John R Dolan
email: john.dolan@imev-mer.fr

On Saville-Kent's "A Manual of the Infusoria"

Introduction

William Saville- Kent (1845Kent ( -1908) ) had a most remarkable and productive life. He was, successively, an amateur naturalist, an assistant at the British Museum, a curator in a number of public aquariums, the would-be founder of two marine stations, Superintendent of Fisheries in Tasmania in Australia, and finally Commissioner of Fisheries in Queensland in Australia (Harrison 1998). Given his diverse roles, it is perhaps not surprising that his works, published over a nearly 40-year period, covered a wide variety of topics. His first article was on a hydra (i.e., [START_REF] Kent | Polyzoon from Victoria Docks[END_REF] and his last publication was on color photography (Saville- [START_REF] Saville-Kent | The photography of colour as applied to obtaining correct colour records of natural history subjects[END_REF]. A true polymath, holding no diplomas, he authored four major scientific works on very distinct topics. His first book was "A Manual of the Infusoria: Including a Description of All Known Flagellate, Ciliate, and Tentaculiferous Protozoa, British and Foreign, and an Account of the Organization and the Affinities of the Sponges", from here on Manual of Infusoria, (Saville [START_REF] Kent | Infusoria as parasites[END_REF]Kent -1881)). His second book was "Handbook of the Marine and Freshwater Fishes of the British Islands (including an enumeration of every species)" (Saville [START_REF] Kent | Handbook of the Marine and Freshwater Fishes of the British Islands:(including an Enumeration of Every Species)[END_REF]. The third was "The Great Barrier Reef of Australia; Its Products and Potentialities" (Saville-Kent 1893), and the last was "The Naturalist in Australia" (Saville- [START_REF] Saville-Kent | The Naturalist in Australia[END_REF]. The work on the Great Barrier Reef is no doubt the one for which he is best known. A relatively complete account of the life and work of William Saville-Kent, including summaries of both his troubled adolescence, and his protistological work, can be found in Esteban et at. (2002).

Here the focus will be on the story of his major work of interest to protistologists, his Manual of Infusoria. Commonly cited as 1880-1881 as the title page indicates (Fig. 1), we will see that it was actually published over a three year period. It has been relatively neglected in histories of protistology. It is cited but not mentioned in the text of Cole's history of protozoology [START_REF] Cole | The History of Protozoology[END_REF]. [START_REF] Corliss | A salute to fifty-four great microscopists of the past : a pictorial footnote to the history of protozoology. Part I[END_REF] remarked that it was more or less eclipsed by Bütschli's monographs (Bütschli 1880(Bütschli -1882(Bütschli , 1883(Bütschli -1887)). In Leadbeater and McCready's account of the beginnings of protozoology in Britain [START_REF] Leadbeater | How it all began[END_REF], Saville-Kent is credited with contributing to the debates around the relationship of sponges and protists, but the value of the Manual of Infusoria at the time of its publication, or today, received little attention. As the complete title reflected, Saville-Kent attempted to describe all known species of 'infusoria'. Infusoria originally were the microscopic forms which appeared in 'infusions' of vegetable matter described by early microscopists such as Leeeuwenhoek and Joblot (Dolan 2022a). For Saville-Kent, the infusoria consisted of the largely heterotrophic taxa: flagellates, rhizopods, ciliates and 'tentaculiferous' (suctoriarelated). As we will shall see, he specifically excluded autotrophic taxa such as diatoms and desmids. When he embarked on the very ambitious project, he had little apparent experience, or expertise, with regard to microscopic organisms. It was a landmark work, the first book on protists in English with original content. In reality, there were few authoritative manuals in English at the time.

Before Saville-Kent's Manual of Infusoria appeared, the most widely used guide was Pritchard's venerable "A History of Infusoria" a constant companion of many early naturalists such as the American Jacob Bailey (Dolan 2022b). It was a replacement for Pritchard's earlier "The Natural History of Animalcules" (Pritchard 1834) that was based on O. F. Müller's "Animalcules Infusoria Fluviatilia et Marina" [START_REF] Müller | The Natural History of Animalcules: Containing Descriptions of All the Known Species of Infusoria; with Instructions for Procuring and Viewing Them, etc[END_REF]). Pritchard's 'Infusoria' book, in its first edition [START_REF] Pritchard | A History of Infusoria, Living and Fossil: Arranged According to "Die Infusionsthierchen[END_REF], was largely a translation of Ehrenberg's "Die Infusionsthierchen" [START_REF] Ehrenberg | Die infusionsthierchen als vollkommene organismen. Ein blick in das tiefere organische leben der natur[END_REF] with his illustrations copied and re-arranged. In the fourth and last edition [START_REF] Pritchard | A History of Infusoria, including the Desmidiace and Diatomace, British and foreign, 4th ed. Enlarged and Revised[END_REF], the text had been updated to include consideration of the work of Dujardin, Claparède and Lachmannn, etc., but the illustrations of flagellates and ciliates were still largely the same as those in the first 1841 edition, taken from Ehrenberg 1838. Thus, Pritchard's Infusoria it was quite outdated in Saville-Kent's time and there was a need for a "new" English language manual. The essay that follows consists of three major parts. The first section will cover the mysterious genesis of Saville-Kent's "Manual of Infusoria", the gradual appearance of the work, its reception at the time, and the contents will be briefly described. The second section will provide a consideration of the value of Saville-Kent's manual throughout the years since its publication to the present era in terms of citation data, and discuss the difficulties encountered as variants of Saville-Kent's name have been used. Finally, some misconceptions concerning Saville-Kent's name will be addressed, and suggestions as to how his works should be cited will be given.

The Mysterious Genesis and Gradual Appearance of "A Manual of the Infusoria"

In the first sentence of the preface to "Manual of Infusoria" (dated May 1882), Saville-Kent states that ten years had passed since he decided to concentrate his microscopical efforts on studies of the infusoria, but he gave no hints as to why he decided to focus on the 'infusoria'. It appears to be a curious action considering that at that point in time, based on his publication history, he had shown relatively little apparent interest in microscopic organisms. By late 1872, Saville-Kent had published quite extensively on various invertebrate taxa, but particularly on corals and sponges, and he could be credited with a certain expertise on both of the groups. He had published nine papers on sponges, seven papers on corals, and one paper on the relationship between sponges and corals (see the supplementary file for bibliographic details). In contrast, Saville-Kent's work on infusoria consisted of only two papers [START_REF] Wm | On some new infusoria from the Victoria Docks[END_REF][START_REF] Kent | Notes on Prof. James Clark's flagellate infusoria, with description of new species[END_REF]. Indeed, the first sentence of the second paragraph of the preface gives the impression that the project of a new manual on infusoria may have been suggested to him and it provided justification for restricting the manual to flagellate, ciliate, and suctoria-related taxa:

"It was in the first instance suggested that this Manual should be based upon the same lines as the, at the time, only other English treatise devoted to the subject, ' A History of the Infusoria/ by Andrew Pritchard, the fourth and last edition of which was published so long since as the year 1861; that it should include in a similar manner an account of the several distinct groups of microscopical organisms known as the Rotifera, Desmidiaceae, Diatomaceae, and other Protophytes which form, as being a reproduction of Ehrenberg's ' Infusionsthiere/ so conspicuous a feature of Mr. Pritchard's book. It soon became apparent, however, that to compass so comprehensive a task with any degree of efficiency would extend the size of this treatise far beyond convenient limits, and that indeed more than sufficient material for a work on the same scale as the one above-named had accumulated in connection with the Infusoria in the most limited and restricted sense as represented by the Flagellate, Ciliate, and Tentaculiferous Protozoa."

In 1872, Saville-Kent held a low salary position in the British Museum as an "assistant', and he had just married (Harrison 1998). If Saville-Kent sought to increase either his reputation or his income or both, authoring a book may well have been his idea. However, given that he had little evident expertise with regard to 'infusoria', the idea of producing a new 'infusoria manual', may have originated not with Saville-Kent, but rather a publisher. In 1872, the first public announcement that a new "Manual of Infusoria", was in preparation appeared in the October 31st issue of Nature; it was not by Saville-Kent but by the publisher John Van Voorst (Anon. 1872), and it showed that the taxa to be covered in the work had already been defined (Fig. 2). John Van Voorst was a very well regarded publisher of books and journals and long time Fellow of the Linnean Society [START_REF] Anon | John Van Voorst[END_REF]. Van Voorst did not publish the Manual of Infusoria. Exactly why is unknown. One might speculate that he dropped the project early on as it does not appear in any of the listings of 'works in progress' in catalogues included in the end papers of books published by Van Voorst from 1872-1874 that are available in the Biodiversity Heritage Library. The year following Van Voorst's announcement, 1873, was quite eventful for Saville-Kent. His changes in employment were chronicled in the pages of Nature. In January of 1873, he left his position at the British Museum to become curator at the Brighton Aquarium (anon. 1873a) but his tenure was short-lived as he resigned in July due to a dispute with directors of the aquarium (Anon 1873b) and he returned to London. By February of 1874, Saville-Kent was said to be moving to New York City to become Superintendent of a new aquarium to be built in Central Park (Anon. 1874a). One can speculate that, with the changes in jobs and locations, Saville-Kent made little progress on the Manual of the Infusoria in 1873, and Van Voorst may have consequently dropped the project.

Regardless of who was to be the publisher, in June of 1874 there appeared evidence that Saville-Kent was actively working on the manual project. A notice was published in the "Microscopy" section of Hardwicke's Science Gossip, shown in Fig. 3A (Anon. 1874b). It was a call for information concerning new forms and stated that his treatise was 'to be shortly published', which turned out be overly optimistic. The eventual publisher, David Bogue, first announced "A Manual of the Infusoria" by 'Mr. W. Saville Kent' in a notice of 'soon to appear' books in 1879 in the October 16th issue of Nature (a full seven years after Van Voorst's announcement!). In the notice, it stated that the work would likely be ready by March 1880, and no mention was made that it would actually be published in separate parts, or installments (Anon. 1879). In reality, the first part of the manual did not appear until late 1880. Advance copies of some of the plates destined to be part of the first installment were provided to Hardwicke's Science Gossip, and a short but glowing review, presumably by the editor John Taylor, appeared in the September 1880 issue (Anon. 1880a), and is shown in Fig. 3B. With regard to the enthusiasm of the review, and the fact that it was based on receiving advance copies of the plates, it should be noted that by 1880, Hardwicke's Science Gossip was published by David Bogue. The review included the first mention that work would be published in installments. Once again, the publication schedule given, parts to appear at monthly intervals, was overly optimistic. However, the price per part was accurate, ten shillings and six pennies. In today's currency each part, which included plates, would sell for about €75. Thus the price of the complete work, sold unbound in 6 parts, and would actually require 3 years to acquire, would run to about €450 in today's currency. From 1874, when Saville-Kent asked for information on new infusoria, to 1880 when the first parts of the manual appeared, he was not occupied exclusively with protists. He continued to publish extensively on a variety of topics. For example, he published three papers, and a letter, on observations of giant squids and two papers on sponges (see supplementary file). He published only two papers in which he described new species of protists (Saville [START_REF] Kent | A new field for the microscopist[END_REF][START_REF] Kent | Infusoria as parasites[END_REF].

The first installment of the Manual of Infusoria appeared in 1880. It consisted of the substantial general introductory material of 143 pages with first eight plates. In Britain, it was reviewed quite positively in Hardwicke's Science Gossip (Anon. 1880b) and in the Popular Science Review (Anon. 1881a), perhaps not surprisingly, as both serials were published by David Bogue. However, very positive reviews also appeared in the foreign press. The first installment was reviewed in The American Naturalist (Anon. 1880c), and in the Annales de la Sociéte Belge de Microscopie [START_REF] Deby | A Manual of the Infusoria[END_REF]. While the quality of the plates was praised, oddly, the reviews did not mention the fact that none of the plates were referred to in the text. The first eight plates included illustrations of many new species of choanoflagellates (Fig. 4). No mention was made of plate legends but presumably they were included with the plates. As the work was being sold by subscription, it seems likely that including plates with each 'installment', even if the plates had no relation to the portion of text published, was a means to keep interest in the work among subscribers. The second and third installments also included eight plates each. They were supposedly published in 1880 but may have actually been available only in early 1881 (Fig. 5). The fourth installment was not described, but was listed in the 'Books Received" in the July 1881 issue of Hardwicke's Science Gossip. It presumably consisted of about 150 pages of text, and plates 25 to 32, covering the last group of flagellates, the dinoflagellates, and the beginning of the sections on ciliates. It appears not to have been reviewed in the journals that carried reviews of the previously published parts.

Part five was reviewed in the November 1881 issue of Hardwicke's Science Gossip and described as 'just been published', and stating "The interesting genus Vorticella is treated at great length" (Anon. 1881e). The sixth and final part was reviewed in the August 1882 issue of the same journal. The review concluded with the rather odd sentence, given the considerable cost of the complete work, "It will be but a short time before this book has increased in money value; for, all things considered, it is by far the cheapest and most comprehensive work connected with Microscopy we have ever seen." [START_REF] Anon | Manual of the Infusoria-Part VI[END_REF]. One could speculate that the review was written by the publisher!

The complete work was reviewed by Edwin Ray Lankester, in a three page article published in Nature that appeared on April 26th 1983 [START_REF] Lankester | A Manual of the Infusoria[END_REF]. From the review, it appears that in 1882, Bogue published a complete edition, with the preface dated May 1882, consisting of two volumes of text and a volume of plates with Saville-Kent's famous frontispiece (Fig. 6). Lankester's review likely carried a great deal of weight as he was a very important figure in his time among biologists. He was the editor of the Quarterly Journal of Microscopical Science, and had a recognized expertise with regard to protists (e.g., Goodrich 1929). Lankester's review was overall quite positive. It pointed out, at some length, that the work was much more than an accurate review of work up to the time of publication; it incorporated new observations and descriptions of many new species, especially among the flagellates. Lankester however begged to differ with Saville-Kent with regard to the opinions expressed by the latter concerning the phylogenetic relationships of flagellates with sponges and ciliates with worms. However, Lankester also stated that opinions as to possible relationships with metazoan taxa did not detract from the obvious value of Saville-Kent's Manual of the Infusoria. (1917,1941,1942,1945,1958,1958,1962). The overall average number of citations per year is 5.7 ±4.65. To date (November 2023), it has been cited over 800 times. Many of the works citing Saville-Kent's manual are taxonomic works and reference is made to his species descriptions, both of species he described as new, as well as those previously described but whose presence he documented. For many species, occurrence details were noted such as in his text on Monosiga fusiformis (page 331): "... found congregated upon the carapace and ovisacs of a species of Cyclops obtained from one of the water-fowl ponds in the Zoological Gardens, Regent's Park, in May 1879." Many such details proved valuable.

The citation data are undeniable evidence of the enduring value of the manual.

Citing Saville-Kent's Publications

As stated above, Saville-Kent's works have been cited in a variety of manners. One could say it is largely his own fault as he used different forms of his name. In the years before 1870 and until mid 1870, he signed his articles "S.W. Kent" or "Wm. S. Kent". He starting using "W. Saville Kent" in August of 1870 (Saville Kent 1870). In 1885 he began using "W. Saville-Kent " (Saville-Kent 1885), but not consistently. It was only from 1887 onward, that he usually, but not always, signed his works W. Saville-Kent.

Further confusing exactly how his works should be cited is an odd contention that his 'real name' was W. S. Kent (he was born William Saville Kent) and that he started using Saville Kent, and later Saville-Kent, to disassociate himself from the enormous family scandal of the murder of his young half-brother in 1860 [START_REF] Ehrenberg | Die infusionsthierchen als vollkommene organismen. Ein blick in das tiefere organische leben der natur[END_REF][START_REF] Harrison | Oxford Dictionary of National Biography[END_REF]. Saville-Kent's sister, Constance, eventually confessed to the killing, years after the event. The spectacular murder, and Saville-Kent's possible involvement in the crime, has been explored in several books in the year since (i.e., [START_REF] Stapelton | The Great Crime of 1860[END_REF][START_REF] Rhode | The Case of Constance Kent[END_REF][START_REF] Bridges | The Tragedy at Road Hill House[END_REF][START_REF] Taylor | Cruelly Murdered, Constance Kent and the Killing at Road Hill House[END_REF][START_REF] Summerscale | The Suspicions of Mr[END_REF][START_REF] Kyle | A Greater Guilt: Constance Emilie Kent and the Road Murder[END_REF], and is still today a mystery. However, the murdered child, born Francis Savill Kent, was commonly called 'Francis Savill Kent' in press accounts at the time. So, using Saville Kent or Saville-Kent, as a family name could only call more attention to a possible connection with the family scandal than simply using "Kent". As to why he began using Saville Kent in mid 1870, one can only speculate that when he reached 25 years of age he sought to establish a new professional identity. He remains today an enigmatic character. His only known portrait (Fig. 8) was published in 1897 as the frontispiece in his book "The Naturalist in Australia" in which, curiously, infusoria or protozoa are completely absent. Today, it is commonly recognized that one has the right, but not the obligation, to use one's birth name. William Saville-Kent was clearly his ultimate choice as he was buried under the name (Fig. 9). However, when citing a published work, one needs to provide accurate information, including the authorship shown on the publication, to allow the cited work to be located by a reader. While one must respect the right of an author to be called as they desire, actual reference listings should use the authorship as listed on the work cited. The contention put forward here is that William Saville-Kent's works should be cited as they were signed. The early works should be cited as in texts as "Kent xxxx" and listed in the reference listed under "Kent, W.S. xxxx" or Kent, "Wm.S. xxxx" as appropriate. Works published after mid-1870 should cited in text as "Saville Kent xxxx" and the reference listed under "Saville Kent, W. xxxx". The works from 1885 onward should be cited in the text cited as "Saville-Kent xxxx", as appropriate, and in the references under "Saville-Kent, W. xxxx". A compromise of sorts has been attempted here as the man is always given as Saville-Kent but in citations in the text to a particular publication the last name as he used in the publication is used and in the references listings as well.

Fig. 1 .

 1 Fig. 1. The title page of Saville-Kent's Manual of Infusoria. The publication years given, '1880-1881' are inaccurate. The last part was actually published in 1882, and the preface is dated 'May 1882'. Hence, full references to the work should probably include the note 'published 1880-1882' (as is done here). Original image from the Biodiversity from the Biodiversity Heritage Library : https://www.biodiversitylibrary.org/item/40541#page/9/

Fig. 2 .

 2 Fig. 2. The announcement of books in progress by the publisher Van Voorst in October of 1872 (Anon. 1872) included a description of the contents of a new "Manual of the Infusoria" by Mr. W. Saville Kent.

Fig. 3 .

 3 Fig. 3. A public notice that Saville-Kent was soliciting material for his Manual of the Infusoria appeared in Hardwicke's Science Gossip in the June issue of 1874, (Anon. 1874). He sought information and specimens of new and interesting forms (3A). The first 'review' of Saville-Kent's manual (Anon. 1880a) appeared in Hardwicke's Science Gossip in the September issue of 1880, and was based on advance copies of some of the plates (3B).

Fig. 4 .Fig. 5 .

 45 Fig. 4. Plate 3 showing choanoflagellates was part of the first installment published. The actual size of the box enclosing the figures is 22 x 15.4 cm. According to the plate legend, most of species were shown were new: Codosigma cymosa (3-7), Codosigma candelabrrum (8-9), Salpingoeca minuta (10-12), Salpingoeca pyxidium (16), Salpingoeca ampulla (17-21), Lagenoeca cuspidata (25), Salpingcoea petiolata (26), Polyoeca dichotoma (27-28). Original image from the Biodiversity Heritage Library: https://www.biodiversitylibrary.org/item/40503#page/23

Fig. 6 .

 6 Fig. 6. The famous frontispiece of the volume of plates of Saville-Kent's Manual of Infusoria, the only color illustration. It shows a choanoflagellate fed carmine particles. The arrows indicate the direction of the feeding current resulting from the movement of the flagella. The actual size of the box enclosing the figures is 22 x 15.4 cm. The illustration is an enlarged, detailed version of fig. 19 in plate 3 of "A new field for the microscopist" (Saville Kent 1878). The image is a scan of the author's personal copy.

Fig. 7 .

 7 Fig. 7. The cumulative number of publications published, by year, containing the phrase "A Manual of the Infusoria" demonstrating usage of the manual since 1883 to November 2023.

Fig. 8 .

 8 Fig. 8. The only known portrait of William Saville-Kent appeared, with his signature, as the frontispiece of his book "A Naturalist in Australia" (Saville-Kent 1897).

Fig. 9 .

 9 Fig. 9. The gravestone of William Saville-Kent in the All Saints Churchyard at Milford-on-Sea. Image courtesy of Barry Leadbeater; the added inset is an enlargement showing more clearly that the family name is Saville-Kent.

Acknowledgments

David J. S. Montagnes parted with his copy of Saville-Kent's Manual of Infusoria for an improbably small sum, and my possession of his copy inspired this essay. Barry Leadbeater kindly provided the image of the gravestone of Saville-Kent as well as valuable encouragement and aid. and provides a temporal coverage of his works greater than those of Harrison (1998) or Esteban et al. (2002). Furthermore, the file is spreadsheet allowing sorting by: authorship form used, year, journal/publication type, and major subject.

Supplementary Material

To aid in the referencing of works by Saville-Kent, a supplementary file is included with this essay. It lists all the publications by Saville-Kent that I was able to inspect and verify the authorship form he used. The supplementary file is not a complete listing of Saville-Kent's works as some works listed in the bibliographies provided in Harrison (1998) and [START_REF] Ehrenberg | Die infusionsthierchen als vollkommene organismen. Ein blick in das tiefere organische leben der natur[END_REF] could not be located. However the file contains 88 references


Clément Dombry
email: clement.dombry@math.univ-poitiers.fr

Christian Mazza
email: isse.christian.mazza@unifr.ch

Vincent Bansaye
email: bansaye@polytechnique.edu

Phenotypic diversity and population

Keywords: branching process in random environment, phenotypic diversity, Lyapounov exponent, optimal strategy, exctinction, typical genealogy. AMS Subject classification. Primary: 60J80 Secondary: 60K37, 62D25

published or not. The documents may come

Introduction

Organisms adapt to fluctuating environments by regulating their intrinsic dynamics, and adjust their phenotypes or traits to the random environment. Observations reveal that most cell populations are heterogeneous, that is, are composed of various phenotypes. In [START_REF] Haccou | Optimal mixed strategies in stochastic environments[END_REF], [START_REF] Kussel | Phenotypic diversity, population growth and information in fluctuating environments[END_REF] and [START_REF] Thattai | Stochastic gene expression in fluctuating environments[END_REF] basic models are provided to explain this heterogeneity: Time is continuous, and the related stochastic processes can be seen as birth processes for the different phenotypes with migration processes between them in fluctuating environments. We first present the models considered in [START_REF] Kussel | Phenotypic diversity, population growth and information in fluctuating environments[END_REF], and then turn to our new setting. In [START_REF] Kussel | Phenotypic diversity, population growth and information in fluctuating environments[END_REF], the trait space T is finite, and the environment space E is such that E = T , the idea being that trait i has the fastest growth rate in environment e = i. The migration rate matrix H (k) = (H (k) ij ) i,j∈T gives the switching rates H (k) ij from phenotype j to i in environment e = k. The changing environment is modeled as an alternating renewal process. The migration rates H (k) ij defines possible strategies to overcome the uncertain future. [START_REF] Kussel | Phenotypic diversity, population growth and information in fluctuating environments[END_REF] considered two basic strategies: stochastic switching and responsive switching. In the first case, the organisms decide to switch the phenotypes independently of the running environment, they do not use sensors, so that the migration rates can be written as H (k) ij = H ij . Responsive switching assumes strong use of sensors, at a certain cost, and the extreme way of responding to environmental changes is obtained by assuming migration rates of the form H (k) kj = H m when j = k, and H (k) ij = 0 when i = k and j = i. In this last situation, organisms opt always for the most favourable phenotype in the running environment, that is for the trait having the largest growth rate. Denoting by X i (t) the mean number of organisms having phenotype i at time t, the average X(t) is solution of the random differential equation dX(t)/dt = A E(t) X(t), where A E(t) denotes the rate matrix composed of the various growth and migration rates in environment E(t) at time t. The authors then estimate the related Lyapounov exponent ln(N (t))/t as t → ∞, where N (t) = i∈E X i (t) is the average total population size. The computations are performed under the assumption that the environment changes slowly, so that, the process has the time to attain an equilibrium during each phase, see [START_REF] Kussel | Phenotypic diversity, population growth and information in fluctuating environments[END_REF] for more details. A similar model was also considered by [START_REF] Thattai | Stochastic gene expression in fluctuating environments[END_REF] for two phenotypes and two environments. The model was studied using Monte-Carlo simulations, which indicate that responsive switching is sometimes not the optimal strategy. The average proportion of cells having the fastest growth rate is maximized for strategies using a sensing mechanism allowing migration to the unfavourable state, leading to heterogeneous populations, providing in this way a rationale for population diversification. These models were then studied analytically in [START_REF] Gander | Stochastic gene expression in switching environments[END_REF]. Modifications of these models were also considered more recently in [START_REF] Visco | Switching and growth for microbial populations in catastrophic responsive environments[END_REF] where catastrophic events are modeled.

In the present work, we treat similar problems using discrete time multitype branching processes in random environments, denoted by MBPRE in what follows, which offer alternatives to the above birth with migration processes. The advantage of multitype branching processes is that one can separate more clearly the birth and migration phases. We can for example treat in this way populations composed of organisms where birth occurs in a very precise period. When this is not the case, discrete time modeling still provides relevant informations for population growth. We will also consider the problem of finding optimal strategies to maximize the random Lyapounov exponent. Our mathematical approach also permits to treat situations where the trait and environment spaces T and E are continuous: all of our results are valid in very general situations, like the models considered in [START_REF] Haccou | Optimal mixed strategies in stochastic environments[END_REF]. To be useful for scientists not familiar with too advanced mathematics, we first illustrate some results and provide examples when both T and E are finite.

Results

Assume that T = {t 1 , • • • , t q } and that E = {e 1 , • • • , e p }. The process of interest (Z n ) n≥0 is written as a random vector Z n = (Z t n ) t∈T , where Z t n denotes the number of organisms having trait t ∈ T at generation n. The transition between generations n and n + 1 is modeled as a two step procedure. First, for given t ∈ T , each of the i = 1, • • • , Z t n organisms having trait t gives birth to a random number of descendants given by a random variable ξ t,ωn i,n , where the index ω n ∈ E models the environmental state at generation n. In the second phase, each of the j = 1, • • • , ξ t,ωn i,n new individuals is assigned to a random trait τ t,ωn i,j,n ∈ T . As usual in such processes, we assume that all of these random variables are independent, with i.i.d. ξ t,ωn i,n and i.i.d. τ t,ωn i,j,n . We further assume that the environmental process (ω n ) n≥0 is fixed, and is supposed to be the realization of a stationary1 and ergodic process2 taking values in E. Let Υ t,e and π t,e be distributions of the ξ t,e i,n and τ t,e i,j,n for a trait t ∈ T in environment e ∈ E. We will often use the first moment m t,e of Υ t,e , giving the average number of descendants for an organism having trait t in environment e. Optimal strategies will be considered for fixed distributions Υ t,e , that is, we will maximize the population growth rate as a function of the distributions π t,e . Sections 3, 4 and 5 consider extinction and optimal growth questions. Section 6 deals with typical genealogies, that is, we consider the question of finding the typical lineage of an individual chosen randomly at generation n. We also illustrate these notions both for finite spaces T and E and for continuous ones.

As stated in the previous Section, the authors in [START_REF] Kussel | Phenotypic diversity, population growth and information in fluctuating environments[END_REF] distinguish between stochastic and responsive switching. In the first case, the distributions π t,e on T depends on t but not on e, so that we can write π t,e ≡ π t . Concerning the second family of strategies, π t,e depends on e but not on t. We will also distinguish between several natural situations, namely between strategies involving or not a sensing mechanism, and strategies π t,e depending or not depending on t, called hereditary and non-hereditary in what follows. We illustrate some results when both T and E are finite, and for 1. non-hereditary strategies with no sensing mechanism, where π t,e ≡ p, for some probability measure on T , 2. and for non-hereditary strategies using a sensing mechanism, of the form π t,e = p e , where p = (p e ) e∈E is a family of probability measures on T . This second family contains responsive switching strategies.

We will describe the optimal set for non-hereditary strategies, that is, we will characterize the set of distributions (π t,e ) yielding the fastest growth rate. We assume that the random environment (ω n ) n≥0 is a stationary process taking values in E.

1.1.1

The non-hereditary with no sensing mechanism case

We first prove in Proposition 1 that

lim n→∞ 1 n log E ω [|Z n |] = γ(p),
where the Lyapounov exponent γ(p) is given by

γ(p) = E[log m p,ω0 ],
and m p,e is the first moment or average value of the mean distribution Υ p,e = T Υ t,e p(dt).

Let P(T ) be the set of probability measures on T and γ * = sup{γ(p); p ∈ P(T )} be the top Lyapounov exponent. We denote by P * the subset of P(T ) containing the distributions p maximizing the Lyapounov exponent, i.e. γ * = γ(p). We show in Proposition 3 that p ∈ P * if and only if

E m t,e m p,e ν 1 (de) ≤ 1, ∀t ∈ T .
where ν 1 is the law of the stationary random environment. A strategy is called pure if it is concentrated on a single t ∈ T , that is, takes the form p = δ t . It then follows that a pure strategy p = δ t is optimal if and only if

E m t ′ ,e m t,e ν 1 (de) ≤ 1, ∀t ′ ∈ T . When T = {t 1 , • • • , t q } and E = {e 1 , • • • , e p },
with q ≥ p, we also prove that there is a unique maximizer in P * which is supported by a set containing at most p elements.

As a further illustration, we consider the simplest case when p = q = 2.

1. p * = δ t2 and γ * = ν 1 (e 1 ) log(m t2,e1 ) + ν 1 (e 2 ) log(m t2,e2 ) when

ν1(e1)mt 1 ,e 1 mt 2 ,e 1 + ν1(e2)mt 1 ,e 2 mt 2 ,e 2 ≤ 1, 2. p * = δ t1 and γ * = ν 1 (e 1 ) log(m t1,e1 ) + ν 1 (e 2 ) log(m t1,e2 ) when ν1(e1)mt 2 ,e 1 mt 1 ,e 1 + ν1(e2)mt 2 ,e 2 mt 1 ,e 2 ≤ 1.
3. Otherwise p * is given by

p * = ν 1 (e 1 )m t2,e2 m t2,e2 -m t1,e2 + ν 1 (e 2 )m t2,e1 m t2,e1 -m t1,e1 δ t1 + ν 1 (e 1 )m t1,e2 m t1,e2 -m t2,e2 + ν 1 (e 2 )m t1,e1 m t1,e1 -m t2,e1 δ t2 with γ * = log |m t1,e1 m t2,e2 -m t1,e2 m t2,e1 | -ν 1 (e 1 ) log |m t2,e2 -m t1,e2 | -ν 1 (e 2 ) log |m t2,e1 -m t1,e1 |
+ν 1 (e 1 ) log(ν 1 (e 1 )) + ν 1 (e 2 ) log(ν 1 (e 2 )).

As a numerical example, consider the case ν 1 (e 1 ) = ν 1 (e 2 ) = 1/2 and m t1,e1 = 1.5, m t2,e1 = 0.6, m t1,e2 = 0.6, m t2,e2 = 1.5.

The two environments are equiprobable, with type t 1 better fitted to environment e 1 , and a symmetry in the model. In this case, the Lyapounov exponent for the population without mutation is given by γ(δ t1 ) = γ(δ t2 ) = 0.5(log(1.5) + log(0.6)) ≈ -0.053.

This implies that both types are subcritical and that the corresponding homogeneous populations extinct almost surely. The optimal strategy is then p * = (0.5, 0.5) and the corresponding Lyapounov exponent is γ * = log(0.5 * (1.5 + 0.6)) ≈ 0.049.

With this optimal strategy, γ * > 0 and the population explodes with positive probability almost surely. This is one of the simplest example when polymorphism is a necessary condition for survival (see section 3.3).

1.1.2

The non-hereditary with sensing mechanism case Assume that π t,e = p e for some strategy p = (p e ) e∈E . It turns out that the relevant piece of environment is given by the pair process

ω (2) = ((ω n-1 , ω n )) n≥1 ∈ E 2 ,
m pe 1 ,e2 = T m t,e2 p e1 (dt).
When the following integral exists

γ(p) = E 2 log(m pe 1 ,e2 )ν 2 (de 1 , de 2 ),
we prove in Proposition 7 that

lim n→∞ n -1 log E ω [|Z n |] = γ(p) almost surely.
Let γ * * be the optimal growth rate, that is, the supremum of γ(p) among all possible families p, and denote by P * * the related set of optimal strategies. Let ν e1 (de 2 ) be the conditional distribution of ω 2 given that ω 1 = e 1 . We prove in Proposition 9 that p is optimal if and only if

E m t,e2 m pe 1 , e 2 ν e1 (de 2 ) ≤ 1, ∀t ∈ T , ν 1 (de 1 ) a.
s. An interesting consequence is that there is no gain to be expected using a sensing mechanism when the random environment has some independence property: If ω 1 and ω 2 are independent, then γ * = γ * * . The information one can gather on the present environmental state using sensors does not help when dealing with future events. Next, we develop further the simplest case when E = {e 1 , e 2 } and T = {t 1 , t 2 }. We suppose that the distribution for the environment ν is a Markov chain with transitions

ν(ω n+1 = e 1 |ω n = e 1 ) = 1 -q 1 ν(ω n+1 = e 2 |ω n = e 1 ) = q 1 ν(ω n+1 = e 1 |ω n = e 2 ) = q 2 ν(ω n+1 = e 2 |ω n = e 2 ) = 1 -q 2
where q i ∈ (0, 1) denotes the probability that the environment switches in the next step when it is currently in state e i . The sequence ω is then ergodic and stationary distribution

ν 1 = q 2 q 1 + q 2 δ e1 + q 1 q 1 + q 2 δ e2 .
We suppose furthermore that the Markov chain is started from the steady state, i.e. ω 0 has distribution ν 1 , so that the sequence ω is stationary. Then, the conditional distributions of ω 1 given ω 0 is precisely given by the transition of the Markov chain. The optimal strategy with sensing is then such that 1. p * * e1 = δ t2 when

(1-q1)mt

p * * e1 = (1 -q 1 )m t2,e2 m t2,e2 -m t1,e2 + q 1 m t2,e1 m t2,e1 -m t1,e1 δ t1 + (1 -q 1 )m t1,e2 m t1,e2 -m t2,e2 + q 1 m t1,e1 m t1,e1 -m t2,e1 δ t2 ,
with similar equations for p * * e2 . Rather than giving a general formula, we consider the numerical case when q 1 = q 2 = q so that the stationary distribution is given by ν 1 (e 1 ) = ν 1 (e 2 ) = 1/2 and we use the same values for the number of offspring as in Section 1.1.1, that is we set m t1,e1 = 1.5, m t2,e1 = 0.6, m t1,e2 = 0.6, m t2,e2 = 1.5.

Using the above results, we obtain that the optimal strategy p * * is given by: p * * e1 = δ t1 and p * * e2 = δ t2 when q ≤ 2 7

p * * e1 = δ t2 and p * * e2 = δ t1 when q ≥ 5 7

p * * e1 = 5 -7q 3 δ t1 + 7q -2 3 δ t2 and p * * e2 = 7q -2 3 δ t1 + 5 -7q 3 δ t2 if 2 7 ≤ q ≤ 5 7 .
We see three different environmental regimes, corresponding to low, intermediate and high frequency switching rates. When the environment fluctuates slowly, with q ≤ 2 7 , the optimal strategy is pure and corresponds to what [START_REF] Kussel | Phenotypic diversity, population growth and information in fluctuating environments[END_REF] called responsive switching. In the intermediate regime, the optimal strategy is a mixture of pure strategies. In the high frequency regime where environment fluctuates quickly (q ≥ 5/7), the optimal strategy is a pure one, where organisms being in the favorable state are pushed to the unfavorable state.

We deduce the optimal growth rate

γ * * =    log 3 2 -q log 5 2 if q ≤ 2 7 q log q + (1 -q) log(1 -q) + log 21 10 if 2 7 ≤ q ≤ 5 7 log 3 5 -q log 5 2 if q ≥ 5 7
.

When q = 1/2, the sequence ω is an independent sequence, so that γ * * (1/2) = γ * (1/2) ≈ 0.049, the optimal strategy is a strategy without sensing. When q ≈ 1 (resp. q ≈ 0), the environment in the next step is very likely to stay the same (resp. to switch), so that we can determine with high probability which type will be fitted in the next generation. We observe indeed that gives the size of the offspring of the i-th individual with trait t at time n in environment ω n , and τ t,ωn i,j,n gives the trait of the j-th descendant of this individual.

lim q→0 γ * * (q) = lim q→1 γ * * (q) = log 3 2 ≈ 0.176.
Let us give now the formal definition of the process.

Let (Ω, F , P) be a probability space and T be a metric space and E be a Polish space. For each pair (t, e) ∈ T × E, let Υ t,e and π t,e be distributions on N and T respectively and suppose that Υ t,e has a finite first moment denoted by m t,e .

Let ω = (ω n ) n≥0 be a E-valued stationary ergodic random process with distribution ν on E N . Denoting by N T the set of almost null N-valued functions on T and (1 t ) t∈T the canonical basis, the stochastic process Z n = (Z t n : t ∈ T ) with values in N T can be defined as:

       Z 0 = N 0 , Z n+1 = t∈T Z t n i=1 ξ t,ωn i,n j=1 1 τ t,ωn i,j,n , n ≥ 0, (1)
where

• N 0 is a N T -
valued random variable giving the population at time 0, respectively,

• {ξ i,n ; i ≥ 1, n ≥ 0} and {τ i,j,n ; i ≥ 1, j ≥ 1, n ≥ 0}
• N 0 , {ξ i,n ; i ≥ 1, n ≥ 0}, {τ i,j,n ; i ≥ 1, j ≥ 1, n ≥ 0} and ω = (ω n ) n≥0 are independent.
Thus Υ corresponds to the offspring distribution and π to the distribution of trait of the offspring. Both distributions a priori depend on the trait of the parent and on the environment. We put |Z n | = t∈T Z t n , n ≥ 0 (an empty sum is equal to zero).

Our attention will be focused on the role of the trait distribution π and how it affects the evolution of the population. The offspring distribution Υ will hence be considered as given and fixed, whereas the trait distribution π will be considered as a parameter. The intuitive idea is that i) the local fitness of an individual is determined by its trait and its environment and corresponds to its mean number of descendants, ii) the intergenerational variability of the traits has to be tuned so as to maximize the global fitness of the whole population.

The trait distribution π can be seen as the strategy used by the population to maximize its growth. We measure the performance of the strategy π by the long term growth of the corresponding population

(Z n ) n≥0 : let γ(π) = lim n→∞ n -1 log E ω,N0 [|Z n |] .
Here E ω,N0 denotes conditional expectation given the environment and the initial population. As will be precised in the sequel, fairly general assumptions ensures that this limit exists and does not depend on the environment and initial population. Note that it is important here to consider the quenched model (i.e. conditionally on the environment) and note the averaged one (i.e. averaged on all possible environments): the criteria γ(π) measures if the population behaves well in a typical environment. An averaged criteria would be biased by unlikely environments where the population grows unusually faster. Several mechanisms can induce the intergenerational variability of the traits and leads to different assumption on the trait distribution π.

offspring depends neither on the trait of the parent nor on the environment. We thus suppose that Z n evolves according to model (1) with π t,e ≡ p for some distribution p on T , and let π = π(p) be the corresponding product trait distribution.

Reduction to a simple BPRE

In some sense, the non-hereditary assumption makes the structure of the population trivial: the trait distribution is given by p whatever the past evolution of the process was. In mathematical terms, we take advantage of some stochastic independence: Lemma 1. For any n ≥ 1, the population structure is conditionally independent of the past population process given the size of the population, i.e.

Z n |Z n | (Z 0 , • • • , Z n-1 ) |Z n |.
Proof. It is easily seen from the assumptions on the model ( 1) and from the non-hereditary assump-

tion π = π(p) that the distribution of Z n given (Z 0 , • • • Z n-1
) and |Z n | is equal to the distribution of 1≤i≤|Zn| 1 τi with τ i an iid sequence with distribution p. This distribution does not depend on (Z 0 , • • • , Z n-1 ), this proves the conditional independence.

It is worth noting that the result also holds for the quenched model (i.e. conditionally on the environment ω). The Lemma implies that the distribution of the population process (Z n ) n≥1 is easily recovered from the size process (|Z n |) n≥1 . This latter process turns out to be a simple BPRE and this allows us to compute the performance γ(p) of the strategy p. Conditionaly to ω, the expected population size at time n is

E ω [|Z n |] = E ω0 [|Z 1 |] n-1 k=1 m p,ω k with m p,e the first moment of Υ p,e . If γ(p) = E [log m p,ω0 ] exists, then lim n→∞ n -1 log E ω [|Z n |] = γ(p) almost surely. Proof. According to Lemma 1, given (|Z 1 |, • • • , |Z n |)
, the population Z n has the same distribution as 1≤i≤|Zn| 1 τi with τ i an iid sequence with distribution p. Intuitively, the i-th individual has type τ i chosen randomly on T with distribution p. The size of the next generation |Z n+1 | is then 1≤i≤|Zn| ξ τi,ωn n,i where ξ τi,ωn n,i is the offspring of the i-th individual of type τ i in environment e n . From this two step procedure i) random choice of the type t, ii) reproduction with random offspring in environment e, we obtain the effective offspring distribution Υ p,e in environment e: it is the mixture of the offspring distributions Υ t,e , with mixing distribution p. This proves the branching property for (|Z n |) n≥1 . The other properties follow: the branching property implies the recursive formula

E ω [|Z n+1 |] = E (ω0,••• ,ωn-1) [|Z n |]m p,ωn
and we get the formula for E ω [|Z n |] follows. Taking the logarithm, we have

n -1 log E ω [|Z n |] = n -1 log E ω0 [|Z 1 |] + n -1 n-1 k=1 log m p,ω k
which converges to γ(p) almost surely according to the ergodic theorem and the integrability assumptions.

For the sake of simplicity, we suppose in what follows that the initial population consists in a single individual with random trait with distribution p. The whole process (|Z n |) n≥0 is then a simple BPRE with offspring distribution Υ p and initial condition |Z n | = 1. Simple branching processes in random environment have been introduced by Smith-Wilkinson [START_REF] Wilkinson | On branching processes in random environments[END_REF] and Athreya-Karlin [START_REF] Athreya | On branching processes with random environments, I: extinction probabilities[END_REF][START_REF] Athreya | On branching processes with random environments, II: limit theorems[END_REF] and have been studied rather intensively since then [START_REF] Tanny | Limit theorems for branching process in a random environment[END_REF][START_REF] Geiger | Limit theorems for subcritical branching processes in random environment[END_REF][START_REF] Afanasyev | Criticality for branching processes in random environment[END_REF]. We recall here some important results concerning the asymptotic behaviour of such processes (see e.g. the classification Theorem in [START_REF] Tanny | Limit theorems for branching process in a random environment[END_REF]): it states conditions under which either the population becomes extinct or explodes at geometric rate.

To avoid the trivial case of a constant population, we suppose that Υ p,ω0 is not almost surely equal to δ 1 . We suppose also that γ(p) = E [log m p,ω0 ] exists and is finite. We say that extinction occurs if Z n → 0 as n → ∞ (in which case the sequence vanishes eventually), and that the population survives otherwise. Let q(ω) = P ω [Z n → 0] be the probability of extinction given the environment.

Theorem 1. (classification theorem)

1. In both subcritical case γ(p) < 0 and critical case γ(p) = 0, the population becomes extinct almost surely, i.e. P [q(ω) = 1] = 1

2. In the supercritical case γ(p) > 0, if furthermore E [-log (1 -Υ p,ω0 ({0}))] < ∞, then the popula- tion can survive with positive probability in almost every environment, i.e. P [q(ω)

< 1] = 1.
Furthermore, conditionally on nonextinction, the population explodes at a geometric rate:

lim n→∞ n -1 log Z n = γ(p) almost surely on {∀n ≥ 0, Z n > 0}.

Optimal strategies

We now focus on optimal strategies: i.e. what choice of the distribution p allows for the fastest growth of the population ? As discussed in Section 2, the performance of the strategy π = π(p) is measured by the Lyapounov exponent γ(p). In the non-hereditary case with no sensing, an explicit formula for γ(p) has been derived in Proposition 1. The question naturally arise to determine the supremum of γ(p) for p varying in P(T ) the space of distribution on T and the set of optimal strategies p * in the case when this supremum is reached. Let γ * = sup{γ(p); p ∈ P(T )} be the optimal growth rate and P * = {p * ∈ P(T ); γ(p * ) = γ * } the set of optimal strategies. A strategy p is called pure if p = δ t for some t ∈ T (i.e. all individual in the population have the same trait t) or mixed otherwise. One ask further whether optimal strategies are pure or mixed. We now precise conditions under which existence of optimal strategies are ensured. We suppose that:

(C 1 )
there is some M > 0 such that m t,e ≤ M for all (t, e) ∈ T × E,

(C 2 )
for all e ∈ E, the application t → m t,e is continuous on T , (C 3 ) for all e ∈ E and ε > 0, there is a compact set K ⊂ T such that m t,e ≤ ε for all t ∈ T \ K Condition (C 1 ) is rather relevant from the biological point of view since an individual could hardly have arbitrary high number of offsprings in a fixed amount of time. Conditions (C 2 ) and (C 3 ) are related with the topology of T : individuals with close traits are supposed to have approximately the same behaviour, and traits close to infinity are supposed to have a poor fitness.

Proposition 2. Under conditions (C 1 ) -(C 3 ), optimal strategies exists and form a closed convex set i.e. the set P * is nonempty closed and convex in P(T ) endowed with the topology of weak convergence. If furthermore the family M = {e → m t,e ; t ∈ T } ⊂ L ∞ (E, ν 1 ) is linearly independent, then the optimal strategy is unique.

Proof. First, we prove that the map γ : P(T ) → R defined by

γ(p) = E log (m p,e ) ν 1 (de) with m p,e = T m t,e p(dt) (2)
is concave and upper semi-continuous on P(T ) with respect to the topology of weak convergence. Let This implies γ(λp

1 + (1 -λ)p 2 ) ≥ λγ(p 1 ) + (1 -λ)γ(p 2 ).
We then prove that P * is nonempty closed and convex. The closeness and convexity properties are straightforward since P * can be seen as the level set {p ∈ P(T ); γ(p) ≥ p * } of the concave uppersemicontinuous application γ. It remains to check non-emptiness and we use compactness arguments. In the case when T is a compact space, then P(T ) is also compact with respect to the weak topology and the upper semicontinuous map γ reaches its maximum on P(T ) so that P * is non-empty. In the case when T is non-compact, we consider its compactification T = T ∪ {∞}. We extend the definition of m by m ∞,e = 0 for all e ∈ E. Condition (C 3 ) ensures that this extension is continuous on T . Then the map γ extends on the compact space P( T ), is uppersemicontinuous and hence reaches its maximum at some point p * ∈ P( T ). It remains to check that p * ({∞}) = 0 so that p * can be seen as an element of P(T ). It is straightforward since p * ({∞}) > 0 would imply γ(p * ) = -∞.

At last, we prove the uniqueness in the case when the family M is linearly independent. Let p * 1 and p * 2 be two points where γ reaches its maximum. Using the strict concavity of the logarithm, we see that necessarily m p *

1 ,e = m p * 2 ,e ν 1 (de)-almost everywhere. Using the linear independence, this in turn implies

p * 1 = p * 2 .
The following characterization of optimal strategies can be useful: Proposition 3. A strategy p is optimal if and only if E m t,e m p,e ν 1 (de) ≤ 1, ∀t ∈ T .

Proof. The strategy p is optimal if and only if γ((1ε)p + εp ′ )γ(p) ≤ 0 for all p ′ and 0 < ε < 1.

Using concavity and differentiability, this is equivalent to

d dε [γ(p + ε(p ′ -p)] |ε=0 ≤ 0,
which can be rewritten as

d dε E log (m p,e + ε(m p ′ ,e -m p,e )) ν 1 (de) |ε=0 = E m p ′ ,e -m p,e m p,e ν 1 (de) ≤ 0.
Thus a necessary and sufficient condition is

E m p ′ ,e m p,e ν 1 (de) ≤ 1 , ∀p ∈ P(T ).
It is easily seen that it is equivalent to test the condition for p ′ = δ t , t ∈ T and this proves the announced result.

As a direct application of Proposition 3, we can answer the question wether there is a pure optimal strategy: Corollary 1. The pure strategy p = δ t is optimal if and only if

E m t ′ ,e m t,e ν 1 (de) ≤ 1 , ∀t ′ ∈ T .
This gives us a simple criterion for the existence of optimal pure strategies.

Extinction

According to Theorem 1, on the one hand the population becomes extinct in the case when γ(p) ≤ 0 and on the other hand (under additional technical conditions) the population survives and explodes with positive probability when γ(p) > 0. For the population's survival, strategies p with γ(p) > 0 are of highest importance. This leads to the definition of the set S of strategies that allow the population to survive:

S = {p ∈ P(T ); γ(p) > 0}.
The following property holds:

Proposition 4. If γ * ≤ 0, the set S is empty. If γ * > 0, S is a nonempty convex set containing P * .
Proof. From the definition of γ * , γ(p) ≤ γ * for all strategy p. Hence S is empty if γ * ≤ 0. If γ * > 0, then every optimal strategy p * ∈ P * satisfies γ(p * ) = γ * > 0 and hence P * ⊂ S. The map p → γ(p) is concave function so that the level set S = {p ∈ P(T ); γ(p) > 0} is convex.

In some cases, a striking phenomenon may happen: no pure strategy can allow for survival, i.e. every homogeneous population with a single trait t suffers from extinction; but some mixed strategies may prevent from extinction, i.e. some polymorphic populations may survive forever. In this case, we should say that polymorphism is a necessary condition for survival. This phenomenon occurs when γ(δ t ) ≤ 0 for all t ∈ T whereas γ * > 0.

The intuitive idea is the following: extinction occurs when the environment is bad for almost all the individuals in the population and hence a diversification of the traits in the population should imply a smaller number of environments that are bad for almost all individuals. This can be seen as a consequence of the concavity property: suppose that for all trait t, γ(δ t ) ≡ γ, i.e. homogeneous population have the same growth rate no matter the trait t. Then for any strategy p, γ(p) ≥ T γ(t)p(dt) = γ, i.e. any polymorphic population has a better growth rate than any homogeneous population. See the examples below for further illustration of this phenomenon.

Example: finite-dimensional case

We consider the case when the vectorial space spanned by the family M = {e → m t,e , t ∈ T } in L ∞ (E, ν 1 ) has finite dimension denoted by d. This occurs in particular as soon as: -either the environment space is finite: E = {e 1 , • • • , e p } and ν 1 is a discrete measure such that ν 1 (e i ) > 0 for all i ∈ {1, • • • , p}. In this case L ∞ (E, ν 1 ) is of dimension p and d ≤ p; -or the trait is finite: T = {t 1 , • • • , t q }. Then d ≤ q and conditions (C 1 ) -(C 3 ) are automatically satisfied. For convenience, we require furthermore that the following conditions holds:

(H) for any pairwise distinct t 1 , • • • , t d ∈ T , the family of functions {e → m ti,e ; 1 ≤ i ≤ d} is linearly independent in L ∞ (E, ν 1 ).
This property will be convenient because it ensures the uniqueness of the optimal strategy. More precisely, Proposition 5. Suppose that spanM has dimension d and that conditions (C 1 ) -(C 3 ) and (H) hold. Then:

• there exists a unique optimal strategy p * ∈ P(T ),

• p * is a discrete probability on T supported by at most d different types.

Proof. Existence of an optimal strategy is a consequence of Proposition 2. However it is worth noting that in this finite-dimensional case, it is a consequence of standard analysis. We denote by Conv(M) the closed convex hull of M in L ∞ (E, ν 1 ) which can be seen as a closed convex set in a d-dimensional vector space. Note that Conv(M) is equal to the set of functions {m p : e → m p,e ; p ∈ P(T )} where m p,e is defined in equation ( 2). Introduce the application Θ : Conv(M) → [-∞, ∞) defined by Θ(m) = E log(m(e))ν 1 (de) (with the convention that log(u) = -∞ if u ≤ 0). With these notations, we easily see that γ(p) = Θ(m p ) and maximizing γ on P(T ) is equivalent to maximizing Θ on Conv(M). Next we observe that the function Θ is uppersemicontinuous and strictly concave, so that it reaches its maximum at a unique point m * on the compact set conv(M). Furthermore the gradient of Θ never vanishes so that the extremum must be reached on a boundary point of Conv(M) and m * ∈ ∂Conv(M). It is worth noting that the application Θ has always a unique minimizer m * in the above proof. Condition (H) ensures that there is a unique strategy p * such that m p * = m * . In absence of this condition, there might be several mixing distribution p such that m p = m * and then several optimal strategies.

An appealing particular case is the case when both E = {e 1 , • • • , e p } and T = {t 1 , • • • , t q } are finite with q ≥ p. In this case L ∞ (E, ν 1 ) is of dimension d = p, and a generic configuration will always satisfy assumption (H) (in the sense that an exact linear relation between the types is very unlikely from a biological point of view). Then, according to Proposition 5, there exists a unique optimal strategy mixing at most p different types. This shows that the number of types involved in the optimal strategy is less than the number of different environments. This is reminiscent from a rule in ecology stating that the number of species in an ecosystem is bounded above by the numbers of different niche in the sense that two species cannot occupy the same niche for a long time (competitive exclusion principle). From a practical point of view, in this finite settings, the optimal strategy can be computed using standard numerical convex optimization (see [START_REF] Bonnans | Numerical optimization, Theoretical and practical aspects[END_REF] for instance). For more illustrations in this setting, see Section 1.1.1.

Example: Gaussian distributions

This example is due to Haccou and Iwasa [START_REF] Haccou | Optimal mixed strategies in stochastic environments[END_REF]: the environment space is the set of real numbers E = R, the environment ω is supposed to be a Gaussian stationary ergodic sequence with stationary distribution

ν 1 = N (µ, σ 2
2 ) the Gaussian distribution with mean µ ∈ R and variance σ 2 2 > 0. The dynamic for the environment is irrelevant in the no-sensing case. The trait space is the set of real numbers T = R and the mean offspring number of an individual of trait t in environment e has the gaussian form

m t,e = C 2πσ 2 1 exp - (t -e) 2 2σ 2 1
for some parameters C > 0 and σ 2 1 > 0. In environment e, individuals with trait value t = e are the best fitted. Note that conditions (C 1 ) -(C 3 ) are fulfilled so that Proposition 2 holds: optimal strategies exist. Furthermore, the family M = {e → m t,e ; t ∈ T } is linearly independent so that unicity hold: there is a unique optimal strategy p * depending a priori on µ, σ 1 , σ 2 and C.

First we easily compute the fitness of a pure strategy

γ(δ t ) = E log(m t,e )ν 1 (de) = log C - 1 2 log(2πσ 2 1 ) - (µ -t) 2 + σ 2 2 2σ 2 1
and the optimal pure strategy is equal to δ µ . Then according to Corollary 1, we test if this pure strategy is optimal in the set of all mixed strategies. For this we compute

E m t,e m µ,e ν 1 (de) = E exp µ 2 -t 2 + 2e(t -µ) 2σ 2 1 ν 1 (de) = exp (t -µ) 2 (σ 2 2 -σ 2 1 ) 2σ 4 1
and check that it is less than one when σ 2 ≤ σ 1 . Hence if the fluctuations of the environment are small (σ 2 ≤ σ 1 ), then the pure strategy δ µ is optimal.

Gaussian strategies p = N (µ p , σ 2 p ) yield easy computations: recalling that a Gaussian mixture of Gaussian distributions is a again Gaussian, we compute

m p,e = T m t,e p(dt) = C 2π(σ 2 1 + σ 2 p ) exp - (µ p -e) 2 2(σ 2 1 + σ 2 p )
and then

γ(p) = E log(m t,e )ν 1 (de) = log C - 1 2 log(2π(σ 2 1 + σ 2 p )) - (µ -µ p ) 2 + σ 2 2 2(σ 2 1 + σ 2 p )
.

We find that γ(p) is maximal (among Gaussian strategies) for µ p = µ and

σ 2 p = 0 if σ 2 2 ≤ σ 2 1 σ 2 2 -σ 2 1 if σ 2 2 > σ 2 1 .
We have seen that this is the optimal strategy in the case when σ 2 2 ≤ σ 2 1 . This is still the case when σ 2 2 > σ 2 1 : we compute indeed for t ∈ T and

p * = N (µ, σ 2 2 -σ 2 1 ) E m t,e m p * ,e ν 1 (de) = E C m t,e de = 1
and this characterizes the optimal strategy according to Proposition 3.

We have proven so far that: Proposition 6. the optimal strategy is

p * = δ µ if σ 2 2 ≤ σ 2 1 N (µ, σ 2 2 -σ 2 1 ) if σ 2 2 ≥ σ 2 1 , (3)
and the optimal growth rate

γ * = log C -1 2 log(2πσ 2 1 ) - σ 2 2 2σ 2 1 if σ 2 2 ≤ σ 2 1 log C -1 2 log(2πσ 2 2 ) -1 2 if σ 2 2 ≥ σ 2 1 . (4)
An interesting quantity is the relative gain of the best mixed strategy over the best pure strategy: it gives an indication of the strength of the selection pressure on mixed as opposed to pure strategies. From the above computation, denoting by χ = σ 2 2 /σ 2 1 , we get

γ(p * ) -γ(δ µ ) = 0 if χ ≤ 1 1 2 (χ -1 -log χ) if χ ≥ 1
This is the log-ratio of the expected long-term growth rates of individuals playing the different types of strategies. It is non-decreasing with respect to χ: when the environmental variance σ 2 2 is large compared to σ 2 1 , there is a strong advantage in playing a mixed strategy.

Next we illustrate the phenomenon discussed in section 3.3 when polymorphism is a necessary condition for survival. This happens when the optimal pure strategy δ µ leads to almost sure extinction of the population (γ(δ µ ) ≤ 0) whereas the optimal mixed strategy p * allows the population to survive (γ * > 0). This phenomenon occurs when σ 2 > σ 1 for every C belonging to the following non empty interval:

1 2 log(2πσ 2 2 ) + 1 2 < log C ≤ 1 2 log(2πσ 2 1 ) + σ 2 2 2σ 2 1 .

The non-hereditary case with sensing mechanism

We now study the case of non-hereditary traits when some sensing-mechanism is available: we mean that the trait distribution of the offspring does not depends on the trait of the parent, but does depend on the environment because the individuals get some information about the environment they evolve in and are able to adapt suitably the trait distribution of their offspring. We thus suppose in this section that Z n evolves according to model ( 1) with π t,e ≡ p e for some family p = (p e ) e∈E of distributions on T and let π = π(p) be the corresponding product distribution.

Reduction to a simple BPRE

In a similar way as in the non-hereditary case with no sensing, the non-hereditary assumption makes the structure of the population very simple: the trait distribution at time n is given by the strategy the parents followed at time n -1 in environment ω n-1 , that is p ωn-1 . This is the expressed in the following Lemma:

Lemma 2. For any n ≥ 1, the population structure is conditionally independent of the past population process given the size of the population and the environment, i.e.

Z n (|Z n |, ω n-1 ) (Z 0 , • • • , Z n-1 ) (|Z n |, ω n-1 ).
Proof. It is easily seen from the assumptions on the model ( 1) and from the non-hereditary assumption π = π(p) that the distribution of Z n given (Z 0 , • • • Z n-1 ), |Z n | and ω n-1 is equal to the distribution of |Zn| i=1 1 τi with τ i an iid sequence with distribution p ωn-1 . This distribution does not depend on (Z 0 , • • • , Z n-1 ), this proves the conditional independence.

The Lemma implies that given the environmental sequence ω, the distribution of the population process (Z n ) n≥1 is easily recovered from the size process (|Z n |) n≥1 . Once again, this latter process turns out to be a simple BPRE and this allows us to compute the performance γ(p) of the strategy p. Let ω (2) = ((ω n-1 , ω n )) n≥1 denote the pair-environment with values in E 2 . It is also stationary and ergodic and we denote by ν 2 its stationary distribution, which is the distribution of the pair (ω 1 , ω 2 ). Proposition 7. The size process (|Z n |) n≥1 is a simple branching process in environment ω (2) with offspring distribution

Υ p,(e1,e2) = T Υ t,e2 p e1 (dt).
Conditionaly to ω, the expected population size at time n is e1,e2) . Suppose the following integral exists,

E ω [|Z n |] = E ω0 [|Z 1 |] n-1 k=1 m pω k-1 ,ω k with m pe 1 ,e2 = T m t,e2 p e1 (dt) the first moment of Υ p,(
γ(p) = E 2 log m pe 1 ,e2 ν 2 (de 1 , de 2 ), then lim n→∞ n -1 log E ω [|Z n |] = γ(p) almost surely. Proof. According to Lemma 2, given (|Z 1 |, • • • , |Z n |)
and ω n-1 , the population Z n has the same distribution as |Zn| i=1 1 τi with τ i an iid sequence with distribution p ωn-1 . Intuitively, the i-th individual has type τ i chosen randomly on T with distribution p ωn-1 . The size of the next generation

|Z n+1 | is then |Zn| i=1 ξ τi,ωn n,i
where ξ τi,ωn n,i is the offspring of the i-th individual of type τ i in environment ω n . From this two step procedure i) random choice of the traits t according to p ωn-1 , ii) reproduction with random offspring in environment ω n , we obtain the effective offspring distribution Υ pω n-1 ,ωn in environment ω (2) n = (ω n-1 , ω n ): it is the mixture of the offspring distributions Υ t,ωn , with mixing distribution p ωn-1 . Other properties are proved as in Proposition 1.

Note that the classification Theorem 1 applies in this case as well and gives criteria for extinction or explosion of the population.

Optimal strategies

We now describe the set of optimal strategies when sensing is allowed. Is is also interesting to evaluate the gain between optimal strategy with or without sensing mechanisms. Optimality when sensing mechanism are allowed will be denoted with a double asterix whereas we keep a simple asterix for optimality without sensing. Let γ * * = sup{γ(p); p ∈ P(T ) E } be the optimal growth rate when sensing is allowed and P * * = {p ∈ P(T ) E ; γ(p) = γ * * } be the set of optimal strategies. Let ν e1 (de 2 ) be the conditional distribution of ω 2 given ω 1 = e 1 , so that ν 2 (de 1 , de 2 ) = ν 1 (de 1 )ν e1 (de 2 ). Note that conditional distributions are well-defined since E is assumed to be a Polish space. From the previous section, the set of optimal strategies without sensing is denoted by P * . It depends implicitly on the environment distribution ν 1 and we write P * (ν 1 ) to emphasize this dependence. Suppose assumptions of Proposition 2 hold. Then optimal strategies with sensing are related with optimal strategies without sensing in the following way: Proposition 8. A strategy p is optimal if and only if p e1 ∈ P * (ν e1 ) ν 1 (de 1 ) almost everywhere.

If conditions (C 1 ) -(C 2 ) hold, then optimal strategies exists and form a closed convex set i.e. P * * is nonempty closed and convex. If the family M = {e → m t,e ; t ∈ T } is linearly independent, there is a unique optimal strategy p * * .

Proof. Using the explicit formula for γ(p) given in Proposition 7 and conditional probabilities, we compute

γ(p) = E 2 log m pe 1 ,e2 ν 2 (de 1 , de 2 ) = E ν 1 (de 1 ) E log m pe 1 ,e2 ν e1 (de 2 ) = E γ(p e1 , ν e1 )ν 1 (de 1 )
whith γ(p e1 , ν e1 ) = E log m pe 1 ,e2 ν e1 (de 2 ) the growth rate associated with strategy p e1 in environment ν e1 . Hence γ(p) is optimal if we choose p e1 so that γ(p e1 , ν e1 ) is maximal, i.e. p e1 ∈ P * (ν e1 ) almost surely. The properties of P * * and the uniqueness follow from the similar results for P * (cf Proposition 2).

Then, the characterization of optimal strategies in the no-sensing case given in Proposition 3 directly extends to strategies with sensing as follows: Proposition 9. A strategy with sensing p ∈ P(T ) E is optimal if and only if

E m t,e2 m pe 1 ,e2
ν e1 (de 2 ) ≤ 1, for all t ∈ T and ν 1 (de 1 )almost everywhere.

Proof. The characterization follows directly from Proposition 3 and Proposition 8 together.

An interesting corollary states that no gain has to be expected from sensing mechanisms if the environment has some independence property. More precisely, Corollary 2. Suppose that ω 1 and ω 2 are independent (i.e. ν 2 = ν 1 ⊗ ν 1 ), then the optimal growth rate with or without sensing are equal, i.e. γ * = γ * * . Proof. In this product case, the conditional distribution are trivial, i.e. ν e1 ≡ ν 1 almost surely. Hence an optimal strategy is such that p * * e1 ∈ P * (ν 1 ) and hence γ(p * * e1 , ν 1 ) = γ * . Integrating with respect to ν 1 (de 1 ), we obtain

γ * * = E γ(p * * e1 , ν 1 )ν 1 (de 1 ) = γ * .
This proves the result.

Note that this result is rather intuitive: the sensing mechanism gives to the individual some information about the current environment state, but from the independence property, this is not useful for inference to the future environment state; hence the information is useless to decide which traits will be well-fitted in the next environment.

Example: Finite dimensional case continued

The results for optimal strategies without sensing developed in Section 3.4 together with Proposition 8 allow us to easily deduce the following properties for optimal strategies with sensing in the finite dimensional case.

Proposition 10. Suppose that spanM has dimension d and that conditions (C 1 ) -(C 3 ) and (H) hold. Then there exists a unique optimal strategy with sensing p * * ∈ P(T ) E such that p * * (e) = p * (ν e ) where p * (ν e ) is the optimal strategy without sensing from associated when the environment has marginal distribution ν e .

Recall furthermore from Proposition 5 that p * (ν e ) is a discrete probability measure on T with at most d extremal types (that may depend on e). See Section 1.1.2 for more examples.

Exemple: Haccou and Iwasa's example continued

This is the sequel of subsection 3.5, the example by Haccau and Iwasa. Recall that the environment is given by a Gaussian stationary ergodic sequence ω = (ω n ) n≥0 with stationary distribution ν 1 = N (µ, σ 2 2 ). Let ρ ∈ (-1, 1) be the pair correlation ρ = corr(ω 0 , ω 1 ). The case ρ = 0 corresponds to independent environments, i.e. ν 2 = ν 1 ⊗ ν 1 . Otherwise dependence holds and standard Gaussian computations give the conditional distribution ν e1 = N (µ+ρ(e 1 -µ), (1-ρ 2 )σ 2 2 ). A particular realization of such a sequence is the Ornstein-Uhlenbeck sequence defined by

ω 0 = µ + σ 2 η 0 ω n = µ + ρ(ω n-1 -µ) + 1 -ρ 2 σ 2 η n , n ≥ 1 ,
for i.i.d. standard normal innovations (η n ) n≥0 (Gaussian white noise). We have seen in section 3.5 that the optimal strategy without sensing is Gaussian (possibly degenerated) and we have given explicit formulas for the parameters (3) and for the corresponding growth rate (4). According to Proposition 8, we deduce the optimal strategy when sensing is allowed: Proposition 11. There exists a unique optimal strategy with sensing, which is denoted by p * * and satisfies ν 1 (de)-a.e.

p * * e = δ µ+ρ(e-µ) if (1 -ρ 2 )σ 2 2 ≤ σ 2 1 N (µ + ρ(e -µ), (1 -ρ 2 )σ 2 2 -σ 2 1 ) if (1 -ρ 2 )σ 2 2 ≥ σ 2 1 . ( 5
)
The corresponding optimal growth rate is given by

γ * * = log C -1 2 log(2πσ 2 1 ) - (1-ρ 2 )σ 2 2 2σ 2 1 if (1 -ρ 2 )σ 2 2 ≤ σ 2 1 log C -1 2 log(2π(1 -ρ 2 )σ 2 2 ) -1 2 if (1 -ρ 2 )σ 2 2 ≥ σ 2 1
Finally, we can evaluate the relative gain of strategies with sensing over strategies without sensing: it gives an indication of the benefit that can be expected from sensing mechanisms. Let χ = σ 2 2 /σ 2 1 . Then

γ * * -γ * =    1 2 ρ 2 χ if χ ≤ 1 1 2 log χ -1 2 (1 -ρ 2 )χ + 1 2 if 1 ≤ χ ≤ (1 -ρ 2 ) -1 -1 2 log(1 -ρ 2 ) if χ ≥ (1 -ρ 2 ) -1
.

It is worth noting that this is an increasing function of the square correlation ρ 2 : this indicates that the more correlated the random environment is, the more useful sensing mechanisms are. The intuitive idea is that higher correlations allows for more accurate prevision for the next environment and hence for a better fitted offspring trait distribution in the environment to come.

The hereditary case

In the hereditary case, the trait distribution π t,e ∈ P(T ) might depends on the trait of the parents. This dependency makes the study of the Lyapounov exponent γ = γ(π) much more difficult because no reduction to a simple branching process in random environment is available. We have no explicit formula for γ(π) in this case and determining the optimal strategy π * and the optimal growth rate γ(π * ) might be very challenging. Nevertheless, we propose an interesting representation of the "finite time" growth rate

γ n (ω, π 0 ) = n -1 log E ω,π0 [|Z n |]
in environment ω and initial population consisting of a single individual with random trait with distribution π 0 . The representation is in term of a functional of the T -valued Markov chain T = (T n ) n≥0 in environment ω such that:

P π0,ω (T 0 ∈ • ) = π 0 ( • ) P π0,ω (T k ∈ • |T 0 = t 0 , • • • , T k-1 = t k-1 ) = π t k-1 ,ω k-1 ( • ) , 1 ≤ k ≤ n . (6)
The Markov chain T n in random environment ω is time-heterogeneous because the transitions depend on time n through the value of the environment ω n . However, in the no-sensing case when the transitions π t,e ≡ π t do not depend on e, the Markov chain T is time-homogeneous. The result is the following: Proposition 12. The finite time growth rate in environment ω and initial population consisting in a single individual with trait distributed according to π 0 is given by

γ n (ω, π 0 ) = n -1 log E ω,π0 [M n (T, ω)] with M n (T, ω) = n-1 k=0 m T k ,ω k .
For the sake of clarity and conciseness, this proposition will be proved together with Theorem 2 below. To our best knowledge, there is no simple way to deal with the asymptotics behaviour of γ n and this issue might be challenging. We provide a lower bound for the growth rate γ n (ω, π 0 ) that might be more tractable. The lower bound for γ n (ω, π 0 ) is obtained using Jensen's inequality (with the concave function log):

γ n (ω, π 0 ) = n -1 log E ω,π0 [M n (T, ω)] ≥ n -1 E ω,π0 [log M n (T, ω)].
The advantage here is that the lower bound

n -1 E ω,π0 [log M n (T, ω)] = n -1 n-1 k=0 E ω,π0 [log m T k ,ω k ]
can be written as an additive functional of the environment ω and we can then use the ergodic theorem to control the convergence. Note that this technique can be refined using changes of measures. Let πt,e any kernel family and denote by T the T -valued Markov chain in environment ω starting from distribution π 0 and with transitions given by πt,e (given by equations similar to (6)). We suppose that π t,e is absolutely continuous with respect to πt,e , i.e. π t,e (dt ′ ) = f t,e (t ′ )π t,e (dt ′ ) where f t,e stands for the density of π t,e with respect to πt,e . Then, using changes of measures, we have

E ω,π0 [M n (T, ω)] = E ω,π0 [ Mn ( T , ω)] with Mn ( T , e) = n-1 k=0 m Tk ,ω k f Tk ,ω k ( Tk+1 ).
Using this, the lower bound becomes

γ n (ω, π 0 ) = n -1 E ω,π0 [log Mn ( T , ω)] ≥ n -1 n-1 k=0 E ω,π0 log m Tk ,ω k f Tk ,ω k ( Tk+1 ) .
6 Typical genealogies: a mean field approach 6.1 Convergence of the typical genealogy in the infinite population limit As explained in Baake and Georgii [START_REF] Baake | Supercritical multitype branching processes: the ancestral types of typical individuals[END_REF][START_REF] Baake | Mutation, selection, and ancestry in branching models: a variational approach[END_REF], the evolution of a branching population can be studied from two possible perspectives: either forward or backward in time. So far, we have focused on the first point of view and mainly studied the growth rate of the population after a large numbers of generations. By way of contrast, the backwards or retrospective aspect of the population concerns the lineages extending back into past from the presently living individuals and asks for the characteristics of the ancestors along such lineages. We now turn to this second perspective and wonder what is the typical lineage or genealogy (backward in time) of an individual chosen at random in the n-th generation. Some definitions are needed here. The right formalism to keep track of the genealogy is the formalism of labeled rooted trees and forests, where the trees stands for the descendence of each ancestor represented by a root, and labels keep track of the traits of the individuals. However we keep this formalism to its minimum. Let G n denote the population at the n-th generation. To each individual g ∈ G n , we associate its lineage or genealogy ℓ(g) = (t 0 , • • • , t n ) ∈ T n+1 with the interpretation that t n is the trait of g, and t k the trait of his ancestor in the k-th generation G k , 0 ≤ k ≤ n -1. The typical genealogy is defined as the genealogy of an individual chosen at random in the n-th generation. This obviously requires the n-generation to be non empty, in which case we adopt the convention that the typical genealogy is ∅. The distribution of the typical genealogy is given by

π n = 1 cardGn g∈Gn δ ℓ(g) if G n = ∅ δ ∅ if G n = ∅ .
This is a random measure on T n+1 ∪ {∅}. We denote by P N,π0,ω the probability measure corresponding to a population evolving in environment ω, starting a time 0 from N individuals with traits i.i.d. with distribution π 0 . In the following, we focus on the typical genealogy in the infinite population limit N → ∞. Recall the definition of the Markov chain T = (T n ) n≥0 given by equation ( 6). We have the following mean field result:

Theorem 2. Under the probability P N,π0,ω , the typical genealogy distribution π n almost surely weakly converges as N → ∞ to the distribution πn,π0,ω defined by

πn,π0,ω (A) = E ω,π0 [M n (T, ω)1 (T0,••• ,Tn)∈A ] E ω,π0 [M n (T, ω)] , A ⊂ T n+1 .
Recall from Proposition 12 that E ω,π0 [M n (T, ω)] = exp(nγ n (π 0 , ω)) is the mean number of individuals in the n-th generation of a population evolving in environment ω and starting from a single individual with random trait with distribution π 0 .

Proof of Theorem 2 and Proposition 12. Let

A = A 0 × • • • × A n be a product subset of T n+1 .
The number of individuals in the n-th generation with genealogy in A is

N n (A) = g∈Gn δ ℓ(g) (A)
and N n (T n+1 ) denotes the total number of individuals in G n . We can see N n as a non-normalized measure and π n is the probability measure associated with by the relation

π n = 1 N n (T n+1 ) 1 {Nn(T n+1 )>0} N n + 1 {Nn(T n+1 )=0} δ ∅ .
From the branching property, the distribution of N n (A) under P N,π0,ω is equal to the sum of N independent copies N i=1 N (i) n (A) under P ⊗N 1,π0,ω . As a consequence of the weak law of large numbers, the distribution of 1 N N n (A) under P N,π0,ω weakly converge to E 1,π0,ω (N n (A)). The results also holds for A = T n+1 and taking the quotient, we see that under P N,π0,ω , π n (A) weakly converge to πn,π0,ω (A) = E 1,π0,ω (N n (A)) E 1,π0,ω (N n (T n+1 )) provided the denominator is non zero. Note that γ n (ω, π 0 ) = n -1 log E 1,π0,ω (N n (T n+1 )).

The theorem and the proposition (letting A = T n ) are then a consequence of For n = 0, E 1,π0,ω (N 0 (A 0 )) = π 0 (A). It remains to note that E 1,π0,ω (N n (T n+1 )) = 0 and this implies that the mass of ∅ vanishes in the limit.

The typical genealogy in the non-hereditary case

When the traits are non-hereditary, the trait distributions π t,e do not depend on t and the mean field typical genealogy distribution πn,ω,π0 has a very simple form. The following Proposition is given in the context of a population with a sensing mechanism π = π((p e ) e∈E ). The no-sensing case corresponds to the particular case when p e ≡ p. Proof. In the non-hereditary case, the Markov chain in random environment ω defined by equation ( 6) is simple because conditionally on the environment ω, the random variables T 0 , • • • , T n are independent with T 0 distributed as π 0 and, for 1 ≤ i ≤ n, T i distributed according to .

This proves the independence property and gives the marginal distributions:

πi (A i ) = E ω,π0 [m Ti,ωi 1 Ti∈Ai ] E ω,π0 [m Ti,ωi ] , 1 ≤ i ≤ n -1.
The interpretation of the above proposition is the following: the mean field typical genealogy in environment ω consists in independent traits T0 , • • • , T n where -the distribution of T0 is a biased version of π 0 (dt 0 ) with bias function equal to m t0,ω0 the mean number of offspring's of an individual of type t 0 , -the distribution of Ti is a biased version of p ωi (dt i ) with bias function equal to m ti,ωi , -the distribution of Tn is p ωi (dt i ) (there is no bias because the offspring of the last generation G n is not involved since the population is considered until time n only).

6.3

The typical genealogy in a hereditary case: Haccou and Isawa's example continued

Recall that T = E = R, ω is a Gaussian stationary ergodic sequence with stationary distribution N (µ, σ Let π 0 = N (µ 0 , s 2 0 ) be the initial distribution. Alternatively, we have the representation in environment ω T 0 = µ 0 + s 0 N 0 ,

T k+1 = α ω k + β ω k T k + θ ω k N k+1 0 ≤ k ≤ n -1,
with N 0 , • • • , N n independent standard normal variable. We introduce the (n + 1) × 1 vectors

T =      T 0 T 1 . . . T n      , N =      N 0 N 1 . . . N n      , A ω =      µ 0 α ω0 . . . α ωn-1     
and the (n + 1) × (n + 1) matrices

C ω =        0 0 • • • 0 0 β ω0 0 • • • 0 0 0 β ω1 0 0 0 0 0 . . . 0 0 0 0 • • • β ωn-1 0        , S ω =        s 2 0 0 • • • 0 0 0 θ 2 ω0 • • • 0 0 0 0 θ 2 ω1 0 0 0 0 • • • . . . 0 0 0 • • • 0 θ 2 ωn-1       
.

Proposition 1 .

 1 The size process (|Z n |) n≥1 is a simple BPRE with offspring distribution in environment e given by Υ p,e = T Υ t,e p(dt), e ∈ E.

  p n converging weakly to p in P(T ). Using conditions (C 1 ) and (C 2 ), we see that m pn,e → m p,e for all e ∈ E. Define f n (e) = log(M )log (m pn,e ) be non-negative measurable functions on E and apply Fatou's Lemma: we getE lim inf f n (e)ν 1 (de) ≤ lim inf E f n (e)ν 1 (de). Equivalently, lim sup E log (m pn,e ) ν 1 (de) ≤ E log (m p,e ) ν 1 (de).Hence lim sup γ(p n ) ≤ γ(p) and this states the upper semicontinuity of the application γ. The concavity of the application γ on P(T ) is a direct consequence from the concavity of the logarithm and the linearity of integration: first for p 1 , p 2 ∈ P(T ), λ ∈ [0, 1] and e ∈ E, we have log m λp1+(1-λ)p2,e = log (λm p1,e + (1λ)m p2,e ) ≥ λ log (m p1,e ) + (1λ) log (m p2,e ) .

E 1 ,

 1 π0,ω (N n (A)) = A π 0 (dt 0 ) n-1 k=0 m t k ,ω k π t k ,ω k (dt k+1 ) = E ω,π0 [M n (T, ω)1 (T0,••• ,Tn)∈A ].This last relation is proven by induction: individuals in the n-th generation with genealogy in A are the offsprings with traits in A n of individuals in the n -1 generation with genealogy inA 0 × • • • × A n-1 . This yields E 1,π0,ω (N n (A)) = An-1×An E 1,π0,ω (N n-1 (A 0 × • • • × A n-2 × dt n-1))m tn-1,ωn-1 π tn-1,ωn-1 (dt n ).

Proposition 13 .

 13 In the non-hereditary case, the typical genealogy distribution πn,ω,π0 is the product measure on T n+1 defined by πn,ω,π0 (dt0 , • • • , dt n ) = ⊗ n i=0 πi,ωi (dt i ) (dt 0 ) = mt 0 ,ω 0 mπ 0 ,ω 0 π 0 (dt 0 ), πi,ωi (dt i ) = mt i ,ω i mp ω i-1 ,ω i p ωi-1 (dt i ), 1 ≤ i ≤ n -1, πn,ωn (dt n ) = p ωn-1 (dt n ).

  p ωi-1 . Let A = A 0 × • • • × A n ⊂ T n+1. Using independence and Theorem 2, we compute:πn,π0,ω (A) = E ω,π0 [M n (T, ω)1 (T0,••• ,Tn)∈A ] E ω,π0 [M n (T, ω)] = E ω,π0 [m T0,ω0 1 T0∈A0 ] × • • • × E ω,π0 [m Tn-1,ωn-1 1 Tn-1∈An-1 ]E ω,π0 [1 Tn∈An ] E ω,π0 [m T0,ω0 ] × • • • × E ω,π0 [m Tn-1,ωn-1 ]

2

 2

1 .

 1 It remains to precise the trait distributions π t,e and π 0 . To make the model explicitly solvable, we require the Markov chain T = (T 0 , • • • , T n ) in environment ω to be multivariate Gaussian. This impose that the transition π t,e are of the form π t,e = N (α e t + β e , θ e ) with E[T 1 |ω 0 = e, T 0 = t] = α e + β e t and Var[T 1 |ω 0 = e, T 0 = t] = θ 2 e .

  of stationary measure ν 2 (de 1 , de 2 ).

		Consider
	the average distribution
	Υ p,(e1,e2) =	Υ t,e2 p e1 (dt),
	T
	of expected value

  The boundary of Conv(M) consists in extremal types (type t such that m t : e → m t,e is an extremal point of the convex set Conv(M)) and of k-dimensional faces determined by the convex hull of k + 1 extremal types, with 0 ≤ k ≤ n. If m * belongs to such a k-dimensional face, then m * = m p * where p * is a discrete probability on T supported by the k + 1 corresponding extremal types, with k + 1 ≤ d. The uniqueness property comes then from assumption (H). Indeed, under this assumption, a k dimensional face determined by k +1 extremal types does not contain any other point of M so that the decomposition in barycentric coordinates are unique.

i.e. for all i ∈ N, (w 0 , w 1 , ...) is distributed as (w i , w i+1 , ...)

i.e. for all bounded Borel function f , P(f (w 0 , w 1 , ...) = f (w 1 , w 2 , ...)) = 1 implies f (w 0 , w 1 , ...) is constant.

The non-hereditary case with no sensing mechanism According to the approach set out in the previous paragraph, we begin by studying the simplest case of non-hereditary traits in absence of sensing mechanism, meaning that the trait distribution of the

With these notations, the recursive relation turns into

and this yields T = (Id -C ω ) -1 (A ω + S ω N ).

We deduce the mean µ and covariance matrix Σ for the Gaussian vector

Suppose that S ω is invertible, then T has density

Denote by T = ( T0 , • • • , Tn ) a random vector with distribution πn,ω,π0 . According to Theorem 2, T is a biased version of T and has a density given by

Introducing J n,1 the diagonal matrix J n,1 = (δ 0≤i=j≤n-1 ) 0≤i,j≤n and V n,ω the column vector

We recognize that f T is a multivariate Gaussian density with mean μ and covariance matrix Σ of the form

Identifying both expressions, we obtain after simplification

These computations prove the following result: Proposition 14. In the Haccou and Isawa model with Markov Gaussian environment, the typical genealogy distribution πn,ω,π0 is the Gaussian distribution on R n+1 with mean μ given by (8) and covariance matrix Σ given by [START_REF] Geiger | Limit theorems for subcritical branching processes in random environment[END_REF]. Furthermore, the finite time growth rate γ n (ω, π 0 ) is given by (9).