Simon

Wilson R Lourenc ¸o
email: wilson.lourenco@mnhn.fr

Andrea Rossi
email: andrea.rossi@arachnida.eu

The cave population of Chaerilus Simon, 1877 from Palawan, Philippines, and description of a new species (Scorpiones: Chaerilidae) La population cavernicole de Chaerilus

Keywords:

The study of an important collection of scorpions, belonging to the genus Chaerilus, recently collected from a cave in the Palawan Island, Philippines, allows the clarification of the identity of this population, often misidentified with Chaerilus chapmani Vachon & Lourenc ¸o, 1985 known from caves in the Gunong Mulu National Park in Sarawak (Borneo). Chaerilus agnellivanniorum sp. n. is described from the Puerto Princesa Underground River Cave in Palawan Island based on 14 specimens, males, females, and juveniles. The new species is totally distinct morphologically from Chaerilus chapmani, a true troglobitic species. Chaerilus agnellivanniorum sp. n. may also be a true troglobitic element, but with a less marked degree of regression for several characters. Some comments on the ecology of the new species and on regional biogeography of Borneo and Palawan islands are also proposed.

´tude d'une importante collection de scorpions appartenant au genre Chaerilus, re ´cemment collecte ´e dans une grotte de l'ı ˆle de Palawan, aux Philippines, permet de ´sormais de clarifier l'identite ´de cette population souvent incorrectement identifie ´e comme Chaerilus chapmani Vachon & Lourenc ¸o, 1985 connue des grottes du Gunong Mulu National Park au Sarawak (Borne ´o). Chaerilus agnellivanniorum sp. n. est de ´crite de la grotte de Puerto Princesa, dans l'ı ˆle de Palawan, a `partir de 14 spe ´cimens, ma ˆles, femelles et juve ´niles. Par sa morphologie, la nouvelle espe `ce est globalement distincte de Chaerilus chapmani, une ve ´ritable espe `ce troglobie. Chaerilus agnellivanniorum sp. n. pourrait

Introduction

The family Chaerilidae, with its type genus Chaerilus Simon, 1877, was the subject of numerous publications in recent years, but remains yet poorly-known among extant scorpions; for some historical details, see Lourenc ¸o and Lourenc ¸o and Pham [START_REF] Lourenc ¸o | The genus Chaerilus Simon, 1877 (Scorpiones, Chaerilidae) in Vietnam; description of a new species with comments on possible species-groups[END_REF][START_REF] Lourenc ¸o | More about the genus Chaerilus Simon, 1877 in Vietnam and Cambodia, with descriptions of two new species (Scorpiones: Chaerilidae)[END_REF][START_REF] Lourenc ¸o | The genus Chaerilus Simon, 1877 in Vietnam (Scorpiones; Chaerilidae): a possible case of a vicariant species[END_REF] Recently a second genus within the family, Chaerilourencous Rossi, 2018, was proposed for accommodating two peculiar anophthalmus species [START_REF] Rossi | Chaerilourencous gen. n., a new genus of blind scorpions from Indonesia (Scorpiones: Chaerilidae), Arachnida-Riv[END_REF]. Some recent attempts to full revise the taxonomy of the genus were far from being successful [START_REF] Tikader | The fauna of India, 3, scorpions (Scorpionida: Arachnida)[END_REF][START_REF] Kovar | Revision of family Chaerilidae (Scorpiones), with description of three new species[END_REF], and brought mainly supplementary confusion to the status of several species. These attempts of revision were followed by an increasing number of publications on the Chaerilidae with the description of numerous new species. Some of these publications were properly constructed and illustrated [START_REF] Lourenc ¸o | The genus Chaerilus Simon, 1877 (Scorpiones, Chaerilidae) in Vietnam; description of a new species with comments on possible species-groups[END_REF][START_REF] Lourenc ¸o | More about the genus Chaerilus Simon, 1877 in Vietnam and Cambodia, with descriptions of two new species (Scorpiones: Chaerilidae)[END_REF][START_REF] Lourenc ¸o | The genus Chaerilus Simon, 1877 in Vietnam (Scorpiones; Chaerilidae): a possible case of a vicariant species[END_REF][START_REF] Lourenc ¸o | Description of two new species of Chaerilus Simon, 1877 (Scorpiones, Chaerilidae) from Laos and Vietnam[END_REF][START_REF] Zhu | The chaerilid scorpions of China (Scorpiones: Chaerilidae)[END_REF][START_REF] Di | A new species of Chaerilus Simon, 1877 (Scorpiones, Chaerilidae) from China[END_REF][START_REF] Di | The genus Chaerilus Simon, 1877 (Scorpiones: Chaerilidae) in China, with a description of the female of C. tricostatus Pocock[END_REF], whereas others were unequally illustrated and worse, constructed without any historical introductions and/or methods [START_REF] Tikader | The fauna of India, 3, scorpions (Scorpionida: Arachnida)[END_REF][START_REF] Kovar | Five new species of Chaerilus Simon, 1877 from China, Indonesia, Malaysia, Philippines, Thailand, and Vietnam (Scorpiones: Chaerilidae)[END_REF][START_REF] Kovar ˇı ´k | Chaerilus horefeti sp. n. from Vietnam (Scorpiones, Chaerilidae)[END_REF][START_REF] Kovar ˇı ´k | Two new Chaerilus from Vietnam (Scorpiones, Chaerilidae), with observations on growth and maturation of Chaerilus granulatus sp[END_REF]. This absence of introductions and methods apparently indicates a deliberated decision of ignoring previous results.

Although some progress has been made in the study of the Chaerilus species with clarification of the status of some populations [START_REF] Lourenc ¸o | One more new species of Chaerilus Simon, 1877 (Scorpiones, Chaerilidae) from the Island of Halmahera, Indonesia[END_REF][START_REF] Lourenc ¸o | About the species of the genus Chaerilus Simon, 1877 (Scorpiones, Chaerilidae) described by Euge `ne Simon[END_REF], the precise identity of some old or recently described species remain yet unclear. The classification of the Chaerilus species in natural groups was attempted [START_REF] Lourenc ¸o | The genus Chaerilus Simon, 1877 (Scorpiones, Chaerilidae) in Vietnam; description of a new species with comments on possible species-groups[END_REF][START_REF] Lourenc ¸o | More about the genus Chaerilus Simon, 1877 in Vietnam and Cambodia, with descriptions of two new species (Scorpiones: Chaerilidae)[END_REF], but without a definitive conclusion. This is mainly because several species deposited in private collections could not be examined. Many private collections are in fact not accessible to Academic Experts. Consequently, only a global and detailed revision of this genus will bring answers to a number of existent gaps, but this is not the aim of this publication.

Among the polemic discussions that took place during the last two decades [START_REF] Lourenc ¸o | A propos de quelques de ´cisions taxonomiques concernant des scorpions des familles Buthidae, Chaerilidae et Scorpiopidae[END_REF], one concerned the precise status and range of distribution of Chaerilus chapmani Vachon & Lourenc ¸o, 1985, a species described from caves in Sarawak (Borneo). The original material of C. chapmani was collected by members of the Royal Geographical Society during two expeditions that took place in the caves of the Gunong Mulu National Park, Sarawak in 1977-1978 and 1980-1981 [17]. A number of scorpions were untrusted to Prof. Max Vachon, in France, by Dr Philip Chapman, but others were also sent to Dr Oscar Francke, in the USA. At least one of these specimens was deposited within the American Museum of Natural History, NY, USA. In the already cited revision done by Kovar ˇı ´k [START_REF] Kovar | Revision of family Chaerilidae (Scorpiones), with description of three new species[END_REF], this author refers to C. chapmani based only on specimens from Palawan Island collected in St. Paul National Park, in a cave (?) under rocks. The date of February 1997 is proposed, but no collector is indicated. Concerning precisely the prove-nance of these specimens, recently Rossi [START_REF] Rossi | On the status of several species of the genus Chaerilus Simon, 1877 in South-East Asia with the description of a new species from Bali, Indonesia (Scorpiones: Chaerilidae), Arachnida-Riv[END_REF] noted that, despite they are now kept in the inaccessible Kovar ˇı ´k's private collection in Prague, the specimens were originally part of the collections of the Museo di Storia Naturale dell'Universita `di Firenze, Sezione di Zoologia ''La Specola'', Italy (MZUF), and were collected by Dr Marco Borri and Dr Cecilia Volpi during an official field trip organized by the Museum in the Philippines, during the year 1991. The specimens were precisely collected on 2 February 1991 in the caves of St. Paul National Park, Puerto Princesa Underground River, in the point of Rockpile (Dr Borri, personal communication). Dr Borri has a diary with precise notes of all his field trips. Several years later, Kovar ˇı ´k borrowed the whole scorpions' collection of MZUF and kept for his private collection a number of specimens. Among them, there are clearly the Chaerilus specimens from Palawan, which obviously are not present anymore in the Museum's collection. According to the loan register of MZUF, Kovar ˇı ´k borrowed in the year 1997 several lots of scorpions and kept these for his private collection, among many, one specimen from India, one specimen from Somalia, and six specimens from the Philippines. Later Kovar ˇı ´k [START_REF] Kovar | Revision of family Chaerilidae (Scorpiones), with description of three new species[END_REF][START_REF] Kovar | The presence of Chaerilus chapmani Vachon & Lourenc ¸o, 1985 (Scorpiones: Chaerilidae) on the Palawan Island (Philippines)[END_REF] cited these six Chaerilus specimens, reporting the following data: ''Philippines, Palawan Island, St. Paul National Park, 2 4 juvs., II.1997, FKCP (the author's collection). All specimens were found under rocks in a cave.'' Two points need to be better explained. [START_REF] Lourenc ¸o | The genus Chaerilus Simon, 1877 (Scorpiones, Chaerilidae) in Vietnam; description of a new species with comments on possible species-groups[END_REF] Although MZUF sometimes can accept, after request, to donate specimens from the Museum's collection to private collections, this is possible only if the requested specimen/ specimens are not the unique samples of the species in the collections. In this case, as in additional ones to be explored, it is clear that Kovar ˇı ´k kept for himself, and his inaccessible private collection, all specimens of the rare Chaerilus from the Palawan caves. It is obvious that he did not explain the real situation, when he did the request to the curator who was not an expert in arachnology. (2) It is interesting, and also alarming, to note that Kovar ˇı ´k changed the data of the specimens. According to Dr Borri, we know that the specimens were collected by him and Dr C. Volpi on 2 February 1991, but Kovar ˇı ´k reported that they were collected in February 1997, approximately when he got them on loan. Besides he did not indicate the collectors. Thus a serious doubt can be addressed: are all data published by Kovar ˇı ´k real and trustable? The inevitable answer can only be negative. Moreover the responsibility of co-authors and editors involved in such publications could suggest a form of complicity.

In a note concerning possible biased taxonomic decisions, Lourenc ¸o [START_REF] Lourenc ¸o | A propos de quelques de ´cisions taxonomiques concernant des scorpions des familles Buthidae, Chaerilidae et Scorpiopidae[END_REF] refuted the possible distribution of a true troglobitic species in two islands largely separated e ´galement repre ´senter un ve ´ritable e ´le ´ment troglobie, mais avec un moindre de ´gre ´e de re ´gression pour plusieurs caracte `res. Des commentaires sont e ´galement ajoute ´s sur l'e ´cologie de la nouvelle espe `ce et sur la bioge ´ographie de Borne ´o et de l'ı ˆle de Palawan. by the sea. The presence of C. chapmani out of the Island of Borneo was once again rejected by Lourenc ¸o [START_REF] Lourenc ¸o | Description du ma ˆle de Chaerilus agilis Pocock, 1899 (Scorpiones, Chaerilidae)[END_REF] while discussing about the ecology of some cave species of Chaerilus, but based also on the fact that, in his revision, Kovar ˇı ´k [START_REF] Kovar | Revision of family Chaerilidae (Scorpiones), with description of three new species[END_REF] had not examined any of the specimens of C. chapmani belonging to the original series. These remarks led immediately to a reply [START_REF] Kovar | The presence of Chaerilus chapmani Vachon & Lourenc ¸o, 1985 (Scorpiones: Chaerilidae) on the Palawan Island (Philippines)[END_REF], in which this author insisted about the presence of C. chapmani in Palawan, basing his conclusions on two comparative photos of the female specimen of C. chapmani deposited in the American Museum of Natural History, NY, with a female deposited in his private collection. Although this 'comparative taxonomic analysis' sounded mostly irrelevant in face of solid ecological arguments associated with cave adaptation, a new re-analysis was not possible in the absence of available material from Palawan. The present study of a relevant number of freshly collected specimens in a cave in Palawan (Fig. 1) leads us to confirm the taxonomic differences between the two populations in Borneo and Palawan, and a new species is described here.

Methods

Illustrations and measurements were made with the aid of a Wild M5 stereomicroscope that has a drawing tube (camera lucida) and an ocular micrometer. Measurements follow Stahnke [START_REF] Stahnke | Scorpion nomenclature and mensuration[END_REF] and are given in millimeters. Trichobothrial notations follow Vachon [START_REF] Vachon | E ´tude des caracte `res utilise ´s pour classer les familles et les genres de Scorpions (Arachnides). 1. La trichobothriotaxie en arachnologie. Sigles trichobothriaux et types de trichobothriotaxie chez les Scorpions[END_REF], and morphological terminology mostly follows Hjelle [START_REF] Hjelle | Anatomy and morphology[END_REF]. This species was described based on 1 male holotype, 1 female allotype, 2 males and 1 female paratypes collected in caves located in the Gunong Mulu National Park in Sarawak (Island of Borneo). The species was properly diagnosed and described with full illustrations. Consequently, a redescription does not appear as necessary. For details about dates and collectors, refer to Vachon & Lourenc ¸o [START_REF] Vachon | Lourenc ¸o, Scorpions cavernicoles du Sarawak (Borne ´o). Chaerilus chapmani sp. n. (Chaerilidae), scorpion troglobie[END_REF] (Figs. 2 and3.

Chaerilus agnellivanniorum sp. n. (Figs. 4567) Diagnosis. Species of moderate to small size in relation to the other species of the genus, reaching 36.8 and 40.9 mm in total length, respectively for male and female. Carapace moderately narrowed toward the anterior edge in both sexes; anterior margin weakly emarginated; carinae and granulations moderate to strong, better marked on females; furrows moderately deep in both sexes. Metasomal carinae strong; better marked on females; ventral carinae reduced on segment I, moderately marked on segment II; latero-ventral and ventral carinae on segments IV-V composed of stronger granules that can be moderately spinoid. Dentate margins of fixed and movable fingers of pedipalp chela with 6-7 rows of granules in both sexes, but better defined in males. Pectinal teeth count 4-4 in males and 3-3 in females, without exceptions. Genital operculum plates have an oval to suboval shape. Trichobothriotaxy of type B, orthobothriotaxic.

Relationships: Chaerilus agnellivanniorum sp. n. shows several morphological similarities with Chaerilus chapmani described from Sarawak. Both species probably derived from a common ancestral proto-element. The new species can, however, be readily distinguished from C. chapmani by a number of features: (i) different morphometric values, in particular for pedipalp segments that are bulkier, (ii) carination and granulations on carapace, tergites, metasomal segments, and pedipalps clearly better marked, in particular on females (iii) smaller number of pectinal teeth; exclusively four on males and three on females, (iv) median eyes and two pairs of lateral eyes of normal size and never absent, (v) more conspicuous chetotaxy on pedipalps.

Patronym: the name honours Dr Paolo Agnelli (MZUF) and Dr Stefano Vanni (MZUF), who organised the scientific expedition in Palawan, The Philippines. ARPC = Andrea Rossi collection, Massa, Italy. Description based on holotype and paratypes (measurements in Table 1)

Coloration: Basically yellow to reddish-yellow. Carapace yellow on males and reddish-yellow on females.

Tergites with the same colour as the carapaces and some diffused confluent yellow pigmentation. Metasomal segments yellow, paler on males; carinae darker on females. Telson yellow with the aculeus reddish-yellow to reddish. Chelicerae yellow without any variegated spots; fingers darker with reddish teeth. Pedipalps reddish-yellow to dark reddish; chela fingers darker, almost blackish. Legs yellow to slightly reddish-yellow. Venter and sternites yellow with some infuscations; pectines pale yellow.

Morphology: Carapace moderately narrowed anteriorly in both sexes; anterior margin weakly emarginated; carinae and granulations moderate to strong, better marked on females; furrows moderately deep in both sexes. Two pairs of lateral eyes, and a pair of median eyes; all eyes of normal size without any reduction or regression; median eyes about twice the size of the lateral eyes; median eyes anterior to the centre of the carapace. Tergites with moderate to strong granulations better marked in the female; carinae weak. Sternum pentagonal, longer than wide; genital plates of the operculum oval to sub-oval in shape. Pectinal teeth count 4-4 in the male holotype, 3-3 in the female paratype (see diagnosis for variation). Sternites almost smooth, with some granulations in the female; spiracles small and round-shaped; carinae absent from VII. Metasomal segment I wider than long; segments II to V longer than wide. All the carinae strongly granular; ventral carinae reduced on I, moderately marked on II; segments IV and V with carinae composed of stronger moderately spinoid granules. Vesicle very elongated with a pear-like shape, smooth, with a short aculeus. Pedipalps: femur with five carinae; internal having spinoid granules. Patella with seven carinae; dorsal and dorso-external moderately granular; dorso and basal-internal with strong spinoid granules. Chela with eight carinae, moderately to strongly granular; internal carinae with strong spinoid granules. Tegument with few granulations. Finger shorter than manus, with 6-7 rows of granulations on the dentate margins. Chelicerae characteristic of the family Chaerilidae [START_REF] Vachon | De l'utilite ´, en syste ´matique, d'une nomenclature des dents des che ´lice `res chez les Scorpions[END_REF]. Trichobothriotaxy type B; orthobothriotaxic [START_REF] Vachon | E ´tude des caracte `res utilise ´s pour classer les familles et les genres de Scorpions (Arachnides). 1. La trichobothriotaxie en arachnologie. Sigles trichobothriaux et types de trichobothriotaxie chez les Scorpions[END_REF]; femur with 9 trichobothria, patella with 14, and chela with 14. Legs with pedal spurs moderately developed. Tarsi with two rows of spiniform setae.

Ecological remarks about Chaerilidae species living in caves

A few species of Chaerilidae are known from caves [START_REF] Lourenc ¸o | Re ´vision des connaissances sur les scorpions cavernicoles (troglobies) (Arachnida, Scorpiones)[END_REF] but only two species have their status of true troglobitic confirmed: Chaerilourencous sabinae and Chaerilus chapmani [START_REF] Vachon | Lourenc ¸o, Scorpions cavernicoles du Sarawak (Borne ´o). Chaerilus chapmani sp. n. (Chaerilidae), scorpion troglobie[END_REF][START_REF] Lourenc ¸o | Chaerilus sabinae, nouvelle espe `ce de scorpion anophthalme des grottes de Matampa en Inde (Scorpiones, Chaerilidae)[END_REF]. Other species are suspected of having a subterranean way of life, mainly because found inside caves, but more formal evidence may yet be necessary to demonstrate this aspect. These are the cases of Chaerilus cavernicola Pocock, 1893 found in the Ngalau cave in Sumatra and Chaerilus agilis Pocock, 1899 collected in the Selangor cave from Malaya. These species remain, however, extremely rare; only recently, the male of C. agilis was finally described [START_REF] Lourenc ¸o | Description du ma ˆle de Chaerilus agilis Pocock, 1899 (Scorpiones, Chaerilidae)[END_REF][START_REF] Pocock | Descriptions of some new species of scorpions[END_REF].

The condition of Chaerilus chapmani as a true troglobitic element is unequivocal. This species was found exclusively in the deepest sections of the caves in Gunong Mulu National Park and no individual of this species was found outside the caves. Besides both adult and juveniles specimens have been found inside the caves, suggesting therefore that the entire cycle takes place in parallel with a subterranean way of life [START_REF] Vachon | Lourenc ¸o, Scorpions cavernicoles du Sarawak (Borne ´o). Chaerilus chapmani sp. n. (Chaerilidae), scorpion troglobie[END_REF].

Concerning the new species described at present, both adult and juvenile forms were equally being found in very deep sections of the cave. This could suggest a permanent subterranean way of life, although many morphological traits do not show major regressions.

All specimens of Chaerilus agnellivanniorum sp. n. were collected deeply in the caves of 'Natuturingam' (the local Filipino name), better known as Puerto Princesa Underground River Cave (Fig. 1), more than 1 km from entry and more or less near the subterranean river (Fig. 8). In the map, the two entrances are indicated as 'daylight' and 'outflow'. The scorpions were found at the ground level, on rocks and on guano. They were present in both areas with much guano and in areas with small amounts of guano. The scorpions were never under rocks, but always on them. Besides, they were not present in vertical rock walls. Also a great amount of other arthropods were found and collected in the caves, such as Arachnida, Myriapoda, Insects, Crustacea.

It is interesting to underline that the specimens of Chaerilus agnellivanniorum sp. n. did not fluoresce under UV lamp while researchers were collecting them in the caves (Fig. 9). The experiment was repeated also with the preserved specimens. This confirms the most interesting discovery by Lourenc ¸o [START_REF] Lourenc ¸o | Fluorescence in scorpions under UV light; can chaerilids be a possible exception?[END_REF], which is supported even by an additional test on a recently described species, Chaerilus merendinoi Rossi, 2018, from Bali, Indonesia [START_REF] Rossi | On the status of several species of the genus Chaerilus Simon, 1877 in South-East Asia with the description of a new species from Bali, Indonesia (Scorpiones: Chaerilidae), Arachnida-Riv[END_REF].

Biogeographic considerations

Palawan Island is stretched between Mindoro Island in the northeast and Borneo Island in the southwest. Palawan is about 450 km long and 50 km wide. With an area of 12,188.6 km 2 , it is the fifth largest island of the country and the 64th in the world. For its peculiar position, between Filipino islands and Indonesian islands, it was, and still is, the subject of important studies in biogeography and zoogeography. Since it is in a particular geographic position, it may have served as a springboard to the oceanic Philippines for some organisms, while simultaneously representing a dead-end Sundaic peninsula for others [START_REF] Esselstyn | Integrating phylogenetic and taxonomic evidence illuminates complex biogeographic patterns along Huxley's modification of Wallace's Line[END_REF]. According to the most recent discoveries, using multiple approaches, it seems evident that Palawan and Borneo were not connected by a land bridge at 17,000 year BP. Voris and Reis & Garong [START_REF] Voris | Maps of Pleistocene sea levels in Southeast Asia: shorelines, river systems and time durations[END_REF][START_REF] Reis | Late Quaternary terrestrial vertebrates from Palawan Island, Philippines, Palaeogeogr[END_REF] noted that if Palawan was really connected to the Greater Sundas and mainland Southeast Asia via land-bridges during the last two major glacial periods (ca. 18,000 and 160,000 year BP), then it should be assessed through the fossil record because species indicative of main-land fauna, such as large carnivores, may have occurred on Palawan during the Middle and Late Pleistocene. In reality, there are neither fossil nor modern tiger (Panthera tigris), clouded leopard (Neofelis nebulosa), or sun bear (Helarctos malayanus) documented from Palawan, but there are from Borneo, Java, Sumatra, and Bali, thus it was suggested [START_REF] Reis | Late Quaternary terrestrial vertebrates from Palawan Island, Philippines, Palaeogeogr[END_REF] that these large carnivores may have not dispersed from the Sunda Shelf in the Pleistocene. Already Heaney [START_REF] Heaney | Zoogeographic evidence for Middle and Late Pleistocene landbridges to the Philippine Islands[END_REF] showed that Palawan and Borneo are now separated by a water depth of about 145 m and taking into account the fact that the sea level was reduced in the past by only 120 m, a land bridge really did not exist between these two islands. In conclusion, as already suggested by Lourenc ¸o [START_REF] Lourenc ¸o | A propos de quelques de ´cisions taxonomiques concernant des scorpions des familles Buthidae, Chaerilidae et Scorpiopidae[END_REF] for the new species here described, and also as outlined by Rossi [START_REF] Rossi | Chaerilourencous gen. n., a new genus of blind scorpions from Indonesia (Scorpiones: Chaerilidae), Arachnida-Riv[END_REF][START_REF] Rossi | On the status of several species of the genus Chaerilus Simon, 1877 in South-East Asia with the description of a new species from Bali, Indonesia (Scorpiones: Chaerilidae), Arachnida-Riv[END_REF] for other Chaerilids species, to deal with the taxonomy of scorpions from this area, which has a very complex geological past, it is fundamental to take in consideration all aspects of biogeography and not only those based on morphology (typology). The idea that a single species of small land arthropods such as Chaerilus celebensis can be present in largely separated islands and territories such as Luzon, Sulawesi and continental Thailand, never connected by landbridges, can hardly be accepted [START_REF] Rossi | Chaerilourencous gen. n., a new genus of blind scorpions from Indonesia (Scorpiones: Chaerilidae), Arachnida-Riv[END_REF]. Moreover, the presence of the same Chaerilus species in two different cave systems from two islands not connected in the late Pleistocene can only be rejected [START_REF] Lourenc ¸o | A propos de quelques de ´cisions taxonomiques concernant des scorpions des familles Buthidae, Chaerilidae et Scorpiopidae[END_REF].

Table 1

Measurements (in mm) of the male holotype and female allotype of Chaerilus chapmani (MNHN) and male holotype and female paratype of Chaerilus agnellivanniorum sp. n. (MZUF-MNHN).

Chaerilus chapmani Chaerilus agnellivanniorum

Total length (including telson)

  3. Taxonomic treatment Chaerilidae Pocock, 1893 Chaerilus Simon, 1877 Chaerilus chapmani Vachon & Lourenc ¸o 1985 (Figs. 2 and 3)

Fig. 1 .

 1 Fig. 1. Schematic map of Puerto Princesa Underground River Cave, also known as the caves of 'Natuturingam'.

Fig. 2 .

 2 Fig. 2. Chaerilus chapmani, male holotype. Habitus, dorsal and ventral aspects.

Fig. 3 .

 3 Fig. 3. Chaerilus chapmani, female paratype. Habitus, dorsal and ventral aspects.

Fig. 4 .

 4 Fig. 4. Chaerilus agnellivanniorum sp. n., male holotype. Habitus, dorsal and ventral aspects.

Fig. 5 .

 5 Fig. 5. Chaerilus agnellivanniorum sp. n., female paratype. Habitus, dorsal and ventral aspects.

Fig. 6 .

 6 Fig. 6. Chaerilus agnellivanniorum sp. n., male holotype. A. Metasomal segments IV-V and telson, lateral aspect. B. Chelicera, dorsal aspect. C. Cutting edge of movable finger with granulations.

Fig. 7 .

 7 Fig. 7. Chaerilus agnellivanniorum sp. n., female paratype. Trichobothrial pattern. A-B. Chela, dorso-external and ventral aspects. C-E. Patella, dorsal, external and ventral aspects. F. Femur, dorsal aspect.

Fig. 9 .

 9 Fig. 9. Chaerilus agnellivanniorum sp. n., male paratype, alive in natural habitat.

Fig. 8 .

 8 Fig. 8. Interior view of Puerto Princesa Underground River Cave, showing the typical habitat.

Acknowledgements

We are most grateful to Dr Paolo Agnelli (MZUF) and Dr Stefano Vanni (MZUF) for information on the 2017 field expedition to Palawan and documentation on the caves, to Dr Alessio Romeo for photos of habitat and alive specimens, to Dr Marco Borri (MZUF) for information on the scorpions collected in the 1991 expedition, to Prof. Jo De Waele (University of Bologna, Italy) for help in collecting scorpions. Field investigations were organized by ''La Venta Esplorazioni Geografiche'' of Treviso, Italy, in collaboration with the ''TagBalay Foundation'' of Puerto Princesa (Palawan) and the ''Puerto Princesa Underground River-Protected Areas Management Bureau (PPUR-PAMB)'', as part of a project promoted by the Philippines Government named ''Support for Sustainable Eco-Tourism in the Puerto Princesa Underground River Area''. Our thanks go also to Miss E ´lise-Anne Leguin, (MNHN), Paris, for assistance in the preparation of photos and plates.


L Éo Girardin
email: leo.girardin@math.u-psud.fr

Vincent Calvez
email: vincent.calvez@math.cnrs.fr

Florence D Ébarre
email: florence.debarre@normalesup.org

CATCH ME IF YOU CAN: A SPATIAL MODEL FOR A BRAKE-DRIVEN GENE DRIVE REVERSAL

Keywords: 2000 Mathematics Subject Classification. 35K57, 37N25, 92D10, 92D25 long-time behavior, gene drive, brake, predator-prey, strong Allee effect

HAL is

Introduction

With the development of CRISPR-Cas9 genetic engineering, population management using gene drives has become a realistic possibility. The technique consists in artificially biasing the inheritance of a trait of interest in a target population [START_REF] Esvelt | Emerging Technology: Concerning RNA-guided gene drives for the alteration of wild populations[END_REF]. Such biased inheritance is due to the presence of an artificial self-replicating element expressing a DNA-cutting enzyme (the Cas9 endonuclease), such that initially heterozygous individuals (i.e. carrying the drive construct on one chromosome and a wild-type sequence on the homologous chromosome) produce almost exclusively drive-carrying gametes instead of 50% of drive-carrying gametes, as expected under Mendelian segregation (see Figure 1.1). The proportion of drive-carrying gametes depends on the conversion efficiency, perfect conversion meaning that an initially heterozygous individual produces 100% of drive-carrying gametes. Thanks to its supra-Mendelian transmission to offspring, a drive can spread in a population even if it confers a significant fitness cost [START_REF] Deredec | The population genetics of using homing endonuclease genes in vector and pest management[END_REF][START_REF] Tanaka | Spatial gene drives and pushed genetic waves[END_REF][START_REF] Unckless | Modeling the manipulation of natural populations by the mutagenic chain reaction[END_REF]. Potential applications for human health and agriculture include the modification of mosquito populations to make them resistant to malaria or the eradication of agricultural pest species [START_REF]Gene Drives on the Horizon: Advancing Science, Navigating Uncertainty, and Aligning Research with Public Values[END_REF]. The drive wins over the wild-type allele due to gene conversion (such that individuals born as heterozygotes only produce drive-carrying gametes).

The brake wins over the drive due to gene conversion and better fitness. The wild-type allele wins over the brake because of greater fitness.

However, while very promising, the technique is not risk-free. A drive could have off-target effects, or spread in a non-target population. For instance, a drive can be introduced on an island to eradicate a local rat population, but the dispersal of drive-carrying individuals to the mainland or to another island would also threaten those populations [START_REF] Esvelt | Conservation demands safe gene drive[END_REF][START_REF] Rode | Population management using gene drive: molecular design, models of spread dynamics and assessment of ecological risks[END_REF]. The effects of population modification using a drive may also have unexpected consequences on other species, e.g., predators or competitors. More generally, it is important to be able to control the spread of a drive, and to stop it if necessary. To this end, a "brake" construct was proposed, that does not contain the cas9 gene (and is hence unable to convert a wild-type allele), but that is able to target the very cas9 sequence contained in a drive construct, and therefore to convert a drive allele into a brake allele in a (drive/brake) heterozygote [START_REF] Wu | Cas9-triggered chain ablation of cas9 as a gene drive brake[END_REF]. The construct has been shown to not only stop a drive, but also in some cases lead to the recovery of the original wild-type population [START_REF] Michael | Evaluating strategies for reversing CRISPR-Cas9 gene drives[END_REF].

Most models of gene drive consider well-mixed populations (except meta-population and partial differential equations (PDE) models, [START_REF] Beaghton | Gene drive through a landscape: Reaction-diffusion models of population suppression and elimination by a sex ratio distorter[END_REF][START_REF] Noble | Current CRISPR gene drive systems are likely to be highly invasive in wild populations[END_REF][START_REF] Tanaka | Spatial gene drives and pushed genetic waves[END_REF]). Here we consider the influence of space and limited dispersal, and ask whether a brake construct is able to stop the spatial spread of a drive. More precisely, we introduce a minimalist PDE model of two interacting sub-populations combining spatial diffusion and Mendelian population genetics. The two sub-populations correspond to the subset of the gene pool with the drive allele, and to the subset with the brake allele (the rest consisting of the wild-type allele). The case of a single sub-population of gene drive was addressed in [START_REF] Tanaka | Spatial gene drives and pushed genetic waves[END_REF]. It was shown that the spatial invasion of the gene drive allele was successful up to a fitness cost a 0.7 associated with the drive allele (see below for details). In the present work, we ask whether the brake, even if lately introduced (with a spatial delay with respect to the gene drive invasion), can catch up the invasion and stop the spatial propagation of the gene drive. Using phase plane analysis and techniques from the theory of reaction-diffusion equations, we show that the drive can be stopped under some conditions on the model parameters. More precisely, we prove that the gene drive frequency will be eventually reduced to zero everywhere, provided that the fitness cost of the drive a is above 0.5 (i.e., the fitness associated with the drive allele is 1 -a < 0.5) and that the fitness cost of the brake is small enough.

1.1. The model. We adopt the following set of notations:

• N ∈ N is the spatial dimension (typically N ∈ {1, 2, 3}); • n ∈ [0, +∞) is the total population density; • u ∈ [0, 1] and v ∈ [0, 1] (0 ≤ u + v ≤ 1)
are the respective frequencies in the population n of the gene drive allele D and the brake allele B (so that the frequency of the wild-type allele O is exactly 1 -u -v); • a ∈ (0, 1) and b ∈ (0, 1) are the respective selective disadvantage (i.e., decreased survival) of the homozygous individuals DD and BB compared to the wild-type, with the assumption a ≥ b, which is biologically relevant 1 ; • h ∈ [0, 1] is the dominance of the brake allele B on the wild-type allele O (in particular, B is dominant if h = 1, recessive if h = 0 and additive if h = 1/2).

We assume that:

(A1) the conversion efficiency of the drive (conversion OD → DD) and of the brake (conversion DB → BB) are perfect 2 ; (A2) gene conversion takes place early in development (e.g., an individual born as OD becomes DD and has the fitness of a DD individual) 3 ; (A3) individuals mate randomly (random union of gametes, according to a uniform law).

Then, following the biological literature [START_REF] Unckless | Modeling the manipulation of natural populations by the mutagenic chain reaction[END_REF][START_REF] Michael | Evaluating strategies for reversing CRISPR-Cas9 gene drives[END_REF], the spatially homogeneous, next-generation discrete system reads:

                   1 -u g+1 -v g+1 = (1 -u g -v g ) (1 -u g -v g + (1 -hb) v g ) 1 -au 2 g + bv 2 g + 2bu g v g -2 (1 -u g -v g ) (au g + hbv g ) , u g+1 = u g ((1 -a) u g + 2 (1 -a) (1 -u g -v g )) 1 -au 2 g + bv 2 g + 2bu g v g -2 (1 -u g -v g ) (au g + hbv g ) , v g+1 = v g ((1 -b) v g + 2 (1 -b) u g + (1 -hb) (1 -u g -v g )) 1 -au 2 g + bv 2 g + 2bu g v g -2 (1 -u g -v g ) (au g + hbv g ) .
The first line corresponds to the frequency dynamics of the wild-type allele, the second line to the drive, and the last line to the brake. For each equation, the numerator on the right-hand side corresponds to the amount of corresponding alleles produced, while the denominator corresponds to the total amount of alleles (i.e., the "mean fitness" in the population). It can be verified that this denominator is exactly such that the first equation is true (in other words, such that the sum of the three frequencies remains identically equal to 1 as time goes on). Since the first equation is redundant, we get rid of it hereafter. These equations can be understood as follows.

1 Depending on the construct, the brake could just convert a drive without affecting its effect on fitness (b close to a, b ≤ a), or at the other extreme the brake could carry a cargo gene restoring wild-type fitness (b close to 0). 2 This is merely for algebraic convenience and the general case will be discussed below in Subsection 4.2.

3 With late gene conversion (typically in the germline), an OD-born individual would have the fitness of an OD. In both cases though, only D gametes are produced by this individual.

• Wild-type alleles (first equation) are carried (i) by all of the gametes produced by OO homozygotes (initially in frequency (1 -u g -v g ) 2 , and with fitness 1), and (ii) by half of the gametes of OB heterozygotes (in frequency 2v g (1 -u g -v g ), and with fitness (1 -hb)). Note that since gene conversion is assumed to be perfect, no wild-type alleles are produced by initially OD individuals. • Drive alleles (second equation) are carried (i) by all of the gametes produced by DD homozygotes (initially in frequency u 2 g and who have a fitness (1 -a)), but also (ii) by all of the gametes produced by initially OD heterozygotes (initially in frequency 2u g (1-u g -v g )), who were immediately converted into DD homozygotes, and hence have fitness (1 -a). Since gene conversion is assumed to be perfect, no drive alleles are produced by initially DB individuals.

• Brake alleles (third line) are carried (i) by all of the BB homozygotes (initially in frequency v 2 g , and who have a fitness (1 -b)), (ii) by all of the gametes produced by initially DB heterozygotes (in frequency 2u g v g ), who were immediately converted into BB homozygotes, and hence have fitness (1 -b), and finally (iii) by half of the gametes produced by OB heterozygotes (in frequency 2v g (1 -u g -v g ), and with fitness (1 -hb)).

• Since only ratios of fitness appear in the equations, the assumption that the wildtype fitness is unitary is done without loss of generality.

Next, we take the spatial diffusion of the individuals of the population n into account and assume:

(A4) the time scale of the diffusion mechanism and the maturation time between two generations are of the same order; (A5) each sub-population diffuses at the same rate.

We are now in position to perform a classical first-order approximation [START_REF] Tanaka | Spatial gene drives and pushed genetic waves[END_REF] and obtain the following reaction-diffusion PDE system:

       ∂u ∂t -∆u -2∇(log n) • ∇u = u (1 -a) u + 2 (1 -a) (1 -u -v) 1 -(au 2 + bv 2 + 2buv) -2 (1 -u -v) (au + hbv) -1 , ∂v ∂t -∆v -2∇(log n) • ∇v = v (1 -b) v + 2 (1 -b) u + (1 -hb) (1 -u -v) 1 -(au 2 + bv 2 + 2buv) -2 (1 -u -v) (au + hbv) -1 ,
where in general u, v and n are functions of time t and space x. Note that in this system there is no equation on the total population density n, which is therefore entirely unknown at this point. The transport term 2∇(log n) is the signature of gene flow in populations which are not homogeneous in size. This is a consequence of the reformulation of the problem by means of frequencies rather than population densities. To make this connection clear, we point out that the diffusion operator ∂ ∂t -∆ for the population density un is related to the diffusion-transport operator ∂ ∂t -∆ -2∇(log n) for the frequency u:

∂ (un) ∂t -∆ (un) = ∂u ∂t n + ∂n ∂t u -(∆u) n -(∆n) u -2∇u • ∇n = n ∂u ∂t -∆u -2∇(log n) • ∇u +u ∂n ∂t -∆n .
We shall address two types of questions in our study. (i) How do the allelic frequencies evolve when all three genotypes asymptotically coexist? (ii) Under which assumptions is the wild-type restored spatially uniformly? For (i), we will mainly be interested in the propagation speeds of u and v. Since these speeds are strongly impacted by the a priori unknown transport term 2∇ (log n), we will simplify this part of the discussion by assuming that n is spatially homogeneous, so that the transport term vanishes. We are aware this is a very serious restriction but cannot properly conclude without it. On the contrary, for (ii), where we are interested in spatially uniform convergence, the transport term does not matter that much and we can easily handle it, under a mild assumption: (A6) ∇ (log n) is well-defined and uniformly Hölder-continuous with an exponent larger than 1 2 (for instance, uniformly Lipschitz-continuous). This mathematical assumption should not be understood as being too restrictive: it is satisfied if, for instance, n solves a generic population dynamics reaction-diffusion equation with initial condition bounded above and below by positive constants. Of course it is satisfied if n is spatially homogeneous, so that we will in any case assume that the above assumption (A6) is satisfied.

For ease of reading, we define the scalar parabolic operator

P = ∂ ∂t -∆ -2∇(log n) • ∇
as well as the reaction functions

w (u, v) = 1 -au 2 + bv 2 + 2buv -2 (1 -u -v) (au + hbv) , g (u, v) = g 1 (u, v) g 2 (u, v) = (1 -a) u + 2 (1 -a) (1 -u -v) (1 -b) v + 2 (1 -b) u + (1 -hb) (1 -u -v) , f (u, v) = f 1 (u, v) f 2 (u, v) = 1 w (u, v) g (u, v) - 1 1
, so that the system finally reads:

(1.1)

P u v = u v • f (u, v) .
Here, the fact that the 2×2 linear parabolic operator on the left-hand side is the same on both lines is of the utmost importance. Mathematically, it is a necessary and sufficient condition to apply several theorems of the standard parabolic theory, in particular a generalized maximum principle due to Weinberger [START_REF] Hans | Invariant sets for weakly coupled parabolic and elliptic systems[END_REF] that we will indeed use extensively (it is recalled in Appendix A). Biologically, it means that all individuals move in space similarly: the gene under consideration does not affect the motility of the individuals carrying it. Since u and v are frequencies, they should satisfy u+v ≤ 1 (with 1-(u + v) the frequency of the wild-type allele O). Therefore it is natural to define the triangle

T = (u, v) ∈ [0, 1] 2 | u + v ≤ 1 .
We will indeed verify later on that it is an invariant subset of the phase plane. 1.2.1. Coexistence. Our first result deals with the case of coexistence. For technical reasons that will be discussed below, the statement is restricted to spatially homogeneous solutions, i.e. we consider the solutions of the system of ODE with the righ-hand-side of (1.1) only.

In the case where the selective disadvantage of the drive compared to the wild-type is less than a half (a < 1/2), we establish that both sub-populations of drive and brake persist in the long term, as measured by their frequencies which are positive at arbitrary large times. The mathematical statement is as follows.

Theorem 1.1. Let (u, v) be a spatially homogeneous solution of (1.1) with n spatially homogeneous and initial condition in the interior of T.

If a < 1 2 , then coextinction of (u, v) cannot occur. More precisely, lim sup

t→+∞ (u (t) + v (t)) > 0.
In view of numerical experiments, we expect damped oscillations converging to a coexistence state or sustained oscillations approaching a periodic limit cycle or a heteroclinic cycle (see Subsection 2.3 below).

Concerning the spatially heterogeneous problem, describing the invasion of the drive and the brake in a territory occupied by the wild-type allele, we explain in Subsection 4.1 why we expect that the brake always catches up with the drive and that, afterwards, both populations persist in the wake of the joint invasion front. We give evidence that this claim is true if a ≤ 1/4 and the total population n is spatially homogeneous, based on Ducrot-Giletti-Matano [START_REF] Ducrot | Spreading speeds for multidimensional reactiondiffusion systems of the prey-predator type[END_REF]. We believe it remains true even if 1/4 < a < 1/2. Figure 1.3 is a numerical illustration of this claim. Indeed, we observe that the brake catches up with the drive, even if it starts with a space and time delay. However, the drive is "strong" enough to persist, resulting in a joint invasion front followed by a complicated spatio-temporal pattern.

1.2.2. Coextinction. Our second result deals with the case of coextinction. Here we are able to handle spatially heterogeneous solutions and the transport term 2∇ (log n).

We assume that the initial frequencies (u 0 , v 0 ) are distributed such that some individuals carrying the brake allele have been released somewhere, whereas the gene drive allele has not completed its invasion in the whole space yet (only individuals carrying the wild-type allele are present far away in space). Note that we do not need to assume that the brake is released in a particular region of space -say, the region of space already colonized by the gene drive. If the selective disadvantage of the drive is sufficient (a > 1/2) while that of the brake is not too large (b < b ), then the drive goes extinct everywhere, followed by the complete extinction of the brake as well. The threshold for the brake b (a, h) appears for technical reasons at several steps of our argument. Consequently, an explicit value is not straightforward to obtain, and might not be informative. As explained below, we believe that this is only a technical restriction, and that the result should remain true in the whole range b ∈ (0, a). The mathematical statement is as follows.

Theorem 1.2. Let (u, v) be the solution of (1.1) with initial condition (u 0 , v 0 ) ∈ C R N , T satisfying v 0 = 0 and lim

x →+∞ (u 0 , v 0 ) (x) = (0, 0) .

There exists b ∈ (0, a) depending on a and h such that, if a > 1 2 and b < b , then coextinction occurs:

lim t→+∞ sup x∈R N u (t, x) + sup x∈R N v (t, x) = 0.
It is important to note that the convergence to zero of both u and v is uniform in space, that is, we rule out the possibility of a persistent wave of u followed or replaced by a wave in v.

We performed numerical experiments to explore the possible behaviours when b varies in the range (0, a). We observed complete extinction for values of b up to a (Figure 1.4)4 . To complete the numerical investigation, we give some evidence in Subsection 4.1 supporting the conjecture that the coextinction of u and v occurs for all b ≤ a. D) are successive snapshots of the drive u (blue) and the brake v (red) from the time t = 80 at which the brake v is released on the left-hand-side of the domain. (E) is the superposition of many snapshots, in order to visualize the spatio-temporal dynamics. We observe a small, stable, composite wave followed by spatio-temporal oscillations. Therefore the system (1.1) is well-posed in T.

Proof.

The function u → w (u, 0) coincides with u → 1 -au 2 -2au (1 -u), that is u → 1 -a + a (1 -u) 2
, whose minimum and maximum in [0, 1] are respectively 1 -a > 0 and 1. Since the bound w ≤ 1 in T is obvious, this directly shows that the maximum of w in T is indeed 1. It only remains to confirm that the minimum is indeed 1 -a. We prove this claim in two steps: first, we prove that the minimum of w on ∂T is 1 -a; second, we prove that there is no critical point of w in the interior of T.

The

function v → w (0, v) coincides with v → 1 -bv 2 -2hbv (1 -v), that is v → 1 + (2h -1) bv 2 -2hbv. Its derivative with respect to b is 2b{v(h-1)+h(v -1)} ≤ 0. Therefore, its minimum in [0, 1] is attained at v = 1 and is 1 -b > 0. The function u → w (u, 1 -u) coincides with u → 1 -au 2 -b (1 -u) 2 -2bu (1 -u), that is u → 1 -b -(a -b) u 2 , whose minimum in [0, 1] is 1 -a > 0.
Therefore the minimum of w on ∂T is indeed 1 -a > 0. Next, straighforward algebra shows that any critical point (u , v

) ∈ R 2 satisfies a a -(1 -h) b a -(1 -h) b (2h -1) b u v = a hb .
The determinant of the 2 × 2 matrix above is -a (a -b) -(1 -h) 2 b 2 , which vanishes if and only if a = b and h = 1, but in such a case the system reduces to u + v = 1 so that there are no critical points in the interior of T. If a > b or h < 1, the 2 × 2 matrix is invertible and Cramer's rule yields

(u , v ) = 1 -a (a -b) -(1 -h) 2 b 2 a(2h -1)b -hb (a -(1 -h)b) ahb -(a -(1 -h)b) a = 1 a (a -b) + (1 -h) 2 b 2 b(1 -h)(a -hb) a(a -b) This point is in the interior of T if and only if b(1 -h)(a -hb) < (1 -h) 2 b 2 , that is if and only if a -hb < (1 -h)b, that is if and only if a < b.
Hence it is not in the interior of T.

Therefore the minimum of w in T is attained only on the boundary and is 1 -a > 0.

Proposition 2.2. Any solution of (1.1) with initial condition in C R N , T is valued in T at all times t ≥ 0.

Proof. To show that T is an invariant region of the phase space, we use Weinberger's maximum principle [START_REF] Hans | Invariant sets for weakly coupled parabolic and elliptic systems[END_REF]. Indeed, the triangle T is a convex invariant set satisfying the so-called slab condition. Therefore we only have to verify that the reaction term is inward-pointing on the boundary of T, namely:

u 0 • f (u, 0) • 0 1 ≥ 0 for all u ∈ [0, 1], 0 v • f (0, v) • 1 0 ≥ 0 for all v ∈ [0, 1], u 1 -u • f (u, 1 -u) • 1 1 ≤ 0 for all u ∈ [0, 1].
These three conditions are trivially verified (the left-hand side being always zero).

The propagation of the gene drive alone.

In what follows, we fix v = 0 and we briefly review some results described in [START_REF] Tanaka | Spatial gene drives and pushed genetic waves[END_REF] about the dynamics of the gene drive invasion. If v = 0, then (1.1) reduces to

Pu = uf 1 (u, 0) = u -au 2 + (3a -1) u -(2a -1) 1 -a + a (1 -u) 2 = au (1 -u) u -2a-1 a 1 -a + a (1 -u) 2 .
If a ≤ 1 2 , this is a monostable equation. Additionally, a bit of algebra shows that this is an equation of KPP type if and only if a ≤ 1 4 , the KPP property being here understood in the following weak sense: the maximal growth rate per capita corresponds to sparse populations, namely

f 1 (0, 0) = max u∈[0,1] f 1 (u, 0). If a > 1 2
, this is a bistable equation with stable steady states 0 and 1 and unstable intermediate steady state θ = 2a-1 a ∈ (0, 1). All known results on bistable equations, and in particular [START_REF] Paul | The approach of solutions of nonlinear diffusion equations to travelling front solutions[END_REF], can therefore be applied to this case.

In particular, the sign of the following quantity plays a crucial role:

1 0 au (1 -u) u -2a-1 a 1 -a + a (1 -u) 2 du = √ 1 -a a 3/2 arctan a 1 -a - 1 2 - 1 -a a
(the calculation of the integral is not detailed here). Since this is positive if a = 1 2 , negative if a = 1 and since

∂ ∂a a → arctan a 1 -a -1 - a 2 a 1 -a = - (2a -1) a 3/2 4a (1 -a) 3/2 < 0,
there exists a unique a 0 ∈ 1 2 , 1 such that the integral is positive if a ∈ 1 2 , a 0 , zero if a = a 0 and negative if a ∈ (a 0 , 1]. It satisfies the numerical approximation a 0 0.6965.

As a consequence, in the simplified case where P = ∂ t -∆ (i.e. the total population n is spatially homogeneous), solutions u that are initially compactly supported will always go extinct if a ∈ (a 0 , 1] and will spread and invade if a ∈ [0, a 0 ). If a ∈ 1 2 , a 0 , it is also necessary that the initial condition is favorable enough (i.e. larger than 2a-1 a in a wide region -see the role of the initial data in the emergence of a wave in the bistable case [START_REF] Barton | Spatial waves of advance with bistable dynamics: Cytoplasmic and genetic analogues of Allee effects[END_REF], and in particular the existence of bubble-like solutions that can prevent the propagation).

2.3.

Basic phase-plane analysis: spatially uniform stationary states. This subsection is devoted to the stability analysis of spatially homogeneous solutions. Interestingly, tedious computations of Jacobians are unnecessary.

Pv = -bv (1 -v) (h(1 -v) + v(1 -h)) 1 -bv 2 -2hbv(1 -v) .
The right-hand side has exactly the sign of -v(1-v), whence this is a "backward-monostable" equation, where 0 is stable and 1 is unstable. Again similarly, if u + v = 1, the equation satisfied by u reduces to

Pu = -u (1 -u) (1 -b + (a -b) u) 1 -b -(a -b)u 2
and this is also a "backward-monostable" equation. These observations together with the stable and unstable manifold theorems show that, regarding the diffusionless system,

• (0, 0) is a stable node if a > 1
2 and is a saddle if a < 1 2 ; • (1, 0) is a saddle; • (0, 1) is a saddle. Now we turn to the stability of 2a-1 a , 0 (which is relevant only if a > 1 2 so this is assumed below without loss of generality). As (u, v) → (0, 0),

2a-1 a + u v • f 2a -1 a + u, v ∼ 2a-1 a ∂ u f 1 2a-1 a , 0 2a-1 a ∂ v f 1 2a-1 a , 0 0 f 2 2a-1 a , 0 u v with ∂ u f 1 2a-1
a , 0 > 0 (see Subsection 2.2) and

f 2 2a -1 a , 0 = (1 -b + a -b) (2a -1) + (1 -hb) (1 -a) a 1 -a -a 1 -2a-1 a 2 > 0.
Hence ( 2a-1 a , 0) is an unstable node. All this implies that, for the trajectories we have in mind (i.e. contained in the interior of T), convergence to (1, 0), (0, 1) or ( 2a-1 a , 0) is impossible and convergence to (0, 0) is possible if and only if a ≥ 1 2 . Consequently, Theorem 1.1 is proved. 2.3.2. Interior stationary states. Any stationary state in the interior of T is a solution of the following algebraic system:

g 1 (u, v) = g 2 (u, v), g 1 (u, v) = w(u, v),
which can be solved explicitly (g 1 -g 2 is a first-order polynomial while w -g 1 is a secondorder polynomial). We do not perform this resolution here, as it is tedious and useless. We simply point out that:

• on one hand, the proof of Theorem 1.2 will imply directly the nonexistence of such a coexistence state when a > 1 2 and b < b ; • on the other hand, when a < 1 2 , the flow is rotating anticlockwise on the boundary of T, whence by classical phase-plane arguments (e.g. the Poincaré-Bendixson theorem) there exists such a coexistence state. Numerically, we observe that this stationary state is unique and is a stable or unstable spiral, depending on the parameters (see Figure 2.1).

Brake with small selective disadvantage: predator-prey regime.

In what follows, we prove that the expected predator-prey structure is indeed true close to b = 0. In order to do so, it is convenient to understand b as a parameter and to add a subscript b to every object that depends on it. In particular, we rewrite the term f b (u, v) as (2.1) Proof. First, we study the case of b = 0.

f b (u, v) = w 0 (u, v) w b (u, v) f 0 (u, v) + b w 0 (u, v) v (v + 2u + 2h (1 -u -v)) v (v + 2u + 2h (1 -u -v)) -(v + 2u + h (1 -u -v)) with w 0 (u, v) = 1 -au 2 -2au (1 -u -v) = 1 + au 2 -2au + 2auv, g 0 (u, v) = (1 -a) u + 2 (1 -a) (1 -u -v) v + 2u + (1 -u -v) = 2 (1 -a) -(1 -a) u -2 (1 -a) v 1 + u , f 0 (u, v) = 1 w 0 (u, v) g 0 (u, v) - 1 1 = 1 w 0 (u, v) -(2a -1) + (3a -1) u -au 2 -2v (1 -a + au) u (1 + 2a -au -2av) = 1 w 0 (u, v) a (1 -u) (u -θ) -2v (1 -a + au) u (1 + 2a -au -2av) .
By straightforward algebra and definition of θ,

∂ v f 0,1 (u, v) = 2 w 0 (u, v) 2 (-(1 -a + au) w 0 (u, v) -(a (1 -u) (u -θ) -2v (1 -a + au)) au) = 2 w 0 (u, v) 2 -2a (2a -1) u 2 -(1 -a) ≤ - 2
w 0 (u, v) 2 min z∈[0,1] 2a (2a -1) z 2 + (1 -a) .
On one hand, if a ≥ 1 2 , the minimum above is 1 -a. On the other hand, if a < 1 2 , the minimum above is -2a (1 -2a) + 1 -a = 4a 2 -3a + 1 > 0. In all cases, since w 0 ≤ 1 in T,

∂ v f 0,1 (u, v) ≤ - 2 w 0 (u, v) 2 min 1 -a, 4a 2 -3a + 1 ≤ -2 min 1 -a, 4a 2 -3a + 1 < 0.
Similarly,

∂ u f 0,2 (u, v) = w 0 (u, v) (1 + 2a (1 -u -v)) + 2au (1 + au + 2a (1 -u -v)) (1 -u -v) w 0 (u, v) 2 ≥ 1 -a w 0 (u, v) 2 ≥ 1 -a > 0.
Thus (1.1) in the limiting case b = 0 has indeed the claimed predator-prey structure. Finally, thanks to the smooth convergence of f b to f 0 as b → 0 and the preceding uniform estimates, the existence of b 1 > 0 as in the statement is immediate.

Proof of Theorem 1.2

In this section, we prove the main result of the paper. In the whole section, we assume a > 1 2 so the drive has a bistable dynamic and we denote θ = 2a-1 a its intermediate steady state.

Interestingly, when b ≤ b 1 , the system under consideration has the structure of a predatorprey system with Allee effect on the prey. Such systems have been studied in the literature but, apart from numerical simulations that show an incredibly wide and complicated variety of behaviors, almost nothing is known (e.g., [START_REF] Morozov | Bifurcations and chaos in a predator-prey system with the allee effect[END_REF][START_REF] Petrovskii | Allee effect makes possible patchy invasion in a predator-prey system[END_REF][START_REF] Wang | Dynamics and pattern formation in a diffusive predatorprey system with strong allee effect in prey[END_REF]). Fortunately, the analysis of our particular case is indeed possible, thanks to a very simple phase-plane structure (that can easily be observed numerically). More precisely, the very natural idea of the proof is the following: first, use phase-plane analysis and Weinberger's maximum principle [START_REF] Hans | Invariant sets for weakly coupled parabolic and elliptic systems[END_REF] to show that u becomes uniformly smaller than its extinction-persistence threshold θ in finite time, so that u goes extinct independently of the dynamics of v; next, use the predative structure and scalar comparison arguments to show that when its prey u goes extinct, the predator v goes extinct as well 5 .

5 Although this is indeed the main idea, it turns out that technical obstacles arise and therefore we will also show in the first step that v becomes uniformly smaller than a constant smaller than 1.

We will use as in Subsection 2.4 the subscripts b. In order to study more precisely the role of b in f b , we define additionally

r(u, v) = 1 v u v • v (v+2u + 2h (1 -u -v)) v (v+2u + 2h (1 -u -v)) -(v + 2u + h (1 -u -v)) = u (v+2u + 2h (1 -u -v)) v (v+2u + 2h (1 -u -v)) -(v + 2u + h (1 -u -v)) = u (v+2u + 2h (1 -u -v)) (1 -2h) v 2 + (3h -1) v -h + 2 (1 -h) uv -(2 -h) u = u (v + 2u + 2h (1 -u -v)) -(1 -v) ((1 -2h) v + h) -u (2 (1 -h) (1 -v) + h) .
Thus the reaction term of (1.1) reads

(3.1) u v • f b (u, v) = w 0 (u, v) w b (u, v) u v • f 0 (u, v) + bv w 0 (u, v) r (u, v) .
Remark. The function r satisfies, in T, r 1 ≥ 0 with equality if and only if u = 0 and r 2 ≤ 0 with equality if and only if (u, v) = (0, 1). This is mostly obvious but we point out that

(1 -2h)v + h = h(1 -v) + v(1 -h) ≥ 0 cannot vanish if v ∈ (0, 1).
3.1. Geometric lemmas on the phase-plane structure. The following lemmas basically state that, when b is small enough, the flow in the interior of T rotates anticlockwise around the repulsive node (θ, 0). Lemma 3.1 is concerned with the region {u > θ}; Lemma 3.2 is concerned with the line segment {u = θ}; Lemma 3.3 is concerned with the region u < θ, v ≥ 1+a 2a (θ -u) ; Lemma 3.4 is concerned with the instability of (θ, 0). Lemma 3.1.

There exists b 2 ∈ (0, 1] such that, if b ∈ 0, b 2 , then for all µ ≥ 0 and all u ∈ [θ, 1] such that µ (u -θ) ≤ 1 -u, (3.2) u µ (u -θ) • f b (u, µ (u -θ)) • µ -1 ≤ 0. Proof. Let C = {(µ, u) ∈ [0, +∞) × [θ, 1] | µ (u -θ) ≤ 1 -u} . In view of (3.1), (3.2) is trivial if µ = 0 and is otherwise equivalent to u u -θ • f 0 (u, µ (u -θ)) + b (u -θ) w 0 (u, µ (u -θ)) µr 1 (u, µ (u -θ)) r 2 (u, µ (u -θ)) • 1 -1 ≤ 0.
Straightforward algebra and the definition of θ lead to

u u -θ • f 0 (u, µ (u -θ)) • 1 -1 = - u (u -θ) w 0 (u, µ (u -θ)) (1 + a + 2aµ) whence (3.2) is actually equivalent to -u (1 + a + 2aµ) + b µr 1 (u, µ (u -θ)) r 2 (u, µ (u -θ)) • 1 -1 ≤ 0.
Clearly, this is true for all µ ≥ 0 and all u ∈ [θ, 1] if b = 0. Hence from now on we focus on the case b > 0. In such a case, requiring (3.2) for all (µ, u) ∈ C is equivalent to requiring

1 b ≥ sup (µ,u)∈C max 0, µr 1 (u, µ (u -θ)) r 2 (u, µ (u -θ)) • 1 -1 u (1 + a + 2aµ)
.

In view of the signs of r 1 and r 2 in T, the right-hand side is positive, so that the above inequality reduces to

1 b ≥ sup (µ,u)∈C µr 1 (u, µ (u -θ)) r 2 (u, µ (u -θ)) • 1 -1 u (1 + a + 2aµ) > 0.
It only remains to verify the finiteness of this supremum. Let (µ, u) ∈ C. From

µr 1 (u, µ (u -θ)) = (µ+2 -2h (µ + 1)) µu 2 + (2h + 2hµθ -µθ) µu and -r 2 (u, µ (u -θ)) = -(µ+2 -2h (µ + 1)) µu 2 -(µ (4hµθ + 3h + 2hθ -1 -2µθ-2θ) + h -2) u -µ 2 θ 2 + µθ -2hµ 2 θ 2 -3hµθ -h , it follows µr 1 (u, µ (u -θ)) r 2 (u, µ (u -θ)) • 1 -1 = (1 -2h)µ 2 θ -(h + 2hθ -1-2θ)µ + 2 -h u -(1 -2h)µ 2 θ 2 + (1 -3h)µθ -h ,
which reads, as a polynomial of µ,

µr 1 (u, µ (u -θ)) r 2 (u, µ (u -θ)) • 1 -1 = (1 -2h) θ (u -θ) µ 2 +((3h -1) θ -(2hθ + h -1-2θ) u) µ+(2 -h) u+h.
On one hand, if h > 1 2 , the above second-order polynomial of µ is bounded above by some constant, whence sup

(µ,u)∈C µr 1 (u, µ (u -θ)) r 2 (u, µ (u -θ)) • 1 -1 u (1 + a + 2aµ) < +∞.
On the other hand, if h ≤ 1 2 , then we use µ (u -θ) ≤ 1 -u and obtain

µr 1 (u, µ (u -θ)) r 2 (u, µ (u -θ)) • 1 -1 ≤ (1 -2h) θµ (1 -u) + ((3h -1) θ -(2hθ + h -1-2θ) u) µ + (2 -h) u + h ≤ ((1+θ) u -h (u -θ)) µ + (2 -h) u + h, whence sup (µ,u)∈C µr 1 (u, µ (u -θ)) r 2 (u, µ (u -θ)) • 1 -1 u (1 + a + 2aµ) < +∞ is again true. The proof is ended with b 2 = min      1,     sup (µ,u)∈C µr 1 (u, µ (u -θ)) r 2 (u, µ (u -θ)) • 1 -1 u (1 + a + 2aµ)     -1      .
Lemma 3.2 and Lemma 3.3 below are proved by similar (and simpler) considerations. For the sake of brevity, we do not detail the proofs. Note that Lemma 3.3 holds true for any admissible value of b.

Lemma 3.2. There exists b 3 ∈ (0, 1] such that, if b ∈ 0, b 3 , then for all v ∈ [0, 1 -θ], θ v • f b (θ, v) • 1 0 ≤ 0. Lemma 3.3. For all µ ≤ -1+a 2a and all u ∈ [0, θ] such that -µ (θ -u) ≤ 1 -u, u µ (u -θ) • f b (u, µ (u -θ)) • µ -1 ≥ 0.
Next, using the fact that (θ, 0) is an unstable node (see Subsection 2.3), we can prove the following similar lemma. Lemma 3.4. There exists η ∈ 0, 1-θ 4 such that, for all η ∈ (0, η) and all µ > 0:

(1) for all u ∈ [θ -η, θ + η], u min 1, µ 2 (u -θ + η) • f b u, min 1, µ 2 (u -θ + η) • min 1, µ 2 -1 ≤ 0. (2) the straight lines of equation v = µ (u -θ) and v = min 1, µ 2 (u -θ + η) intersect at u = θ + η if µ ≤ 2, θ + η µ-1 if µ > 2,
and the intersection point is in the interior of T. These lemmas will be used to construct a family of convex sets, illustrated in Figure 3.1.

Main proof.

We are now in position to prove the main theorem, Theorem 1.2, whose statement is recalled and precised below.

Theorem. Assume b ≤ min b 1 , b 2 , b 3 .
Let (u, v) be the solution of (1.1) with initial condition (u 0 , v 0 ) ∈ C R N , T satisfying v 0 = 0 and lim

x →+∞ (u 0 , v 0 ) (x) = (0, 0) .

Then coextinction occurs, namely

lim t→+∞ sup x∈R N u (t, x) + sup x∈R N v (t, x) = 0.
Proof. First, for all µ ≥ 0, we define the line segment

S µ = {(ũ, ṽ) ∈ T | ṽ = µ (ũ -θ)} .
For all t ≥ 0, we define the spatial image

I t = (u, v) (t, x) | x ∈ R N ⊂ T.
For all t ≥ 0 and µ ≥ 0, we define the (Euclidean in R 2 ) distance between I t and S µ d (t, µ) = dist (I t , S µ ) .

By standard parabolic estimates and the fact that (0, 0) is a solution of (1.1), the assumptions imply lim

x →∞ (u, v) (t, x) = (0, 0) for all t ≥ 0.

C -1 2 ( 1+a 2a + 1 θ ) = (ũ, ṽ) ∈ T | ṽ ≤ - 1 2 1 + a 2a + 1 θ (ũ -θ) , ṽ ≥ (ũ -θ + η) .
Here we use the inequality θ -1 > 1+a 2a which implies the inequality -1 2 1+a 2a + 1 θ < -1+a 2a . Therefore we have in finite time, say T ≥ T , both u < θ and

v < θ 2 1 + a 2a + 1 θ = 4a 2 -(1 -a) 4a 2 < 1.
Repeating the comparison with the scalar bistable equation

P ũ = ũf b,1 (ũ, 0) ,
with this time x → u T , x as initial condition, it follows that u → 0 uniformly in space asymptotically in time.

In view of the convergence of u, the function

ε : t → sup t ≥t max x∈R N u(t , x)
On one hand, if h > 0, the reaction term (1 -v) (F (v) -Cε (t 0 )) has in [z 0 , 1] a monostable structure: its only zeros are z 0 and 1 with negative derivative at 1 and positive derivative at z 0 . Again by standard comparison, this implies that the solution of Pz = (1 -z) (F (z) -Cε (t 0 )) z(t 0 , x) = v (t 0 , x) > z 0 for all x ∈ R converges uniformly in space to 1 as t → +∞. Finally, by the comparison principle, v converges uniformly to 1, or in other words v converges uniformly to 0.

On the other hand, if h = 0, then (1 -v) (F (v) -Cε (t 0 )) has in (z 0 , 1) one more zero, say z 1 . The steady states z 0 and 1 are unstable whereas z 1 is stable. By arguments similar to the previous case, we obtain

lim t→+∞ inf x∈R N v(t, x) ≥ z 1 .
But z 1 depends continuously on t 0 and converges to 1 as t 0 → +∞. Passing to the limit, we deduce lim

t→+∞ inf x∈R N v(t, x) ≥ 1,
whence v converges uniformly to 0 indeed.

Discussion

In this last section, we discuss the results and suggest new directions of research. More precisely, we discuss different parameter ranges (Subsection 4.1, Subsection 4.2), a different PDE model (Subsection 4.3) and biological implications (Subsection 4.4) 4.1. On different choices of selective disadvantages. In what follows, we give some evidence arguing in favor of coextinction when a > 1 2 (and b ≤ a), versus the existence of a joint front with complex spatio-temporal dynamics when a < 1 2 , see also Figure 1.5. 4.1.1. Monostable drive and brake with small selective disadvantage. In this subsection we assume a ≤ 1 2 so that the drive has a monostable dynamic and b ≤ b 1 so that we are in the predator-prey regime (see Proposition 2.3). We recall that, by Subsection 2.3, global convergence to (0, 0) for the diffusionless system cannot occur if a < 1 2 , as it is a saddle. As showed previously by Figure 2.1, the system is oscillating with either damped oscillations eventually converging to a coexistence state or sustained oscillations approaching a periodic limit cycle or the heteroclinic cycle on the boundary. What about the reaction-diffusion system? In particular, is the monostable drive fast enough to persist ahead of the oscillating front?

First, since v → f 2 (1, v) is v → 2(1 -b) -b(1 -h)v 1 -a + (2h -1)bv 2 + 2(a -b)v -1
and is therefore decreasing with respect to v, the nonlinearity vf 2 (1, v) has a KPP structure. Hence (1.1) is a particular case of the more general framework investigated by Ducrot-Giletti-Matano [START_REF] Ducrot | Spreading speeds for multidimensional reactiondiffusion systems of the prey-predator type[END_REF], provided that a ≤ 1 4 (the prey follows KPP-like dynamics), and the drift term due to a heterogeneous population size is neglected, i.e. P = ∂ t -∆. However, regarding the part of their analysis we are interested in here, we believe the KPP condition and the absence of drift could be relaxed. In any case, the following discussion can be rigorously justified when a ≤ 1 4 and n is spatially homogeneous following the arguments in [START_REF] Ducrot | Spreading speeds for multidimensional reactiondiffusion systems of the prey-predator type[END_REF]. This boils down to the comparison between the spreading speed c of u invading the wildtype type 0 alone (cf. Subsection 2.2) and the spreading speed of v invading the gene drive u ∼ 1 separately, namely 2 f 2 (1, 0) = 2 1+a-2b 1-a . In the case a ≤ 1 4 , the sole gene drive verifies a KPP condition ensuring that the propagation occurs at the explicit speed 2

f 1 (0, 0) = 2 √ 1 -2a. It is immediate to see that 2 √ 1 -2a < 2 1+a-2b
1-a , so that u is unable to evade, and will eventually be caught up by v. In the case a ∈ 1 4 , 1 2 , the spreading speed c can be estimated by comparison with a well-chosen KPP equation, namely with a KPP equation whose reaction term is larger than or equal to u → uf 1 (u, 0) in [0, 1]. It can be verified easily that the minimal KPP reaction term satisfying this condition is exactly defined in [0, 1] as

u → u max z∈[u,1] f 1 (z, 0). Denoting α = max u∈[0,1]
f 1 (u, 0), the KPP speed associated with this reaction term is exactly 2 √ α, so that the spreading speed c of u invading 0 satisfies c ≤ 2 √ α. To estimate α itself, we use the fact that, since u → f 1 (u, 0) is not decreasing, α is exactly such that the second-order polynomial equation in the variable u

α 1 -a + a (1 -u) 2 = a (1 -u) (u -θ)
admits a positive double root. After some algebra, we find that the corresponding discriminant vanishes if and only

if α = 1- √ a 2 √ a , whence the estimate c ≤ 2 √ α finally reads c ≤ 2 1 - √ a √ a .
Subsequently, we consider the quantity

2 1 + a -2b 1 -a -2 1 - √ a √ a .
It turns out that there exists a 1,b ∈ 0, 1 2 , increasing with respect to b ∈ [0, a] and satisfying a 1,a = 1 9 , such that this quantity is positive if and only if a > a 1,b . Consequently, if 1 2 ≥ a ≥ 1 4 > a 1,b , we expect that u is never able to evade v and a joint invasion front is observed.

Due to the diffusion and possible instabilities, we cannot really expect simple sustained or damped oscillations in the wake of the joint invasion front. Numerically, we observe very complicated spatio-temporal patterns (see Figure 4.1). 4.1.2. Brake and drive with the same large selective disadvantage. When b is close to a > 1 2 , the dynamics are more complicated.

In the limiting case a = b with h = 1, the following properties can be established (quite directly). Of course they persist in a neighborhood of a = b, h = 1.

(1) The dynamics in T are no longer of predator-prey type. The system remains predative outside of the neighborhood of (0, 0) delimited by the graph of but inside this neighborhood, the system is now cooperative. Even though (0, 0) might be globally attractive inside the cooperative region (and this should actually be easy to establish), general solutions of the reaction-diffusion system might never enter uniformly this region. In order to prove such a property, we need a Lyapunov function or a family of contractant sets, as in the proof of Theorem 1.2. We did not manage to perform such a construction. (2) If in addition a < 2 3 , then there is a (unique) spatially uniform interior stationary state at

u → 1 2 2 -u -(2 -u) 2 + 4 (u -θ) , 0
(u , v ) = a 4(1 -a) , 2 -3a 4(1 -a) .
In view of Figure 4.2, this steady state is, regarding the diffusionless system, an unstable spiral. This is one of the main obstacles encountered when trying to construct a Lyapunov function or a family a contractant sets.

Because of these obstacles, we did not manage to find an analytical proof of the coextinction. Nevertheless, in view of numerical experiments, it remains conjectured to hold true. . When h = 1, there is a coexistence state that is an unstable spiral. In both cases, (0, 0) seems to be globally attractive.

4.2. On imperfect conversion efficiencies. When the conversion efficiencies of the drive and the brake are no longer assumed to be perfect, the system reads instead

       Pu = u (1 -a) u + (1 -c B ) (1 -h BD b) v + (2c D (1 -a) + (1 -c D ) (1 -h DO a)) (1 -u -v) w c D ,c B (u, v) -1 , Pv = v (1 -b) v + (2c B (1 -b) + (1 -c B ) (1 -h BD b)) u + (1 -hb) (1 -u -v) w c D ,c B (u, v) -1 .
Here c D , c B , h DO and h BD are all in [0, 1] and are respectively the conversion efficiency of the drive, the conversion efficiency of the brake, the dominance of D on O and the dominance of B on D. The normalizing mean fitness w c D ,c B (u, v) is, similarly to the numerator, a mere algebraic modification of the perfect case.

It is therefore quite clear that the various uniform estimates on the reaction term we derived near the limit b ∼ 0 should remain true for values of (c D , c B ) close enough to (1, 1) and up to a slight modification of the threshold a = 1 2 . Hence both Theorem 1.1 and Theorem 1.2 are expected to remain true in this framework. We leave this extension for future work. 4.3. On a slightly different model. When writing down the model, if we follow Nagylaki [START_REF] Nagylaki | Conditions for the existence of clines[END_REF] instead of following Tanaka et al. [START_REF] Tanaka | Spatial gene drives and pushed genetic waves[END_REF], we obtain a different system: where w OO is the fitness of wild-type homozygous individuals (it can no longer be assumed to be unitary without loss of generality). In short, the reaction term for u and v is multiplied by w OO w (u, v) and an explicit equation on n appears.

            
It turns out that we can study the subsystem satisfied by (u, v) exactly as before (handling n as data) and obtain exactly the same results provided w OO is a positive constant. The only (very small) differences appear in the algebraic expression of three thresholds, namely:

(1) the KPP threshold for the sole drive becomes a = 1 3 ; (2) the positivity threshold for the bistable speed of the sole drive becomes a = 2 3 (exactly);

(3) the cooperativity neighborhood when a = b, h = 1 is now delimited by the straight line u + v = θ. Note that, in this model, our coextinction result implies that, asymptotically in time, the population density n has an exponential growth with rate w OO -1.

The more general case where the wild-type fitness w OO ≥ 0 is non-constant (for instance, when it is a nonnegative function of (u, v, n) that accounts for an Allee effect or a saturation effect) is left as an open problem. Nevertheless, let us point out that if the predator-prey structure at b = 0 is preserved (e.g., w OO depends only on n), then the coextinction result still holds true (namely, there are only wild-type individuals in the long run) and this might simplify a lot the asymptotic population dynamics. 4.4. On biological implications. Whether the fitness cost associated to the drive allele, a, is greater or lower than 1 2 , is critical for the outcome of the model. The a = 1 2 value is also critical in a well-mixed population in the absence of brake: when a < 1 2 , only the drive-only equilibrium (u = 1) is locally stable, but when a > 1 2 , the wild-type-only equilibrium (u = 0) becomes locally stable too: which equilibrium is eventually reached depends on the initial frequency of drive in the population (bistability). This frequency has to be higher than a threshold for the drive to fix. This bistability regime corresponds to what Tanaka et al. [START_REF] Tanaka | Spatial gene drives and pushed genetic waves[END_REF] call "socially responsible drives". [START_REF] Tanaka | Spatial gene drives and pushed genetic waves[END_REF] indeed showed that such drives could be stopped by barriers of finite width. Our results confirm that drives in the bistability parameter range (in our case, with a > 1 2 ) are more "socially responsible" than drives without threshold (a < 1 2 ): in the former case, a brake can stop the spread of a drive, while in the latter case, an indefinite co-invasion takes place, according to our model.

Should artificial gene drives be implemented in nature for the control of wild populations, these results indicate that it would be preferable to design them so that they fall in the bistability parameter range. Such drives would be more controllable than drives without introduction thresholds. Having such control on parameters will obviously be much more complicated empirically, than it is theoretically. In addition, in the bistability parameter range, the threshold for successful drive introduction can be very high, meaning that very high numbers of drive-carrying individuals would need to be reared.

Our macroscopic model ignores the effect of stochasticity by averaging all quantities. The spatial spread of the drive and the brake in individual-based models modeling smaller population sizes and possibly accounting for stochasticity remains to be investigated.

As our model uses a diffusion approximation, the location of the point of introduction of the brake does not affect our results. With non-diffusive models, for instance individualbased models, but also and more importantly in reality, this factor will probably affect the efficiency of the control of a gene drive by a brake.

Figure 1 . 1 .

 11 Figure 1.1. The presence of a brake induces rock-paper-scissors dynamics.The drive wins over the wild-type allele due to gene conversion (such that individuals born as heterozygotes only produce drive-carrying gametes). The brake wins over the drive due to gene conversion and better fitness. The wild-type allele wins over the brake because of greater fitness.

1. 2 . 2 ,Figure 1 . 2 .

 2212 Figure 1.2. Possible behaviors investigated in the forthcoming pages, depending on the values of (a, b). The parameter h has a fixed arbitrary value in this diagram. The threshold b depends on a and h and its graph as a function of a is unknown.

Figure 1 . 3 .

 13 Figure 1.3. Numerical solution of (1.1) in the case a = 0.45 < 1/2, b = 0.35 < a, h = 0.5, n spatially homogeneous (i.e. the drift term ∇ log n vanishes).(A-B-C-D) are successive snapshots of the drive u (blue) and the brake v (red) from the time t = 80 at which the brake v is released on the left-hand-side of the domain. (E) is the superposition of many snapshots, in order to visualize the spatio-temporal dynamics. We observe a small, stable, composite wave followed by spatio-temporal oscillations.

Figure 1 . 4 .

 14 Figure 1.4. Same as in Figure 1.3, but with a = 0.55 and b = 0.45. We observe invasion of the drive by the brake, up to extinction of the drive, followed by the retraction of the brake, up to complete extinction (meaning restauration of the wild-type).

8 Figure 1 . 5 . 1 . 3 .

 81513 Figure 1.5. Further numerical investigation for a larger disadvantage of the brake b > a > 1/2 (biologically unrealistic range). Uniform coextinction remains true for values of b larger than a (A), but no too large (B). Joint persistence of both species associated with complicated spatio-temporal behaviours were observed for intermediate values of b (a = 0.55, b = 0.75, result not shown).

2. 3 . 1 .

 31 Boundary stationary states. Similarly to Subsection 2.2, if u = 0, (1.1) reduces to

Proposition 2 . 3 .

 23 There exists b 1 ∈ (0, 1] such that, if b ∈ 0, b 1 , then (1.1) has a predatorprey structure where u is the prey and v is the predator, namely ∂ v f b,1 < 0 and ∂ u f b,2 > 0 in T. a = 0.4, b = 0.1, h = 0.2. a = 0.4, b = 0.1, h = 0.8.

Figure 2 . 1 .

 21 Figure 2.1.Oscillating trajectories of the homogeneous system with ten random initial conditions and a < 1 2 . Depending on the value of h, the oscillations are either damped (A) or sustained (B). For intermediate values of h (not illustrated here), we find a mixture of both, depending on the initial condition. In all cases, the solutions spend a very long time in the neighborhood of u = 0 (final time of the simulation T = 1000). We refer to Hastings et al.[START_REF] Hastings | Transient phenomena in ecology[END_REF] for a discussion about the relevance of long transient regimes in biology.

Figure 3 . 1 .

 31 Figure 3.1. Each colored polygon is the boundary of a set C µ (from right to left: µ = 0.5, 1, 2, +∞, -1 2

Figure 4 . 1 .

 41 Figure 4.1. Further numerical snapshots of the solution of (1.1) in the case a = 0.45, and b = 0.35, h = 0.5, and n is spatially homogeneous, see Figure 1.3.

  a = 0.6, b = 0.6, h = 1. a = 0.6, b = 0.6, h = 0.1.

Figure 4 . 2 .

 42 Figure 4.2. Trajectories of the diffusionless system with twenty random initial conditions and b = a > 1 2. When h = 1, there is a coexistence state that is an unstable spiral. In both cases, (0, 0) seems to be globally attractive.

  ∂u ∂t -∆u -2∇(log n) • ∇u = w OO u (g 1 (u, v) -w (u, v)) , ∂v ∂t -∆v -2∇(log n) • ∇v = w OO v (g 2 (u, v) -w (u, v)) , ∂n ∂t -∆n = (w OO w(u, v) -1) n,

We actually observed that the extinction threshold is larger than a but smaller than 1 (Figure1.5).Nevertheless, the case b > a is beyond the scope of our assumptions and does not correspond to an relevant case in the context of the biological problem.

Acknowledgments

The authors thank three anonymous referees for valuable comments which lead to an improvement of the manuscript.

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 639638). This work was supported by a public grant as part of the Investissement d'avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, and ANR-14-ACHN-0003-01.

Consequently, for all t ≥ 0 and µ ≥ 0, there exists x t,µ ∈ R N such that d(t, µ) = dist ((u, v) (t, x t,µ ) , S µ ) .

Now, let

and let us show that T < +∞. Without loss of generality, we restrict the analysis to the case T > 0.

For all t ∈ [0, T ], we can define

and, by continuity, we find

Next, by virtue of Weinberger's maximum principle [START_REF] Hans | Invariant sets for weakly coupled parabolic and elliptic systems[END_REF] applied in the convex invariant set T satisfying the so-called slab condition, for all t ∈ (0, T ], (u, v) (t, x t,µt ) ∈ int (T). This directly implies that for all t ∈ (0, T ], µ t > 0.

However, in view of Lemma 3.1 and Lemma 3.4, for any η ∈ (0, η) and any µ > 0, the convex set satisfying a slab condition

is again an invariant set for (1.1). Now we define T 0 = min 1, T 2 and fix η ∈ (0, η) so small that

Applying again Weinberger's maximum principle, we find that for all (t, t ) ∈ [T 0 , T ] 2 such that t > t > T 0 , (u, v) t , x t ,µ t ∈ int (C µt ). Hence the family (C µt ) t≥T0 is decreasing with respect to the inclusion, or in other words the family (µ t ) t≥T0 is increasing.

Assuming by contradiction that T = +∞, we can define

We extend naturally the definition of C µ and S µ to the case µ = ∞:

Then, by standard parabolic estimates, the family

converges locally uniformly up to extraction to an entire solution (u ∞ , v ∞ ) of (1.1) valued in C µ∞ and satisfying (u ∞ , v ∞ ) (0, 0) ∈ S µ∞ . Applying again Weinberger's maximum principle and using the fact that (θ, 0) ∈ C µ∞ , we find that necessarily µ ∞ = +∞. However, by the scalar comparison principle and the predator-prey structure (see Proposition 2.3), u ∞ is a subsolution for the bistable equation

Therefore T < +∞, that is (u, v) enters in finite time the domain C ∞ . By Lemma 3.2 and again by Weinberger's maximum principle applied this time to C ∞ , it actually enters in converges monotonically to 0. Thanks to the predator-prey structure, it follows just as before that, for all t 0 ≥ 0, v is in [t 0 , +∞) × R N a subsolution for the equation P ṽ = ṽf b,2 (ε (t 0 ) , ṽ) .

The Taylor expansion of f b,2 and the predator-prey structure bring forth the existence of a positive constant C > 0 such that, for large values of t 0 ,

Hence the function v = 1 -v satisfies

where

In all cases, F is increasing in 0, 1 2 and maps 0, 1 2 onto 0, F 1 2 . Assume that t 0 is so large that Cε (t 0 ) < F 1 2 and let z 0 ∈ 0, 1 2 be the root of F (z) = Cε(t 0 ) > 0. Assume further that t 0 is so large that t 0 ≥ T and

To conclude, we distinguish two cases, depending on whether h = 0 or not.

Appendix A. Weinberger's maximum principle

Below is recalled the main tool of the proof of Theorem 1.2. For clarity, we temporarily get rid of all our notations and adopt the original ones from [START_REF] Hans | Invariant sets for weakly coupled parabolic and elliptic systems[END_REF].

Theorem A.1 (Weak maximum principle). Let D be a C 1,ν domain in R n with ν ∈ (0, 1), S be a closed convex subset of R m , f (u, x, t) be Lipschitz-continuous in u ∈ S and uniformly Hölder-continuous in x ∈ D and t ∈ [0, T ], with the property that for any outward normal p at any boundary point u of S,

be uniformly parabolic with coefficients uniformly Hölder-continuous with Hölder exponent greater than 1 2 . If u is any solution in D × (0, T ] of the system

and if the values of u on D × {0} ∪ ∂D × [0, T ] are bounded and Hölder-continuous and lie in S, then u (x, t) ∈ S in D × (0, T ].

Theorem A.2 (Strong maximum principle). Let D be an arbitrary domain in R n and S be a closed convex subset of R m such that every boundary point of S satisfies a slab condition. Let f (u, x, t) be Lipschitz-continuous in u, and suppose that if p is any outward normal at a boundary point u , then

be locally uniformly parabolic, and let its coefficients be locally bounded.

If u is any solution in D × (0, T ] of the system