethane (1:1). The resulting product 2-Co was isolated as black crystalline needles (0.094 g, 35% yield). 1 H NMR (300 MHz, CDCl3): δ = 7.15 (dt, J = 8.7, 2.2, 8H), 6.66 (dt, J = 8.9, 2.0, 8H), 3.92 (t, J = 6.5, 8H), 1.85-1.68 (m, 8H), 1.51-1.43 (m, 10H), 1.43-1.27 (m, 16H), 0.92 (t, 12H) ppm. 13


Antoine Perasso
email: antoine.perasso@univ-fcomte.fr

Quentin Richard
email: quentin.richard@univ-fcomte.fr

Implication of age-structure on the dynamics of Lotka Volterra equations

Keywords: Lotka Volterra, nonlinear EDP, stability, bifurcation, semigroup theory. 1

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.

Introduction

The relationships between a predator and its prey are subject of numerous studies in ecology. Since the first mathematical model describing over time such trophic interactions was introduced (see [START_REF] Lotka | Elements of physical biology[END_REF], [START_REF] Volterra | Fluctuations in the abundance of a species considered mathematically[END_REF]), the so-called Lotka Volterra equations, predator-prey models are still a wide subject of study in population dynamics. Most of them are described using ODEs and sometimes using PDEs when adding a diffusive part to model spatial dispersion properties (see for instance [START_REF] Hosono | Traveling waves for the lotka volterra predator-prey system without diffusion of the predator[END_REF], [START_REF] Langa | Permanence and asymptotically stable complete trajectories for nonautonomous lotka-volterra models with diffusion[END_REF] and more generally the book of Murray [START_REF] Murray | Mathematical Biology, Biomathematics[END_REF]).

Furthermore, it can be important to take into account an age-structure in the dynamics of the considered interacting species, the prey size being recognized as a key factor of a selective predation [START_REF] Solomon | The natural control of animal populations[END_REF].

The literature about predator-prey models with age-structure offers several compartmental ODE models (see [START_REF] Al-Omari | A stage-structured predator-prey model with distributed maturation delay and harvesting[END_REF], [START_REF] Falconi | Stability and global dynamic of a stage-structured predator-prey model with group defense mechanism of the prey[END_REF], [START_REF] Fang | Stage-structured models of intra-and inter-specific competition within age classes[END_REF], [START_REF] Liu | Recent progress on stage-structured population dynamics[END_REF] and references therein), but to our knowledge very few articles take into account a continuous age variable: the first model was proposed in [START_REF] Gurtin | On predator-prey interactions with predation dependent on age of prey[END_REF] by Gurtin and Levine, in which they prove that under some specific assumptions on the demographic parameters, the model can be transformed into a more classical ODE problem. Then most of the following articles on this subject used the same kind of assumptions on the parameters and consequently get some stability results through the ODE models (see Levine [START_REF] Levine | On the stability of a predator-prey system with egg-eating predators[END_REF], Thompson [START_REF] Thompson | Predator-prey interactions: egg-eating predators[END_REF], Coleman and Frauenthal [START_REF] Coleman | Satiable egg eating predators[END_REF], Saleem [START_REF] Saleem | Predator-prey relationships: egg-eating predators[END_REF][START_REF]Egg-eating age-structured predators in interaction with age-structured prey[END_REF] for the specific case of egg-eating predator models, also Cushing and Saleem [START_REF] Cushing | A predator prey model with age structure[END_REF], [START_REF]Predator-prey relationships: Indiscriminate predation[END_REF] and Luo, He and Li [START_REF] Luo | Optimal birth control for predator-prey system of three species with age-structure[END_REF] for more general models). Two articles focused on some qualitative properties of the solutions, that are the existence of a time-periodic solution when assuming a constant mortality rate for the prey by Levine in [START_REF]Bifurcating periodic solutions for a class of age-structured predator-prey systems[END_REF], and boundedness results by Venturino in [START_REF] Venturino | Age-structured predator-prey models[END_REF]. In the case where the predation depends only on the total quantity of prey, Li studied in [START_REF] Li | Dynamics of age-structured predator-prey population models[END_REF] the existence and stability of three biologically meaningful equilibria corresponding to the extinction and the coexistence of both species and the persistence of the prey specie only. However, it is more realistic and relevant to assume that the predation depends on the age of the prey. More recently, based on the model of Cushing and Saleem where the predator is age-structured, two articles analyzed the effect of a maturation period and a delay for a birth process (see Liu [START_REF] Liu | Stability and bifurcation in a predator-prey model with age structure and delays[END_REF][START_REF] Tang | Hopf bifurcation for a predator-prey model with age structure[END_REF]).

Finally, one can also note that some age-structured predator-prey models were also used in epidemiology, coupled with SI models (see Arino, Delgado and Molina-Becerra [START_REF] Arino | Asymptotic behavior of disease-free equilibriums of an age-structured predator-prey model with disease in the prey[END_REF][START_REF] Delgado | Analysis of an age-structured predatorprey model with disease in the prey[END_REF]).

The goal of the present work is to study predator-prey interactions by incorporating a continuous agestructure in the prey densities, leading to a PDE predator-prey model that would generalize the Lotka Volterra equations. Then we exhibit some time asymptotic properties of the solutions through a stability analysis of the PDE problem. These latter properties then allow us to study how the age-structure may alter the dynamics of the classical Lotka Volterra equations (i.e. with linear reponse) described by ODEs.

As introduced by Sharpe and Lotka in [START_REF] Sharpe | A problem in age-distribution[END_REF] and by McKendrick in [START_REF] Mckendrick | Applications of mathematics to medical problems[END_REF], structuring individuals by a continuous age variable leads to the formulation of a linear PDE of transport type. This single PDE model has been extensively studied by Gurtin and MacCamy [START_REF] Gurtin | Nonlinear age-dependent population dynamics[END_REF], Webb [START_REF] Webb | Theory of Nonlinear Age-Dependent Population Dynamics[END_REF], Metz and Diekmann [START_REF] Metz | The Dynamics of Physiologically Structured Populations[END_REF],

Thieme [START_REF] Thieme | Mathematics in Population Biology[END_REF], Perthame [START_REF] Perthame | Transport Equations in Biology, Frontiers in Mathematics[END_REF] and Magal [START_REF] Magal | Structured population models in biology and epidemiology[END_REF]. In our case, the interactions between species due to predation induce a nonlinear part in the PDE problem as described in the following predator-prey system:

                 ∂x ∂t (t, a) + ∂x ∂a (t, a) = -µ(a)x(t, a) -y(t)γ(a)x(t, a), dy dt (t) = αy(t) ∞ 0 γ(a)x(t, a)da -δy(t), x(t, 0) = ∞ 0 β(a)x(t, a)da ∀t ≥ 0, x(0, a) = x 0 (a) ∀a ≥ 0 and y(0) = y 0 , (1)
with x(t, a) and y(t) that are respectively the density of preys at age a ≥ 0 and time t ≥ 0 and the density of predators at time t where:

-α ∈ (0, 1), δ > 0 are constant parameters that respectively denote the assimilation coefficient of ingested preys and the basic mortality rate of the predator; -µ, γ, β ∈ L ∞ + (R + ) are age-dependent functions that represent the basic mortality rate of the preys, the predation rate and the birth rate for the preys.

In all that follows, we assume the following on parameter µ: ∃ µ 0 > 0 such that µ(a) ≥ µ 0 f.a.e. a ≥ 0.

(H1)

A consequence of (H1) is that ∞ µ(a)da = ∞, implying that a → e -a 0 µ(l)dl is a probability function, this latter describing the survival until age a.

Note that, linked to Problem (1), the total population at time t is given by

y(t) + ∞ 0
x(t, a) da and the total ingested preys by the predators by

α ∞ 0 γ(a)x(t, a) da.
To study nonlinear problems with an age-structure, the main approaches that have been developped are the use of Volterra formulation (see [START_REF] Iannelli | Mathematical Theory of Age-structured Population Dynamics[END_REF], [START_REF] Webb | Theory of Nonlinear Age-Dependent Population Dynamics[END_REF]), the use of suns and stars spaces (see [START_REF] Diekmann | Stability and bifurcation analysis of volterra functional equations in the light of suns and stars[END_REF]) and by semigroup theory (see [START_REF] Webb | Theory of Nonlinear Age-Dependent Population Dynamics[END_REF]). In the present work, we focus on semigroup theory, that is a useful tool to study the well-posedness of the problem and also the asymptotic properties of the solution in the nonlinear case.

This article is structured as follows: in Section 2 we tackle the well posedness of the PDE problem and prove that the solution is global in time. These results are obtained by considering model (1) as a semilinear abstract Cauchy Problem and using semigroup theory. Section 3 is devoted to the main results of the article: one can exhibits an explicit formulation of a threshold for the extinction of the total population, by performing a stability analysis of equilibria of Problem [START_REF] Al-Omari | A stage-structured predator-prey model with distributed maturation delay and harvesting[END_REF]. The use of spectral theory for the differential operator of the PDE problem, and some compactness properties of the nonlinear part of the problem are the main arguments to achieve that goal. Results about boundedness and persistence are also proved, using upper bounds of the characteristics of the PDE equation. Some numerical computations that enlighten the theoretical results are performed in Section 4. Finally, some remarks are postponed in Section 4.3.

The reader will find in Appendix a brief reminder about Lotka Volterra model and the link with the age-structured predator-prey model studied in this article.

2 Well Posedness and Positivity

Notations

In all that follows, consider the Banach Spaces X = L 1 (R + ) × R with the product norm and his nonnegative cone defined by

X + = L 1 + (R + ) × R + . We consider the following differential operator A : D(A) ⊂ X → X, D(A) = {(φ, z) ∈ X, φ ∈ W 1,1 (R + ) and φ(0) = ∞ 0 β(a)φ(a)da}, A =   D 0 0 -δ   with Dφ = - dφ da -µφ
and the function f : X → X given by

f (φ, z) =   -zγ(.)φ(.) αz ∞ 0 γ(a)φ(a)da   ,
so that Problem (1) rewrites as the following abstract Cauchy Problem:

         d dt   x(t) y(t)   = A   x(t) y(t)   + f (x(t), y(t)),
(x(0), y(0)) = (x 0 (.), y 0 ) ∈ X.

(2)

Linear Part

To perform an analysis of Problem (2), we start by a study of the differential operator (A, D(A)).

The same arguments that developped in [START_REF] Banks | Transformation semigroups and l 1 -approximation for size structured population models[END_REF] prove that D(A) is a dense subset of X. Furthermore, as proved in [START_REF] Webb | Theory of Nonlinear Age-Dependent Population Dynamics[END_REF] and [START_REF] Perasso | Parameter identifiability for systems described by partial differential equations[END_REF] for such an operator, there exists real values λ 0 and ω 0 satisfying min(λ 0 , ω 0 ) ≥ β L ∞ (R+) -µ 0 such that A -λ 0 I is dissipative and (λI -A) is surjective for every λ ≥ ω 0 . Then the classical Lumer-Phillips theorem implies that A is the infinitesimal generator of a positive C 0 -semigroup (T A (t)) t≥0 that is quasicontractive: T A (t) X ≤ e ωt for all t ≥ 0 and ω ≥ β L ∞ (R+) -µ 0 .

Local existence, Uniqueness and Positivity

Since the nonlinear part f of Problem ( 2) is quadratic, it is clearly a locally Lipschitz continuous function on X. A consequence of [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Differential Equations[END_REF] is that for every (x 0 , y 0 ) ∈ X, there exists t max ≤ +∞ such that Problem

(2) has a unique mild solution u ∈ C([0, t max (x 0 , y 0 )), X) where t max (x 0 , y 0 ) ≤ ∞. Furthermore, this solution is defined in a classical sense whenever (x 0 , y 0 ) ∈ D(A).

We now prove that for any initial condition (x 0 , y 0 ) ∈ X + , the corresponding solution remains nonnegative on [0, t max ). To achieve that goal, we need the two following lemmas. Their proof can be found in the articles [START_REF] Perasso | Infection load structured si model with exponential velocity and external source of contamination[END_REF][START_REF]Asymptotic behavior and numerical simulations for an infection load-structured epidemiological model : Application to the transmission of prion pathologies[END_REF].

Let us define B m = {(φ, z) ∈ X : (φ, z) X ≤ m} for m > 0.

Lemma 2.1. ∃K > 0 such that ∀m > 0 and ∀((

φ 1 , z 1 ), (φ 2 , z 2 )) ∈ B 2 m , we have f (φ 1 , z 1 )-f (φ 2 , z 2 ) X ≤ m K (φ 1 , z 1 ) -(φ 2 , z 2 ) X . Lemma 2.2. ∀m > 0, ∃λ m ∈ R such that (φ, z) ∈ B m ∩ X + ⇒ f (φ, z) + λ m (φ, z) ∈ X + . In fact, it is sufficient to take λ m > m γ L ∞ (R+) . Proposition 2.1. ∀(x 0 , y 0 ) ∈ X + , ∃t max (x 0 , y 0 ) ≤ ∞ such that Problem (2) has a unique mild solution u ∈ C([0, t max (x 0 , y 0 )), X + ).
Proof. Let m > 0 and λ m ≥ m γ L ∞ (R+) . Let us define the differential operator A m = A -λ m I and the function f m = f + λ m I. Then A m is the infinitesimal generator of a positive C 0 -semigroup (T Am (t)) t≥0 on X + that satisfies T Am (t) X ≤ e -(λm+ω)t for every t ≥ 0.

We then define r m = 2 (x 0 , y 0 ) X sup s∈[0,1] T Am (s) > 0, suppose m large enough to have r m ≤ m and we denote X rm + = X + ∩ B rm ⊂ B m . Then considering τ > 0 such that:

τ ≤ min   1, 1 2(Kr m + λ m ) × sup s∈[0,1] T Am (s) X    ,
a consequence of lemmas 2.1 and 2.2 is that the linear operator

G : C([0, τ ], X) → C([0, τ ], X) defined by G(x(t, •), y(t)) = T Am (t).   x 0 y 0   + t 0 T Am (t -s)f m   x(s, •) y(s)   ds is a 1/2-shrinking operator on C([0, τ ], X rm
+ ) that preserves this latter space. The Banach-Picard theorem and some classical time extending properties of the solution then yield the proposition.

Global existence

Theorem 2.3. For all (x 0 , y 0 ) ∈ X + , Problem (2) has a unique mild solution (x, y) ∈ C(R + , X + ).

Proof. Consider (x, y) ∈ C([0, t max ), X + ) the solution of (2) and suppose by contradiction that t max < ∞.

Let us first prove that for every t ≥ 0, x(t, •) L 1 (R+) < ∞.

A direct consequence of the positivity is that ∂ t x(t, a) + ∂ a x(t, a) ≤ -µ(a)x(t, a). It is classical, using the characteristics of the PDE equation, that an implicit formulation of the solution of ∂ t x(t, a) + ∂ a x(t, a) = -µ(a)x(t, a) that satisfies the loopback boundary condition in (1) is given by:

x(t, a) =      x 0 (a -t)e -a a-t µ(s)ds if a ≥ t, ψ(t -a)e -a 0 µ(s)ds if a < t, (3)
where ψ(t) = x(t, 0) satisfies:

ψ(t) = t 0 β(u)ψ(t -u)e -u 0 µ(s)ds du + ∞ t β(u)x 0 (u -t)e -u u-t µ(s)ds du = t 0 ψ(u)β(t -u)e -t-u 0 µ(s)ds du + ∞ 0 β(u + t)x 0 (u)e -u+t u µ(s)ds du. ( 4
)
From equation (4) we define two operators S 1 : L 1 (0, t) → L 1 (0, t) and S 2 :

L 1 (R + ) → L 1 (0, t) for every ψ ∈ L 1 (0, t), φ ∈ L 1 (R + ) and ξ ∈ [0, t] by: S 1 ψ(ξ) = ξ 0 ψ(y)β(ξ -y)e -ξ-y 0 µ(s)ds dy, S 2 φ(ξ) = ∞ 0 φ(y)β(y + ξ)e -ξ+y y
µ(s)ds dy, so we formally get the following representation:

x(t, a) =      x 0 (a -t)e -a a-t µ(s)ds if a ≥ t, (I -S 1 ) -1 S 2 x 0 (t -a)e -a 0 µ(s)ds if a < t.
This latter equality is well defined. Indeed, as proved in [START_REF] Arino | Asymptotic behavior of disease-free equilibriums of an age-structured predator-prey model with disease in the prey[END_REF], S 1 is a Volterra operator, then for all

λ ∈ C \ {0} and ψ ∈ L 1 (0, t) fixed, we have a unique function ϕ ∈ L 1 (0, t) such as (λI -S 1 )ϕ = ψ. Thus (I -S 1 ) -1 is well defined from L 1 (0, t) to L 1 (0, t). Since x 0 ∈ L 1 (R + ) then (I -S 1 ) -1 S 2 x 0 ∈ L 1 (0, t).
So for all t ≥ 0 we have:

x(t, •) L 1 (R+) ≤ ∞ t x 0 (a -t)da + t 0 (I -S 1 ) -1 S 2 x 0 (t -a)da < ∞.
Moreover, straightforward upper bounds imply that for every t ≥ 0, y (t) ≤ αM y(t) γ L ∞ (R+) where

M = max s∈[0,tmax] x(s, •) L 1 (R+) < ∞.
Thus, if t max < ∞, an integration of the differential inequality would lead to y(t) ≤ y 0 e αM tmax γ L ∞ < ∞, implying a contradiction with the fact that we have either

lim t→tmax x(t, •) L 1 (R + ) = ∞ or lim t→tmax |y(t)| = ∞.
Finally t max = ∞ and the solution is global in time.

We finally get existence and uniqueness of a global nonnegative solution. The goal is now to analyze the asymptotic behavior of the solutions.

Stability, persistence and boundedness

Suppose that we have a positive initial condition (x 0 , y 0 ) ∈ X * + , where X * + = {(x 0 , y 0 ) ∈ X + : ∞ 0 x 0 (a)da > 0 and y 0 > 0}. Let us define a 1 by:

a 1 = sup{a ≥ 0 : |supp(γ) ∩ (0, a)| = 0} < ∞. (5)
Remark 1. This definition implies that there exists γ -> 0 and a 2 > a 1 such that a2 a1 γ(a)da ≥ γ -. The case a 1 > 0 translates the fact that the youngest preys are not considered as a resource availability for the predators.

In all that follows, let us consider the following thresholds:

R 0 = ∞ 0 β(a)e -a 0 µ(s)ds da, (6)
R -= a1 0 β(a)e -a 0 µ(s)ds da. ( 7
)
Note that, similarly to the basic reproductive number in the epidemiological case (see [START_REF] Dieckmann | Mathematical epidemiology of infectious diseases[END_REF]), the R 0 value represents the average number of offspring that is produced over the lifetime by one prey and in a context with no predation. This threshold will give a characterization of extinction of the total population.

In the same idea, the R -value represents the offspring produced by one prey from his birth, until it begins to be hunted by the predator. We will get an unboundedness result from this threshold.

Equilibria

We first look for steady points of problem (1).

The point (x * , y * ) ∈ X will be an equilibrium if it is a solution of the following system:

         (x * ) (a) = -µ(a)x * (a) -y * γ(a)x * (a), 0 = αy * ∞ 0 γ(a)x * (a)da -δy * , x * (0) = ∞ 0 x * (a)β(a)da.
An integration then gives:

         x * (a) = x * (0)e -a 0 µ(s)ds-y * a 0 γ(s)ds , x * (0) 1 - ∞ 0 β(a)e -a 0 µ(s)ds-y * a 0 γ(s)ds da = 0, y * α ∞ 0 γ(a)x * (a)da -δ = 0.
Note that R -≤ R 0 and we get the following proposition:

Proposition 3.1. 1. If R 0 < 1 or if (R 0 > 1 and R -≥ 1)
then there is a unique equilibrium that is E 0 = (0, 0);

2. If R 0 = 1 and R -< 1 then for all ξ ∈ [0, ∞), E 1,ξ = (x * 1,ξ , 0) is an equilibrium, where x 1,ξ is defined by x * 1,ξ (a) = ξe -a 0 µ(s)ds . In particular E 1,0 = E 0 ; 3. If R 0 = 1 and R -= 1 then ∀ξ ≥ 0, E 1,
ξ is an equilibrium and E 2,ξ = (x * 2,ξ , ξ) also, where:

x * 2,ξ (a) = x * 2,ξ (0)e -a 0 µ(s)ds-ξ a 0 γ(s)ds , x * 2,ξ (0) = δ α ∞ 0 γ(a)e -a 0 µ(s)ds-ξ a 0 γ(s)ds da -1 ;
4. If R 0 > 1 and R -< 1 then there are the trivial equilibrium E 0 and the positive equilibrium

E 2 = (x * 2 , y * ) = (x 2,y * , y * ) with y * that satisfies: ∞ 0 β(a)e -a 0 µ(s)ds-y * a 0 γ(s)ds da = 1. ( 8
)

Stability

To perform the stability analysis, we exhibit some spectral properties of the differential operator A and of the semigroup {T A (t)} t≥0 . Before proceeding, we introduce some notations and recall some definitions of spectral theory.

Denoting ρ(A) the resolvent set of A : D(A) ⊂ X → X, we define the spectrum σ(A), the point spectrum σ p (A) and the spectral bound s(A) by:

σ(A) = C\ρ(A), σ p (A) = {λ ∈ C, λI -A is not injective}, s(A) = sup{Re λ, λ ∈ σ(A)}.
Denoting L(X) the set of bounded linear operators on X and K(X) the subset of compact operators on X, we then define the essential norm L ess of L ∈ L(X) by:

L ess = inf K∈K(X) L -K X .
We recall that the quotient L(X)/ K(X) is called the Calkin algebra which, when providing the norm

L = L ess ,
where L = L + K(X), is a Banach algebra with unit (see [START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF] and references cited in for details). The

growth bound ω 0 (A) ∈ [-∞, ∞) of A is defined by: ω 0 (A) = lim t→∞ 1 t ln ( T A (t) X ) ,
and the essential growth bound ω ess (A) ∈ [-∞, ∞) of A is:

ω ess (A) = lim t→∞ 1 t ln ( T A (t) ess ) .
The following theorem that gives a characterization of the growth bound using the spectrum of A was proved by Engel and Nagel [START_REF] Engel | One-parameter semigroups for linear evolution equations[END_REF].

Theorem 3.1. The growth bound of A satisfies

ω 0 (A) = max{ω ess (A), s(A)},
and for every ω > ω ess (A) the set σ ω = {λ ∈ σ(A), Re (λ) > ω} is finite and composed of finite algebraic multiplicity elements.

A classical result states that (see [START_REF] Yosida | Functional Analysis[END_REF]):

ω > ω ess (A) ⇒ σ ω ⊂ σ p (A).
Remark 2. Due to the quotient defined previously, the use of the Calkin algebra shows well why the compact operators do not affect the growth bound values. More specifically, one gets ω ess (A+K) = ω ess (A)

for every K ∈ K(X).

The following theorem (see [START_REF] Prüß | Equilibrium solutions of age-specific population dynamics of several species[END_REF] or [START_REF] Webb | Theory of Nonlinear Age-Dependent Population Dynamics[END_REF]), gives some conditions for an equilibrium to be stable.

Theorem 3.2. Consider E an equilibrium of Problem (2). Then the following assertions hold:

1. If ω 0 (A + D E f ) < 0 then E is locally exponentially asymptotically stable for Problem (2); 2. If ω 0 (A + D E f ) > 0 and ω ess (A + D E f ) ≤ 0 then E is unstable for Problem (2).
In the context of Problem (1), the following result holds:

Theorem 3.3. We have ω ess (A + D E f ) ≤ -µ 0 < 0.
To prove Theorem 3.3, we start by a lemma dealing with some compactness properties about the differential of f . One can note that for every E = (x * , y * ) ∈ X, the differential of f at an equilibrium E can be written as:

D E f = (D E f ) 1 + (D E f ) 2 =   -y * γ 0 0 0   +   0 -γx * αy * L γ (•) α ∞ 0 γ(a)x * (a)da   ,
where L γ is the operator defined for some integrable function h on R + by:

L γ : h → ∞ 0 γ(a)h(a)da. (9)
Here is the compactness result:

Lemma 3.4. The function f is in C 1 (X) and the operator (D E f ) 2 is compact. Proof. Since function D E f : X → X is Lipschitz continuous, it is a bounded operator on the Banach space X and so D E f is continuous and f is in C 1 (X).
We now prove that (D E f ) 2 is compact.

Denoting (D E f ) 2 = (G 1 , G 2 ) where G 1 : X → L 1 (R + ) and G 2 : X → R, we can easily check that G 2
has a finite dimensional range and is consequently compact. To prove the compactness of G 1 we use the classical Riesz-Fréchet-Kolmogorov (RFK) criterion in L 1 (see for instance Yosida [START_REF] Yosida | Functional Analysis[END_REF]).

Let h ∈ R + and S ⊂ X be a bounded subset of X. Then there exists M ∈ R + such that (φ, z) X ≤ M for every (φ, z) ∈ S. Denoting by τ h (φ) = φ(• + h) the translation operator in L 1 we then have:

τ h G 1 (φ, z) -G 1 (φ, z) L 1 (R + ) ≤ |z| ∞ 0 |γ(a)x * (a) -γ(a + h)x * (a + h)|da ≤ M τ h (γx * ) -(γx * ) L 1 (R + ) .
Since γ ∈ L ∞ (R + ) and x * ∈ L 1 (R + ) then γx * ∈ L 1 (R + ) and consequently we have:

τ h (γx * ) -(γx * ) L 1 (R+) → h→0 0.
It implies that:

sup (φ,z)∈S τ h G 1 (φ, z) -G 1 (φ, z) L 1 (R+) → h→0 0. ( 10
)
Furthermore we can similarly prove, using the Lebesgue theorem, that sup

(φ,z)∈S ∞ r |G 1 (φ(a), z)|da → r→∞ 0. ( 11
)
The consequence of ( 10)-( 11) and the RFK criterion is the relative compactness of

G 1 (S) in L 1 (R + ).
Finally, G 1 is compact and so is (

D E f ) 2 .
We can now prove Theorem 3.3:

Proof. (Theorem 3.3) Lemma 3.4 implies that for every equilibrium E, ω ess (A + D E f ) = ω ess (A + (D E f ) 1 ), so we have to

prove that ω ess (A + (D E f ) 1 ) ≤ -µ 0 .
Similarly to the proof of Theorem 2.3, we can calculate the concrete expression of the semigroup generated by A + (D E f ) 1 that is:

T A+(D E f )1 (t)   x 0 y 0   =   x 0 (a -t)e -a a-t (µ(s)+y * γ(s))ds 1 {a≥t} + ψ(t -a)e -a 0 (µ(s)+y * γ(s))ds 1 {a<t} y 0 e -δt   ,
where ψ(t) = x(t, 0).

We decompose the operator T A+(D E f )1 in:

T A+(D E f )1 (t)   x 0 y 0   (a) = T 1 (t)   x 0 y 0   (a) + T 2 (t)   x 0 y 0   (a) + T 3 (t)   x 0 y 0   (a)
, with :

T 1 (t)   x 0 y 0   (a) =      (x 0 (a -t)e -a a-t (µ(s)+y * γ(s))ds , 0) if a ≥ t, (0, 0) if a < t. T 2 (t)   x 0 y 0   (a) =      (0, 0) if a ≥ t,
(ψ(t -a)e -a 0 (µ(s)+y * γ(s))ds , 0) if a < t.

T 3 (t)   x 0 y 0   (a) = (0, y 0 e -δt ), ∀t ≥ 0, a ≥ 0.
The operator T 3 is compact because its range is a finite dimensional space.

For the operator T 1 , we get the following upper bound:

T 1 (t)   x 0 y 0   X ≤ ∞ t
x 0 (a -t)e -a a-t µ(s)ds da ≤ e -µ0t ∞ 0

x 0 (u)du and consequently we get:

T 1 (t) X ≤ e -µ0t . (12)
Let us define Ŝ1 : L 1 (0, t) → L 1 (0, t) and Ŝ2 : Thus we get the following expression for the operator T 2 :

L 1 (R + ) → L 1 (0, t) by: Ŝ1 ψ(ξ) = ξ 0 ψ (
T 2 (t)   x 0 y 0   (a) =      (0, 0) if a ≥ t, (I -Ŝ1 ) -1 Ŝ2 x 0 (t -a) e -a 0 (µ(s)+y * γ(s) ds), 0 if a < t.
We can show that Ŝ1 is bounded, so as I -Ŝ1 . Since (I -Ŝ1 ) -1 is well defined and (I -Ŝ1 ) is bijective from L 1 (0, t) in itself, which is a Banach space, then (I -Ŝ1 ) -1 is bounded.

Let us define the operator S 2 : L 1 (R + ) → L 1 (0, t) by:

S 2 φ(ξ) = ∞ 0 φ(y)ce -ξ+y y [µ(s)+y * γ(s)]ds dy,
where c is a positive constant. Using the RFK criterion (see Lemma 3.4), we can prove that S 2 is compact for every c > 0. Indeed, setting h > 0 and taking B a bounded subset of L 1 (R + ) we get, for φ ∈ B:

τ h S 2 φ -S 2 φ L 1 (0,t) ≤ c t 0 ∞ 0 φ(y) e -ξ+y y [µ(s)+y * γ(s)]ds -e -ξ+y+h y [µ(s)+y * γ(s)]ds dydξ ≤ c t 0 ∞ 0 φ(y)e -ξ+y y [µ(s)+y * γ(s)]ds 1 -e -ξ+y+h ξ+y [µ(s)+y * γ(s)]ds dydξ ≤ c t 0 ∞ 0 φ(y) 1 -e -h µ L ∞ dydξ ≤ ct 1 -e -h µ L ∞ ∞ 0 φ(y)dy,
which converges to 0 uniformly on B when h tends to 0 since B is bounded.

Therefore S 2 is compact. Since for c = β L ∞ we have Ŝ2 φ(x) ≤ S 2 φ(x) for all φ ∈ L 1 (R + ) and x ∈ [0, t],

then Ŝ2 is also compact and so is the operator T 2 .

Finally since T 2 and T 3 are compact,

T A+(D E f )1 (t) ess = T 1 (t) + T 2 (t) + T 3 (t) ess = T 1 (t) ess ≤ T 1 (t) X ,
and consequently to [START_REF] Fang | Stage-structured models of intra-and inter-specific competition within age classes[END_REF],

ω ess (A + (D E f ) 1 ) ≤ -µ 0 .

Equilibrium E 0

The differential of f at the point E 0 is the null matrix. So the linearized system to study is u (t) = Au(t).

Using Theorem 3.1 and since ω ess (A) < 0, we just need to study eigenvalues of A. We thus try to solve the following system:

         ∂x ∂t (t, a) = - ∂x ∂a (t, a) -µ(a)x(t, a), y (t) = -δy(t), x(t, 0) = ∞ 0 x(t, a)β(a)da. (13)
We are looking for solutions of the form x(t, a) = x(a)e λt and y(t) = ȳe λt , λ ∈ C. So, after replacing the latter expressions in the first equation of the system (13) then resolving the system, we get:

         x(a) = x(0)e -a 0 [λ+µ(s)]ds , λȳ = -δ ȳ, x(0) = ∞ 0 x(a)β(a)da.
The second equation only admits -δ as eigenvalue, which is real and negative. Then, using the third equation, we obtain the following characteristic equation:

∞ 0 β(a)e -a 0 [λ+µ(s)]ds da = 1.
We thus can show the following theorem, where R 0 is defined in (6):

Theorem 3.5.

1. If R 0 < 1 then E 0 is globally stable.

2. If R 0 > 1 then E 0 is unstable.

Proof.

1. Suppose that R 0 < 1. The real part of the characteristic equation gives:

∞ 0 β(a)e -Re (λ)a cos(-Im (λ)a)e -a 0 µ(s)ds da = 1.

Then, if Re (λ) ≥ 0 we get R 0 ≥ 1 that is absurd, so ω 0 (A) < 0 and E 0 is locally exponentially asymptotically stable from Theorem 3.2. Now we prove the global stability. Using characteristics we get for t ≥ 0:

x(t, a) ≤      x 0 (a -t)e -µ0t if a ≥ t, x(t -a, 0)e -a 0 µ(s)ds if a < t.
Then ψ(t) ≤ t 0 β(u)ψ(t -u)e -u 0 µ(s)ds du + ∞ t β(u)x 0 (u -t)e -µ0t du. Some standard upper bounds give:

ψ(t) ≤ R 0 max z∈[0,t] ψ(z) + β L ∞ x 0 L 1 e -µ0t . Since R 0 < 1, we can consider C ∈]R 0 , 1[.
Then for t = t big enough, we get ψ(t ) ≤ Cψ(t), where ψ(t) = max z∈[0,t ] ψ(z).

Moreover, ∀h > 0, ψ(t + h) ≤ C max z∈[0,t +h] ψ(z) so ψ(t + h) ≤ Cψ(t), consequently ψ(t) = max z∈[0,∞[ ψ(z) and ψ is bounded.

For every h > 0, we get, by separating the integral, the following upper bound:

ψ(t + h) ≤ max u∈[0,t ] ψ(u) β L ∞ e -µ0h µ 0 + R 0 max u∈[t ,t +h] ψ(u) + β L ∞ x 0 L 1 e -µ0(t +h) .
Since the first and the third terms go to 0 when h tends to ∞, then we can find h 1 > 0 such that

∀h ≥ h 1 , ψ(t + h) ≤ C 2 ψ(t).
Using the same method, we can find

h 2 > h 1 such that ∀h ≥ h 2 : ψ(t + h) ≤ C 3 ψ(t).
Then by induction we get: ∀n ∈ N, ∃h n > 0 such that ∀h ≥ h n : ψ(t + h) ≤ C n+1 ψ(t).

Since C < 1, we have lim n→∞ C n+1 = 0 and lim t→∞ ψ(t) = 0.

Then using the implicit formulation (3) of x(t, a) along the characteristics and some basical upper bounds we get:

x(t, .) L 1 (R+) ≤ e -µ0t x 0 L 1 (R+) + e -µ0t t 0 ψ(u)e µ0u du.
Moreover, for every κ > 0, there exists T > 0 such that ψ(t) ≤ κ, ∀t ≥ T and consequently:

x(t, .) L 1 (R+) ≤ e -µ0t x 0 L 1 (R+) + e -µ0t We know that for every κ 2 > 0, there exists T 2 > T such that e -µ0t ≤ κ 2 , ∀t ≥ T 2 . Then we get:

x(t, .) L 1 (R+) ≤ κ 2 x 0 L 1 (R+) + κ 2 T 0 ψ(u)e uµ0 du + κ µ 0 .
Now, take ε > 0 and define:

κ = εµ 0 3 , κ 2 = κ µ 0 min 1 x 0 L 1 (R+) , 1 T 0 ψ(u)e uµ0 du .
Then for every t ≥ T 2 we have x(t, .) L 1 (R+) ≤ ε. Consequently lim t→∞ x(t, .) L 1 (R+) = 0.

Finally, with the second equation of (1), we get lim t→∞ y(t) = 0 and the global stability of E 0 follows.

2. Suppose that R 0 > 1 and define the function g by:

g : λ → ∞ 0
β(a)e -λa e -a 0 µ(s)ds da.

Then g is strictly decreasing, with g(0) = R 0 > 1. There consequently exists λ > 0 such that g(λ) = 1, so ω 0 (A) > 0 and since ω ess (A) ≤ 0, Theorem 3.2 implies that E 0 is unstable.

Remark 3. We can see that in the case R 0 < 1, the total population is bounded.

Equilibria E 1,ξ

Let ξ > 0 and consider the equilibrium E 1,ξ . Then the differential of f at the point E 1,ξ is given by:

D E 1,ξ f =   0 -γx * 1,ξ 0 α ∞ 0 γ(a)x * 1,ξ (a)da   .
The linearized system is thus:

du dt (t) = (A + D E 1,ξ f )u(t).
Once again we just need to study eigenvalues of the operator A + D E 1,ξ , so we study the following system:

         x(a) = x(0)e -a 0 [λ+µ(s)]ds -γ(a)x * 1,ξ (a)ȳ, λȳ = -δ ȳ + αȳ ∞ 0 γ(a)x * 1,ξ (a)da, x(0) = ∞ 0 x(a)β(a)da. ( 14
)
Let us denote :

S = δ α ∞ 0 γ(a)e -a 0 µ(s)ds da -1
.

Then we get the following instability theorem:

Theorem 3.6.

1. If ξ > S then the equilibrium E 1,ξ is unstable.

2. If ξ > 0 then for every > 0 there exists ξ( ) such that E 1, ξ ∈ B(E 1,ξ , ).

Proof.

1. Let ξ > S. The second equation of ( 14) and the definition of x * 1,ξ given in Proposition 3.1 imply:

λȳ = -δ + α ∞ 0 γ(a)ξe -a 0 µ(s)ds da ȳ.
Then defining λ = -δ + α ∞ 0 γ(a)ξe -a 0 µ(s)ds da, we get λ > 0 since ξ > S.

The first and third equations of ( 14) give us:

1 - ∞ 0 β(a)e -a 0 (µ(s)+ λ)ds da x(0) + ξ ȳ ∞ 0 β(a)γ(a)e -a 0 µ(s)ds da = 0.
Since λ > 0, we get ∞ 0 β(a)e -a 0 (µ(s)+ λ)ds da < R 0 . We know by definition of E 1,ξ that R 0 = 1.

Consequently (1 -∞ 0 β(a)e -a 0 (µ(s)+ λ)ds da) > 0. Let ȳ = 1 and define:

x(0) = -ξ ∞ 0 β(a)γ(a)e -a 0 µ(s)ds da 1 - ∞ 0 β(a)e -a 0 (µ(s)+ λ)ds da .
Then, defining x(a) = x(0)e -a 0 [ λ+µ(s)]ds -γ(a)x * 1,ξ (a), we finally find (x, ȳ) = (0, 0) verifying the system ( 14), so we find a positive eigenvalue and the equilibrium E 1,ξ is unstable by using Theorem 3.2.

2. Let ξ > 0 and let > 0. Then, by defining ξ = ξ + µ 0 we get

E 1, ξ ∈ B(E 1,ξ , ). Indeed E 1, ξ -E 1,ξ X = x * 1, ξ -x * 1,ξ L 1 (R+) = | ξ -ξ| ∞ 0 e -a 0 µ(s)ds da ≤ µ 0 ∞ 0 e -µ0a da ≤ .
The latter assertion prevents all equilibria E 1,ξ , ξ ≥ 0 to be locally asymptotically stable.

Equilibrium E 2

We now focus on the equilibrium E 2 = (x * 2 , y * ) that exists if and only if R 0 > 1 and R -< 1. Then the differential of f at the point E 2 is given by:

D E2 f =   -y * γ -γx * 2 αy * L γ (•) δ  
where L γ is defined in (9). Since the linearized system is u (t) = (A + D E2 f )u(t) we thus have to study the following system:

         x (a) = -[λ + µ(a) + y * γ(a)] x(a) -γ(a)x * 2 (a)ȳ, λȳ = αy * ∞ 0 γ(a)x(a)da, x(0) = ∞ 0 x(a)β(a)da.
We finally have to solve the system BX = C, where: In the specific case where γ is constant, we can compute the eigenvalues. This implies that:

B =   b 1 b 2 b 3 b 4   , C =   0 0   and X =   x(0) ȳ   , with :                  b 1 = 1 - ∞ 0 β(a)e -
b 1 b 4 = b 2 b 3 ⇔ b 4 = δγ 0 b 3 αΓλ ⇔ -λ - δy * γ 2 0 λΓ ∞ 0 e -a 0 (µ(s)+γ0y * )ds da - ∞ 0 e -a 0 (µ(s)+γ0y * +λ)ds da = δy * γ 2 0 λΓ ∞ 0 e -a 0 (µ(s)+γ0y * +λ)ds da.
With the expression of Γ, we finally get:

-λ = δy * γ 0 λ and the proof is completed.

Note that similarly to the ODE Lotka Volterra system, we get the existence of imaginary eigenvalues.

In the case where γ is not constant, the analysis is much more complicate to perform. However, we will perform some simulations in the next section to exhibit the different possible behaviors.

Persistence

In this subsection we express some standard notions from mathematical ecology by formulating, in the context of Problem (1), the definition of persistence. First, let us give a proposition, whose arguments of the proof can be found in [START_REF]Asymptotic behavior and numerical simulations for an infection load-structured epidemiological model : Application to the transmission of prion pathologies[END_REF].

Proposition 3.2. Problem (1) induces a continuous semiflow via

φ t : R + × X + → X + (t, z 0 ) → φ t (z 0 ) = (x(t), y(t)),
where (x(t), y(t)) is the unique solution that satisfies (x(0), y(0)) = z 0 .

Let ρ : X → R + be a nonnegative uniformly continuous function on X and consider the composition

σ ρ (t, x) = ρ(φ t (x))
, where φ t is the semiflow defined in Proposition 3.2. Remark that this latter proposition implies that σ ρ is a continuous map from R + × X + to R + . Let us introduce the following notations, that will be used in all the sequel:

σ + ρ (x) = lim sup t→∞ σ ρ (t, x), σ - ρ (x) = lim inf t→∞ σ ρ (t, x).
Let us suppose the following assumption on parameter β:

∃ η 1 > 0, ∃ 0 < a < a < ∞ such that β(a) ≥ η 1 for almost every (f.a.e) a ∈ (a, a) (H2)

and consider the following property that will ensure persistence:

∃ η 2 > 0, ∃ 0 ≤ t 1 < t 2 < a : t2 t1 x 0 (a)da ≥ η 2 . (P)
Without lost of generality, we assume that |t 2 -t 1 | < |a -a| even if we reduce η 2 .

Remark 4.

1. Assumption (H2) means that preys of a certain range of age have a high ability to reproduce.

2. Assumption (P) together with property (H2) means that there is initially a high enough quantity of young preys that will be able to reproduce later.

Let us consider in all that follows the set X P = {(x 0 , y 0 ) ∈ X * + that satisfies (P)}. Here is the definition of persistence for our system. Definition 1. Considering the mapping ρ 1 : (x, y) ∈ X → x L 1 (R+) , we say that: 1. The prey population is uniformly strongly persistent if ∃ε > 0 : ∀(x 0 , y 0 ) ∈ X P , ρ 1 (x 0 , y 0 ) > 0 ⇒ σ - ρ1 (x 0 , y 0 ) ≥ ε.

The prey population is uniformly weakly persistent

when considering σ + ρ1 (x) instead of σ - ρ1 (x).
This definition can be naturally extended to the case of persistence of the predator population, by considering the map ρ 2 : (x, y) ∈ X → y instead of ρ 1 .

We thus can prove the following theorem, where R -is defined in [START_REF] Dieckmann | Mathematical epidemiology of infectious diseases[END_REF].

Theorem 3.8. Suppose that the initial condition (x 0 , y 0 ) ∈ X P . If R -> 1 then prey population and predator population are unbounded functions.

Proof. We know that:

∂ t x(t, a) + ∂ a x(t, a) = -(µ(a) + y(t)γ(a))x(t, a), ∀t ≥ 0, ∀a ≥ 0.
This latter equation leads to:

x(t, a) ≥     
x 0 (a -t)e -a a-t (µ(s)+y(t-a+s)γ(s))ds if a ≥ t, ψ(t -a)e -a 0 (µ(s)+y(t-a+s)γ(s))ds if a < t.

1. First we prove that ∃t such that ∀t ∈ [t , t + a 1 ], ψ(t) > 0 with a 1 defined by [START_REF] Cushing | A predator prey model with age structure[END_REF].

We know that ∀t ∈ [a -t 1 , a -t 2 ] : Either a -t 2 -(a -t 1 ) ≥ a 1 and this step is done, or we continue by defining ε ∈]0, a -t 2 -(a -t 1 )[. Then we prove that ∀t ∈ [2a -t 1 + ε, 2a -t 2 -ε] we have:

ψ(t) ≥ t2 t1 β(u + t)x 0 (u)e -( µ L ∞ +M γ L ∞ )t du, where M = max u∈[0,a-t2] y(u) < ∞ so ψ(t) ≥ σ 1 := η 1 η 2 e -( µ L ∞ +M γ L ∞ )(a-
ψ(t) ≥ σ 2 := η 1 σ 1 (t 2 -t 1 )e -µ L ∞ + M γ L ∞ (2a-t2-ε) > 0, where M = max u∈[0,2a-t2-ε] y(u) < ∞.
If a -t 2 -(a -t 1 ) + (a -a -ε) ≥ a 1 we stop, else we continue.

Since we get each time a bigger interval on which ψ is positive then we get what we wanted.

2. With this suitable t , we can even prove that ∀t ≥ t , ψ(t) > 0.

Indeed, since R -> 0, there exists ∈]0, a 1 [ such that a1 β(u)du > 0.

Then ∀¯ ∈ [0, ] we get:

ψ(t * + a 1 + ¯ ) ≥ t +a1 t +¯ ψ(u)β(t + ¯ + a 1 -u)e -(t +¯ +a1-u) µ L ∞ du > 0.
Doing this step by step, we finally prove that ∀t ≥ t , ψ(t) > 0.

3. Now we prove that lim t→∞ ψ(t) = ∞.

We have ψ(t + a 1 ) ≥ ψ(t)R -where ψ(t) = min u∈[t ,t +a1] ψ(u).

Moreover ∀h > 0 :

ψ(t + a 1 + h) ≥ R -min u∈[t +h,t +a1+h] ψ(u) ≥ ψ(t)R -.
Then ∀h > 0, ψ(t

* + 2a 1 + h) ≥ R -min u∈[t +a1+h,t +2a1+h] ψ(u) ≥ ψ(t)R 2 -.
Once again, by induction and since R -> 1 we prove that lim t→∞ ψ(t) = ∞. Then a lower bound of Problem (1) gives, for every t ≥ 0:

x(t, a) ≥      x 0 (a -t)e -( µ L ∞ +M γ L ∞ )t if a ≥ t, ψ(t -a)e -( µ L ∞ +M γ L ∞ )a if a < t.
Since lim t→∞ ψ(t) = ∞, then ∀M > 0, ∃t > 0 such that ∀t ≥ t : ψ(t) ≥ M .

Consequently, we have for every t ≥ t + a 2 and every a ∈ (a 1 , a 2 ):

x(t, a) ≥ M e -( µ L ∞ +M γ L ∞ a 2 ) =: M C.
From Problem (1) and Remark 1 we deduce that for every t ≥ t + a 2 :

dy dt (t) ≥ α a2 a1 γ(a)x(t, a)da -δ y(t) ≥ αγ -M C -δ y(t) =: Cy(t).
Taking M big enough, we get C > 0.

Finally, an integration of the latter equation gives, for every t ≥ t + a 2 :

y(t) ≥ y(t + a 2 )e -C(t * +a2) e Ct → t→∞ ∞
which is a contradiction with the fact that y is bounded. By definition of a 1 , we can find a 2 such that a 2 -a 1 < a 1 even if we reduce γ -.

Since lim t→∞ x(t, a) = ∞, ∀a ∈ [0, a 1 ] then ∃t > 0 such that ∀t > t and ∀a ∈ [a 1 -(a 2 -a 1 ), a 1 ] :

x(t, a) ≥ δ αγ - e -(a2-a1)( µ L ∞ +K γ L ∞ ) ,
so ∀a ∈ [a 1 , a 2 ] : x(t + (a 2 -a 1 ), a) ≥ δ/(αγ -) and then y (t + (a 2 -a 1 )) ≥ 0.

Finally, taking a time big enough to have both conclusions, we get a contradiction.

We have then lim t→∞ y(t) = ∞ and the proof is completed.

Corollary 1. Prey population and predator population are uniformly strongly persistent.

The following result states that persistence may hold in the case where R -< 1 but under the assumption R 0 > 1.

Theorem 3.9. Suppose that R 0 > 1 and R -< 1. Then the total population of prey and the total population of predator are uniformly weakly persistent.

Proof.

1. First, suppose that given a fixed ε > 0, there exists (x 0 , y 0 ) ∈ X P (that may depend on ε) such that

σ + (x 0 , y 0 ) < ε. Since we have R 0 > 1, then ∃ a > 0 such that C := a 0 β(z)e -z 0 µ(s)ds dz > 1.
Let us take C ∈]1, C[ and define:

M = 1 a γ L ∞ ln C C > 0.
Taking ε small enough and since σ + (x 0 , y 0 ) < ε, we get y(t) ≤ M , ∀t ≥ t big enough.

Moreover y is bounded by a positive constant M and, using the last proof, we get ψ(t) > 0, ∀t ≥ t .

Integrating the first equation of (1) regarding a over K j and supposing x regular enough, we get:

∂ ∂t Kj x(t, a)da + x(t, a j+1/2 ) -x(t, a j-1/2 ) = - Kj [µ(a) + y(t)γ(a)] x(t, a)da.
Then, a implicit Euler's scheme and the integrals estimates gives us:

x n+1 j+1 = x n j+1 + ∆t ∆a x n+1 j 1 + ∆t ∆a + ∆tµ j+1 + ∆ty n+1 γ j+1 , ∀j ≥ 0, ∀n ≥ 1.
Moreover, the boundary conditions gives us x n+1 0 = ∆a j≥1 β j x n j , ∀n ≥ 1. Then we have the following theorem which guarantees positivity of the numerical solution.

Theorem 4.1. If (x 0 , y 0 ) ∈ X + then ∀j ≥ 1, n ≥ 1, we have x n j ≥ 0 and y j ≥ 0.

Simulations

According to biological considerations, let us use the following functions:

1. µ(a) = µ 0 + µ 0 a/(1 + ah) with µ 0 > 0 and h ∈ R * + : the older is the prey, the easier she dies naturally, 2. β(a) = β 0 ae -ca with β 0 , c > 0: the middle-aged preys are the one that reproduce themselves the most, 3. γ(a) = γ 0 (1 -age 1-ga ), with γ 0 > 0, g > 0: the young and the old preys are more easily killed by the predators.

For our simulations we take the parameters: a max = 20, ∆a = 0.1, α = 0.7, δ = 0.1, µ 0 = 0.05, h = 1, c = 1, g = 0.25, γ 0 = 0.5 and we represent the trajectory of the solution with on the x-axis the quantity of predator and on the y-axis the total quantity x L 1 of prey.

If we take β 0 = 1, we have R 0 < 1, so for all (x 0 , y 0 ) ∈ X + the solution will converge to E 0 : Theorem 3.5 (see figure 1).

If we take β 0 = 4 or β 0 = 7, then the simulations make us suppose that the solution is bounded for any positive initial conditions (x 0 , y 0 ) ∈ X P . Moreover we have R 0 > 1, R -< 1 and (H2) is verified since β 0 (a) > 0 f.a.e. a ≥ 0. Consequently of Theorem 3.9, the total populations are uniformly weakly persistent and the solution will either converge to a periodic function if β 0 = 4 (see figure 2) or converge to E 2 if β 0 = 7 (see figure 3). are verified and R -> 1 so consequently to Theorem 3.8: the solutions are unbounded (see figure 4).

Final remarks

All the cases that we have studied theoretically and numerically are gathered in Table 1. One can note that, even when considering a linear functional response, a structure according to the age of the preys provides more complex dynamics of the predator-prey interactions than the Lotka Volterra ODE model.

In particular, the realistic case of extinction of the population may occur. Indeed, we proved that, depending on the age distribution of the fertility rate and of the mortality rate of the preys, the total population tends to disappear. This phenomenon happens when a prey will produce, in average, less than one direct offspring during its lifespan, translated by R 0 < 1. In the opposite case, when R 0 > 1, we proved that under the assumption that the initial prey population is young enough (i.e. assumption (P)) then the total population is uniformly weakly persistent.

The model shows another unexpected behavior: if the young preys "uncatchable" by predators have a high enough ability to reproduce, which is translated by (R -> 1), then both populations explode. This A further work will be dedicated to a deeper analysis of the equilibrium E 2 in order to determine under which conditions it is asymptotically stable, and to theoretically look for the existence of periodic trajectories.

R 0 < 1 R 0 > 1 and (P)

Convergence to E 0 R -< 1 R -> 1
Limit cycle or Convergence to E 2 Unbounded solutions The Lotka Volterra equations are used to describe the dynamics of biological systems in which two species interact. The classical Lotka Volterra ODE system is the following:

         X (t) = a 1 X(t) - a 2 X(t)Y (t), Y (t) = a 3 X(t)Y (t) - a 4 Y (t), X(0) = X 0 and Y (0) = Y 0 , (15)
with a 1 , a 2 , a 3 , a 4 some positive real parameters. For such a system, we have the following well-known result (see [9] or [START_REF] Murray | Mathematical Biology, Biomathematics[END_REF]):

Theorem 4.2. For any positive initial condition (X 0 , Y 0 ), Problem (15) has a unique positive solution that is time periodic.

A consequence of the latter theorem is that for a given positive initial condition, the total population never extincts even when considering an infinite time horizon.

A formal integration with respect to the age variable a explains how the age-structured PDE problem

(1) can be seen as a generalization of the Lotka Volterra equations, as stated in the following proposition: Remark 5. As it was explained in Section 2, D(A) ⊂ W 1,1 (R + ) × R, where D(A) is the domain of the differential operator in Problem (1). Then for any initial condition (x 0 , y 0 ) ∈ D(A), a → x(t, a) remains in W 1,1 (R + ) for every t ≥ 0. A consequence is that for every t ≥ 0, lim a→∞ x(t, a) = 0 and consequently ∞ 0 ∂ a x(t, a)da = -β 0 X(t), and the formal integration w.r.t. a is possible.

  y)β(ξ -y)e -ξ-y 0 [µ(s)+y * γ(s)]ds dy, Ŝ2 φ(ξ) = ∞ 0 φ(y)β(y + ξ)e -ξ+y y [µ(s)+y * γ(s)]ds dy.

T 0 ψ

 0 (u)e uµ0 du + κe -µ0t t T e uµ0 du ≤ e -µ0t x 0 L 1 (R+) + e -µ0t

  a 0 (µ(s)+λ+y * γ(s))ds da, b 2 = δ αΓ ∞ 0 β(a)e -a 0 [µ(s)+y * γ(s)]ds a 0 γ(u)e -λ(a-u) duda, b 3 = αy * ∞ 0 γ(a)e -a 0 (µ(s)+λ+γ(s)y * )ds da, b 4 = -λ -δy * Γ ∞ 0 γ(a)e -a 0 [µ(s)+γ(s)y * ]ds a 0 γ(u)e -λ(a-u) duda, and Γ = ∞ 0 γ(a)e -a 0 [µ(s)+y * γ(s)]ds da.

Theorem 3 . 7 .

 37 If γ(a) = γ 0 , ∀a ≥ 0, then λ = ±i √ y * γ 0 δ. Proof. By solving BX = C, we need to have det(B) = 0 to get a non zero solution X. So, if and only if we have b 1 b 4 = b 2 b 3 . But we see that: )e -a 0 (µ(s)+y * γ(s))ds da -∞ 0 β(a)e -a 0 (µ(s)+λ+y * γ(s))ds da . Then, with (8) we get: )e -a 0 (µ(s)+λ+y * γ(s))ds da = δγ 0 b 1 αΓλ .

  t2) > 0 by Hypotheses (P) and (H2).

4 .

 4 Since ∀a ∈ [0, a 1 ] and ∀t > a : x(t, a) = ψ(t -a)e -a 0 µ(s)ds , then ∀a ∈ [0, a 1 ], lim t→∞ x(t, a) = ∞. Consequently we have lim t→∞ x(t, •) L 1 (R+) = ∞.Now, let us suppose that ∃M > 0 such that ∀t ≥ 0, y(t) ≤ M .

Figure 1 :

 1 Figure 1: Convergence to E 0

Figure 2 :

 2 Figure 2: Convergence to a periodic function

Figure 3 :

 3 Figure 3: Convergence to E 2

Figure 4 :

 4 Figure 4: Unbounded solution

Proposition 4 . 1 .

 41 Suppose that parameters γ, β and µ are independent of the age, given by the following constants: γ(a) = γ 0 , β(a) = β 0 and µ(a) = µ 0 f.a.e. a ≥ 0. Then (X(t), Y (t)) := ( ∞ 0 x(t, a) da, y(t)) is the solution of system (15) with a 1 = β 0 -µ 0 , a 2 = γ 0 , a 3 = αγ 0 , a 4 = δ, X 0 = ∞ 0 x 0 (a)da and Y 0 = y 0 .

  5.Consequently there exists an increasing sequence (t n ) n∈N such that lim n→∞ t n = ∞ and such thatlim n→∞ y(t n ) = ∞. So ∀M > 0, ∃n ∈ N : ∀n ≥ n , y(t n ) ≥ M .Suppose that y(t) does not go to ∞ when t goes to ∞.Then ∃ ε > 0 : ∀n ∈ N, ∃ t > t n such that y(t) < ε. Obviously ∃n ∈ N such that t ∈ [t n , t n +1 ],meaning that y(t n ) ≥ M, y(t) < ε and y(t n +1 ) ≥ M . So, by taking M ≥ K := εe δ(a2-a1) , we can find by the fact that y (t) ≥ -δy(t) and by continuity of y, an interval [t , t +(a 2 -a 1 )] ⊂ [t n , t n +1 ] such that y(t) ≤ K and such that y (t +(a 2 -a 1 )) < 0.

Table 1 :

 1 Different behaviorsAppendix: A Brief Reminder ofLotka Volterra

Consequently, defining t = max{t , t}, we get ψ( t + a ) ≥ ψ(t)C where ψ(t) = C min u∈[ t, t+a ] ψ(u).

Using the last proof with respectively t, a , C instead of t , a 1 , R -: we get lim t→∞ ψ(t) = ∞ then lim t→∞ x(t, •) L 1 (R+) = ∞ and lim t→∞ y(t) = ∞ since y is bounded, which is a contradiction.

Thus we get the persistence result for the predator.

2. Now suppose that for ε > 0 fixed, there exists (x 0 , y 0 ) ∈ X P such that σ + ρ1 (x 0 , y 0 ) < ε. Taking ε small enough we get:

for all t ≥ t big enough, so lim t→∞ y(t) = 0 with the second equation of (1). We then get a contradiction by using the first point. Consequently the proof is completed.

Numerical Simulations

The following numerical simulations, that are performed using the finite volume method, aim at investigating other behavior of the dynamical system (1) under some biologically reasonable parameters.

Numerical Scheme

We define the intervals a ∈ [0, a max ] and t ∈ [0, T ] then we note ∆a and ∆t respectively the age and the time steps. We define a j+1/2 = j∆a and t n = n∆t, for j, n ∈ N then we note K j = (a j-1/2 , a j+1/2 ).

We denote x n j the approximation of the average of x(t n , a) over K j and we compute the initial states:

Then we set α, δ > 0 and once β, µ, γ are chosen, we compute the data:

We define T (γx n ) = ∆a j≥1 γ j x n j . An implicit Euler's Scheme for the second equation of (1) gives:

Short title: Predator-prey model with age structure Names and mailing address of authors:

Antoine PERASSO, antoine.perasso@univ-fcomte.fr

Quentin RICHARD, quentin.richard@univ-fcomte.fr