démographie · 2020. 9. 14. · deep human genealogies reveal a selective advantage to be on an...

BIO2529 Écologie

DémographieGabriel Blouin-Demers Professeur titulaire

• Suite de BIO1530

• Croissance de la population (modèle exponentiel)

• Densité dépendance

• Capacité de support

• Croissance de la population (modèle logistique)

• Exercice

• Applications

ContenuDémographie

Démarche scientifiqueModèles de croissance des populations

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Croissance de la population

Croissance de la populationComposantes

Nt+1 = Nt + B + I - D - E

N = taille de la populationB = naissances (births)I = immigrationD = mortalités (deaths)E = émigration

Croissance de la populationMétapopulation

Population 1

Population 3

Population 2I = 0,40E = 0,40λ = 1,00

I = 0,70E = 0,30λ = 0,95

I = 0,40E = 0,80λ = 1,05

m = 0,30

m = 0,20

m =

0,1

0

m =

0,2

0

m = 0,20

m = 0,50


Nt+1 = Nt + B + I - D - EB = f x Nt

D = (1-ϕ) x NtNt+1 = (f+ϕ)Nt + dispersion

f = féconditéϕ = taux de survie au temps t


Population ferméeNt+1 = (f+ϕ)Nt

• Taux de croissance géométrique pour les organismes qui se reproduisent de façon discrète (p. ex. les plantes annuelles) où les individus sont dans des cohortes distinctes et où il n’y a habituellement pas de chevauchement des générations

• La population croit de façon discrète

• Taux de croissance exponentiel pour les organismes qui se reproduisent de façon continue (p. ex. les bactéries) où il y a chevauchement des générations

• La population croit continuellement et graduellement

Taux géométrique et exponentielCroissance de la population

Croissance de la populationTaux de croissance géométrique

Nt+1 = (f+ϕ)Nt Nt+1 = λ Nt

λ = taux de croissance géométriqueλ = 1 population stableλ < 1 population décroissanteλ > 1 population croissante

Croissance de la populationTaux de croissance géométrique

Nt+1 = λ NtNt = N0 x λ1 x λ2 x λ3… λt

Nt = N0 x λtλ = √Nt / N0t

λ = taux de croissance géométrique moyen de 0 à t

Croissance de la populationTaux de croissance exponentiel

lorsque Δt → 0dN / dt = rN

Nt = N0 x ert

r = pente de la tangente à la courbe de N en fonction de t

r = taux de croissance exponentielr = 0 population stabler < 0 population décroissanter > 0 population croissante

Croissance de la populationTaux de croissance géométrique et exponentiel

ln(λ) = r ou λ = er

e = la base du ln ≈ 2,718

Croissance de la populationCroissance exponentielle

Nt = N0 x er r = 0,10 pente = r

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0 5 10 15 20 25 30 35 40 45 501

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• La croissance exponentielle est une augmentation par une proportion fixe par unité de temps, par exemple 5% par année

• Le temps de doublage est le temps nécessaire pour augmenter de 100% ou par un facteur de 2

• Temps de doublage ≈ 70 ans / % augmentation par an

• Taux de croissance de 5% par année implique un doublage dans 14 ans (70/5)

Temps de doublageCroissance de la population

Temps de doublageD’où provient 70 ans?

Nt = 2 N0

Nt = N0 x ert

Nt / N0 = ert

2 = ert

ln 2 = rt0,69 / r = tdoublage

Temps de doublageCalculs

r %/an tdoublage (années)

0,01

0,02

0,03

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0,10

Densité dépendance

• Si une entité augmente par un facteur constant par unité de temps, éventuellement les augmentations deviennent énormes

Croissance exponentielleCroissance de la population

• Masse d’une bactérie ≈ 10-12 g

• Temps de doublage ≈ 1 heure

• Masse de la Terre ≈ 5,9 x 1027 g

• En 132 doublages, soit en 5,5 jours, la masse de la population bactérienne ≈ la masse de la Terre

Croissance exponentielleCroissance de la population

Deep Human Genealogies Reveal aSelective Advantage to Be on anExpanding Wave FrontClaudia Moreau,1 Claude Bhérer,1 Hélène Vézina,2 Michèle Jomphe,2

Damian Labuda,1,3* Laurent Excoffier1,4,5*

Since their origin, human populations have colonized the whole planet, but the demographicprocesses governing range expansions are mostly unknown. We analyzed the genealogy of morethan one million individuals resulting from a range expansion in Quebec between 1686 and 1960and reconstructed the spatial dynamics of the expansion. We find that a majority of the presentSaguenay Lac-Saint-Jean population can be traced back to ancestors having lived directly on orclose to the wave front. Ancestors located on the front contributed significantly more to the currentgene pool than those from the range core, likely due to a 20% larger effective fertility of womenon the wave front. This fitness component is heritable on the wave front and not in the core,implying that this life-history trait evolves during range expansions.

Most species go through environmental-ly induced range expansions or rangeshifts (1), promoting the evolution of

traits associated with dispersal and reproduction(2). Humans likely colonized the world by aseries of range expansions from Africa (3), pos-sibly with episodes of interbreeding with nowextinct hominins (4, 5), leading to allele frequen-cy and heterozygosity clines from entry pointsinto several continents [e.g., (6, 7)]. Range ex-pansions can also lead to drastic changes in allelefrequencies, sometimes mimicking the effect ofpositive selection in recently colonized habitats(8, 9), through a process called gene surfing (9).Neutral, favorable, or even deleterious mutationscan surf and increase in frequency (10, 11), im-plying that wave fronts may harbor mutationswith a wider range of selective coefficients thancore populations. The evolutionary consequencesof range expansions have been studied in a widearray of species (2, 12), but studies of the dy-namics of range expansions have been generallyrestricted to species with short generation times(13, 14) or to invasive species (15, 16), becauseboth spatial and temporal sampling are requiredto understand the dynamics of wave fronts.

Deep-rooted human genealogies in recentlyexpanded populations may offer an opportunityto study the wave front demographics and itsgenetic consequences on present-day populations.We studied the genealogies reconstructed from

Quebec parish registers that document the recenttemporal and spatial expansion of the settle-ment of the Charlevoix Saguenay Lac-Saint-Jean (ChSLSJ) region, northeast of Quebec City,Canada: a prime example of a recent, fast, andwell-documented range expansion (17) (Fig. 1).The European colonization of Quebec was ini-tiated in 1608 with the foundation of QuebecCity, and the colony was well established by theend of the 17th century (18). The peopling of theCharlevoix region started from Baie-Saint-Paul,and both a rapid demographic growth and the de-velopment of the timber industry promoted furtherexpansions after 1838 up the Saguenay River andthe Lac-Saint-Jean region (SLSJ) (19, 20). Thespatial and temporal dynamics of the peopling ofthe whole ChSLSJ region can be reconstructed by

tracing back the founding events of new localities.As shown in Fig. 1, the inferred colonization pro-cess is a mixture of long-distance settlementscreating an irregular wave front, followed by fur-ther, more progressive, short-range expansions,which then filled gaps and created a more reg-ular wave front.

On the basis of the computation of a wavefront index (WFI) (21), we find that the ancestorsof the Saguenay and the Lac-Saint-Jean peoplelived more often on or close to the wave frontthan expected by chance (WFI, P < 0.001 in bothregions) (fig. S1). Indeed, the very high WFI of0.75 observed in Lac-Saint-Jean corresponds toa situation in which half of the Lac-Saint-Jeanancestors had lived directly on the wave front andthe other half just one generation away from it.In contrast, WFI is significantly lower in theCharlevoix region (P = 0.003) (fig. S1). Theseresults are consistent with different colonizationdynamics of SLSJ and Charlevoix. The wavefront was always widespread in SLSJ where newlocalitieswere continuously settled, whereas it wasmuch smaller in Charlevoixwheremost localitiesremained in the range core until the 20th century(Fig. 1). New immigrants from outside ChSLSJconstituted an important minority of the peoplegetting married, with a greater proportion of im-migrants settling on the wave front than on therange core, especially before 1900 (up to 20% onthe wave front and up to 10% in the range core)(table S2). Generally, more male than female im-migration occurred in all regions, and this biastoward males is significantly higher in the corethan on the wave front (table S3). Nevertheless,the new territories of SLSJ have been largely col-onized by people recruited directly on the wavefront or next to it, not by people from the rangecore (table S4).

1686-1720

1721-1750

1751-1780

1781-1810

1811-1840

1841-1870

1871-1900

1901-1930

1931-1960

LacSaint-Jean

Saguenay River S

aint

Law

renc

e R

iver

Baie-Saint-Paul

Chicoutimi

Fig. 1. Map of the Charlevoix Saguenay Lac-Saint-Jean region showing the range expansion dynamicsand the wave front at different periods. Each filled circle represents a locality, and its color indicates itsage. Localities from the Charlevoix region are indicated by a black dot.

1Centre de Recherche, Hôpital Sainte-Justine, Université de Mon-tréal, 3175 Côte Sainte-Catherine, Montréal, Québec, Canada,H3T 1C5. 2Projet BALSAC, Université du Québec à Chicoutimi,555 Boulevard de l’Université, Chicoutimi, Québec, Canada, G7H2B1. 3Département de Pédiatrie, Université deMontréal, Montréal,Québec, Canada, H3T 1J4. 4Computational and Molecular Pop-ulation Genetics, Institute of Ecology and Evolution, Universityof Berne, Baltzerstrasse 6, 3012 Berne, Switzerland. 5SwissInstitute of Bioinformatics, 1015 Lausanne, Switzerland.

*To whom correspondence should be addressed. E-mail:[email protected] (L.E.); [email protected] (D.L.)

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Ever since the fi rst humans migrated out of Africa almost 2 million years ago, our ances-tors have crisscrossed the globe in search of food, resources, and better lives. Now a remarkable study of the genealogy of 1 mil-lion Canadians has found that the fi rst waves of European settlers to push into the wilder-ness of northeastern Quebec in the 17th and 18th centuries had more children than people who came later or stayed in central towns. The study was presented in a poster by popula-tion geneticists Claudia Moreau and Damian Labuda of the University of Montreal and Laurent Excoffi er of the University of Bern, and the work is published online in Science

(http://scim.ag/cmoreau) this week. Some 30,000 settlers, mostly farmers, who

opened frontiers starting in the 17th century passed on up to four times as many genes to living people than did immigrants who lagged behind in core towns, the researchers found. The work allows geneticists to test computa-tional models of how populations and genes spread when humans move into new territo-ries and multiply rapidly. It “confi rms what many people must have thought: that when conditions were hard, it was better to move and take their chances than to stay in place,” says population geneticist Montgomery Slatkin of the University of California, Berkeley, who was not part of the study.

The team used the comprehensive BALSAC population database of Quebec, which contains marriage, birth, and death records by parish for 5 million people from 1608 to 1970. The researchers were able to trace the pattern of settlement as 84 new par-

ishes opened in the Charlevoix and Saguenay-Lac-Saint Jean regions of Quebec. Starting in 1608, European immigrants, mostly from France, arrived in Quebec City and began to spread northeast along the St. Lawrence River. In 1638, for example, about 15 people moved north from the town of Baie-Saint-Paul, followed by other groups that created an uneven wave front rolling into the terri-tory occupied by Algonquian tribes. By 1838, other settlers had arrived, fi lling in the gaps and evening the front line of settlers, whereas others remained behind in core settlements.

Those families closest to the frontier had the most children. By comparing birth and marriage records, the researchers found that women on the leading edge of the wave had an average of 9.1 children, compared with 7.9 children per family at the core, a boost of about 15%. That increase was amplifi ed in their offspring: An average of 4.9 chil-dren from front-line families married com-pared with 4.1 at the core, a 20% difference. Moving forward to 1960, 40% of living peo-ple’s ancestors had lived on the front wave between 1686 and 1930. “We’re excited that we’ve reconstructed what happens when you expand into a new territory,” Labuda says.

The researchers noticed that women on the front line got married a year earlier than those in core towns and bore children later, giving them a higher reproduction rate. So what was it about life on the frontier that boosted fertility? It could be partly due to cul-tural preferences, such as the benefi ts of hav-ing a large family. Or it might be that “fertility is highest at the wave front because there are

more resources [there],” explains population geneticist Henry Harpending of the Univer-sity of Utah in Salt Lake City, who was not part of the study. “As a place fi lls up, fertility declines in response to fewer resources. This paper shows that is exactly what happened.”

–ANN GIBBONS

MEETINGBRIEFS>>12TH INTERNATIONAL CONGRESS OF HUMAN GENETICS | 11–15 OCTOBER 2011 | MONTREAL, CANADA

Life on the Fertile Frontier

Fruitful founders. A Quebec couple in 1876, with seven of their 14 children (below), gave rise to many descendants, as seen in a 1960 photo of one of their sons with his descendants (left).

X-tra Diversity For AfricansCall it the X factor. Compared with Europe-ans and Asians, Africans have extra diversity on their X chromosomes, according to an invited lecture at the meeting by Cornell Uni-versity geneticist Andrew Clark. The lack of diversity in non-Africans may be the legacy, handed down for thousands of generations, of fewer women than men in the bands of mod-ern humans who fi rst left Africa to colonize the rest of the world, Clark said.

His team used new data from the 1000 Genomes Project to resolve a recent debate about diversity on the X and to begin to explain why non-Africans have so little vari-ation on this sex chromosome compared with the other chromosomes, or autosomes. “We’re seeing a very consistent pattern of reduced X diversity out of Africa,” Clark said. “It shows a greater loss of variation than even in the autosomes out of Africa.”

To evolutionary geneticists, the X chro-mosome has always been something of a rid-dle. Unlike its partner the Y, the X still has its full complement of genes. But because of the way it is inherited—men inherit only one copy from their mothers, whereas women inherit a copy from each parent—the X is

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La famille Bhérer du Saguenay / Lac St-JeanLes femmes avaient en moyenne 9,1 enfants!

We computed the expected number of genesleft by a given ancestor to the current genera-tion [its genetic contribution (GC)] (21) for allancestors of ChSLSJ, distinguishing betweenthose having reproduced on the wave front andthose in the range core (Table 1). We find thatover the entire studied period, individuals on thefront have contributed significantly more genesto the present generation than those in the core, inline with the theory predicting that surfing allelesshould be traced back to ancestors living on orclose to the wave front (22). We find similarresults when we restrict the analysis to the SLSJregion (Table 1), which has been colonized morerecently. Overall, ancestors on the edge contrib-uted 1.2 to 3.9 times more genes to the currentgeneration than ancestors from the core, the oldestancestors generally passing on more genes thanmore recent ones, in keeping with previous results[see, e.g., figure 4 in (23)]. In addition, 40.2% ofall ancestors of the ChSLSJ living between 1686and 1930 were on the wave front, reaching 45.1%for the SLSJ region (Table 1). For SLSJ, thenumber of ancestors living directly on the front orjust one generation away from it even reaches81% (table S4), showing the importance of thismoving edge for this region.

We compared the reproductive success ofwomen on the edge to the ones in the core, con-sidering both the number of their children [fami-ly size (FS)] and the number of their marriedchildren [effective family size (EFS)]. SLSJ fe-male ancestors living on the edge had on average15% more children than core SLSJ female an-cestors (Table 2) (P < 0.001) and 20% moremarried children (P < 0.001). These results showthat women’s fertility was significantly higher onthe wave front than in the range core and that thelarger genetic contribution of ancestors reproduc-ing on the wave front is likely not due purely to a

neutral surfing process but also to a net effect ofpositive selection on the front.

Women on the front overall had a slightlyhigher modal FS value (fig. S2A) and larger EFSdue to a right shift of the whole distribution to-ward higher values (fig. S2B), leading to a largerproportion of women on the front having morethan five married children (40% on the frontversus 26% in the core).We find only a slightlylower mortality rate of children under the age of5 on the wave front (23.6 versus 25.1% in thecore), implying that the increase in EFS comparedwith FS on the front is likely due to facilitatedaccess to reproduction (marriage). Interestingly,women on the front married almost 1 year earlierthan women in the core (Table 2), increasingtheir reproductive life, which may partly explaintheir overall higher fertility. This is in line withCharbonneau’s observations (18) that Quebecwomen had an overall longer reproductive lifecompared with French women of the same period,due to both an earlier age at first child and a laterage for their last child. However, our results sug-gest that the larger fertility of women in Quebecis mainly due to a front effect. An analysis ofcovariance (ANCOVA) reveals that the numberof children per women actually depends signif-icantly both on the age of marriage and on the

spatial location of reproduction (front or core)(P < 0.001 for the two effects) but that there is nointeraction between these factors (P = 0.46). Forthe number of married children, the two factors(P < 0.001) and their interaction (P = 0.02)are significant. We conclude that even thoughwomen on the front reproduce earlier thanwomen from the core, this contrast does not fullyexplain the difference in their fertility. The ad-vantage of being on the wave front remains if weuse less informative criteria to assign individualsto the front in SLSJ (tables S6 and S7), and it isnot due to a higher fertility of new immigrantssettling preferentially on the wave front (table S8).

We compared the fertility of women to theaverage fertility of their offspring (24), using thefact that the regression slope b gives us directly ameasure of heritability as h2 = 2b (25). Awomen’sFS is correlated with that of her own children onthe wave front, with a nonsignificantly differentheritability on the wave front (Fig. 2A) (h2 = 0.22)and in the range core (h2 = 0.16) (ANCOVA testof slope difference, P = 0.07). In contrast, if weconsider the transgenerational correlation in EFS,the correlation is only significant on the wavefront (Fig. 2B) (h2 = 0.24, ANCOVA test ofslope, P < 0.001) and not in the range core (h2 =0.04, ANCOVA test of slope, P = 0.23). The

Table 2. Age of reproduction and number of children of women from SLSJ in the period 1840 to 1900.Note that this table only includes women with known birth dates, such that age at marriage can becomputed.

No. ofwomen

Mean no. ofchildren(FS)

Mean no. ofmarriedchildren(EFS)

Mean age atmarriage

FS ratioWF/RC

EFS ratioWF/RC

Marriageage ratioWF/RC

Wave front (WF) 2663 9.1 4.9 20.51.15*** 1.20*** 0.95***

Range core (RC) 1783 7.9 4.1 21.6***, t test of difference between means; P < 0.001

Table 1. Genetic contribution (GC) of ancestors having lived in the ChSLSJ region to individuals from the 1931 to 1960 generation found anywhere in theQuebecprovince. Note that GCs of different generations are not independent.

Wave front (WF) Range core (RC)Generation

Total GCNo. of

ancestors ingenealogy

Mean GC Total GCNo. of

ancestors ingenealogy

Mean GCAncestors on wave

front (%)Mean GC

ratio (WF/RC)

ChSLSJ1686–1720 19298 48 402.0 612 6 102.0 88.9 3.94***1721–1750 19263 104 185.2 16833 106 158.8 49.5 1.17*1751–1780 22119 196 112.9 25990 373 69.7 34.4 1.62***1781–1810 21696 364 59.6 35613 1069 33.3 25.4 1.79***1811–1840 30504 1383 22.1 27061 1815 14.9 43.2 1.48***1841–1870 56589 6555 8.6 10175 2438 4.2 72.9 2.07***1871–1900 40386 8757 4.6 25619 8784 2.9 49.9 1.58***1901–1930 23370 10034 2.3 44408 26255 1.7 27.7 1.38***Total ChSLSJ 27441 40846 40.2

SLSJ1841–1870 27833 3743 7.4 39 15 2.6 99.6 2.8***1871–1900 33917 7300 4.6 15444 4420 3.5 62.3 1.3***1901–1930 21061 8832 2.4 35777 19726 1.8 30.9 1.3***Total SLSJ 19875 24161 45.1

*, P < 0.05 ***, P < 0.001

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À ce taux, il faudrait≈ 665 ans pour que la descendance d’une personne≈ la population totale de la Terre

Science (2011) 334: 1148-1150

• Quand les populations sont basses par rapport à la capacité de support, la compétition pour les ressources est faible

• La reproduction augmente et la mortalité diminue

• Quand les populations approchent la capacité de support, la reproduction diminue et la mortalité augmente

Réponses démographiquesDensité dépendance

o N P 0

I I Opinion is intended to facilitate communication between reader and author and reader and N reader. Comments, viewpoints or suggestions arising from published papers are welcome.

Discussion and debate about important issues in ecology, e.g. theory or terminology, may I I also be included. Contributions should be as precise as possible and references should be

P 0 kept to a minimum. A summary is not required. o N

Population regulation: a synthetic view

Peter Turchin, Dept of Ecology and Evolutionary Biology, Univ. of Connecticut, Storrs, CT 06269-3043, USA ([email protected]).

Population ecologists continue to debate population

regulation. If anything, the controversy intensified dur-

ing the last decade. Does it mean that our field has not

progressed very far since the days of the "great debate"

between Nicholson and Andrewartha? Three years ago

I suggested that in actuality the broad outlines of a

consensus were emerging (Turchin 1995). What I have

read in the literature since then has only confirmed me

in the opinion that most population ecologists are in

agreement on the major, strategic issues in population

regulation, while the ongoing debate increasingly fo-

cuses on narrow tactical questions. This does not mean,

however, that the intensity of the debate, as well as the

vituperance, has diminished (for some quotes see

Turchin 1995).

What are these broad issues of agreement? Here is

my attempt to formulate them:

1. The central quantity of interest in the analyses of

population regulation is the realized per capita rate

of population change, defined as rt = In Nt/N -_, = ln N - ln N__ l, where In Nt is the natural logarithm of population density at time t.

2. The realized per capita rate of change, rt, is affected by both exogenous and endogenous factors. (Exoge-

nous factors are those that affect population change, but are not themselves affected by population num-

bers. In other words, there is no dynamic feedback

between an exogenous factor and population den-

sity. By contrast, endogenous factors represent dy-

namical feedbacks affecting population numbers,

possibly involving time lags.) Exogenous factors are

not "noise" to be tuned out. They represent important biological processes affecting population

change, and are a legitimate and interesting subject

for study.

3. Some negative feedback between rt and population density (that is, density dependence) is a necessary

(but not sufficient) condition for population

regulation.

4. Population dynamics are inherently nonlinear. A

wide variety of functional relationships between the

expected (or average) r, and population density is possible, both in theory and in practice (see Fig. 1).

The relationship is often monotonic - either convex

or concave - but not necessarily so. There may be

an Allee effect (in which case the rate of change first

increases and then decreases with density), or even

more complex relationships leading to metastable

dynamics. For some ranges of density the expected

per capita rate of change may be flat, with popula-

tion dynamics dominated by exogenous factors (the

so-called "density-vagueness").

5. Rate of population change may be affected not only

by the current population density, but also by

lagged density. In some cases the lag structure of

population regulation is quite complex, with more

than one time delays involved in regulation (Fig.

Id).

6. Finally, a focus on testing null hypotheses against

unspecified alternatives has proved to be unproduc-

tive in investigations of population regulation. In

other words, the interesting question is not whether

we can reject the hypothesis that a population is

"unregulated". A much more fruitful approach, for

example, in the analysis of time-series data is to

estimate the relative strengths of exogenous versus

endogenous contributions to population change, the

lag structure of regulation, and the shapes that

characterize the functional relationship between r, and lagged population densities. Ultimately, we

need to determine which of the alternative ecological mechanisms explains population regulation in

any particular case study, or perhaps what are the

relative contributions of several ecological factors to

regulation.

To summarize, unlike the controversies in the past, the

modern view of population dynamics is synthetic and

OIKOS 84:1 (1999) 153

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Densité dépendanceDonnées empiriques

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Oikos (1999) 84: 153-159

Capacité de support

• K (carrying capacity)

• Le point d’équilibre où la mortalité et la natalité sont égaux, et donc la taille de la population est stable

• λ = 1 (r = 0)

• La capacité de support est variable et dépend de la saison, des ressources, des années, etc.

DéfinitionCapacité de support

• Une ressource essentielle pour la croissance

• Élément nutritif, eau, lumière, espace, proie, etc.

• Une ressource consommée, enlevée de l’environnement

• Un photon capté par une plante n’est plus disponible pour sa voisine

• Une souris consommée par une couleuvre n’est plus disponible pour sa voisine

Facteurs limitantsCapacité de support

Capacité de supportLoi du minimum

Philipp Karl Sprengel(1787-1859)

Justus von Liebig(1803-1873)

• La croissance des plantes n’est pas limitée par la quantité totale de ressources, mais bien par la ressource la plus rare (le facteur limitant)

• La population augmentera jusqu’à ce qu’une ressource devienne limitante

• La loi a aussi été appliquée aux populations animales et aux écosystèmes

Loi du minimumCapacité de support

Loi du minimumComposition chimique d’une plante aquatique

Élément Plante Eau Ratio0 80,5 89 0,90H 9,7 11 0,88C 6,5 0,0012 5 417Si 1,3 0,00065 2 000N 0,7 2,3E-05 30 435

Ca 0,4 0,0015 267P 0,08 1E-06 80 000

Mg 0,07 0,0004 175S 0,06 0,0004 150Cl 0,06 0,0008 75Cu 0,0001 1E-06 100Co 2E-06 5E-08 40

• Pourquoi est-il peu probable que O et H soient limitants?

• Pourquoi les engrais commerciaux contiennent-ils plus d’un nutriment?

• Quels sont les deux nutriments principaux inclus dans les engrais commerciaux?

QuestionsFacteurs limitants

• Si une ressource est limitante, son ajout devrait produire

• Une augmentation du taux de croissance

• Une augmentation de la biomasse

• Un agrandissement de la répartition géographique

ÉvidenceFacteurs limitants

Facteurs limitantsÉvidence que le CO2 est limitant?

Taux

de

phot

osyn

thès

e

0

20

40

60

80

100

Intensité lumineuse (W/m carré)0 250 500 750 1000

[300] ppm[600] ppm

Modèle de croissance logistique

• Croissance exponentielle

• r = constant

• Croissance logistique

• r = f(N)

Croissance exponentielle et logistiqueCroissance de la population

Croissance de la populationModèle de croissance exponentiel

Taux

(Mor

talit

é ou

Nat

alité

)

0,00

0,20

0,40

0,60

0,80

1,00

Taille de la population (N)

0 100 200 300 400 500

Mortalité Natalité

r

Croissance de la populationModèle de croissance logistique

Taux

(Mor

talit

é ou

Nat

alité

)

0,00

0,20

0,40

0,60

0,80

1,00


0 100 200 300 400 500

Mortalité Natalité

K

r

Croissance de la populationModèle de croissance logistique

r

-1,00-0,80-0,60-0,40-0,200,000,200,400,600,801,00


0 100 200 300 400 500

K

Modèle de croissance logistiqueCroissance continue

dN = rN 1 - NKdt

x e-rt

Nt = K1 + - 1K

N0

Applications

ApplicationsModèle d’exploitation logistique

Taille

de

la p

opul

atio

n (N

)

0

200

400

600

800

1000

Temps (t)

0 10 20 30 40 50

r = 0,30

K = 1000

Δt

ΔN

ApplicationsModèle d’exploitation logistique

ΔN

/Δt

0

20

40

60

80


0 200 400 600 800 1000

r = 0,30

K = 1000rK/4

K/2

• Le rendement continuel maximum

• Population = K/2

• Nombre d’individus = rK/4

Modèle d’exploitation logistiqueApplications

Journal of the Fisheries Research Board of Canada (1967) 24: 145-190

ApplicationsMatrice de transition (matrice de Leslie)

Nouveau-né Juvénile Adulte 1 Adulte 2

Nouveau-né 0 0 4,5 6,5

Juvénile 0,67 0,10 0 0

Adulte 1 0,61 0,25 0

Adulte 2 0,64 0,89

fϕ

ApplicationsTableau de vie

Âge Survie annuelle (Sx)

Abondance (Nx+1 = SxNx)

Survie par âge (lx = Nx/N0)

Fécondité (mx) lxmx

e-rx (r = 0,20) lxmxe-rx

0 100 1,00 0 0,00 1,00 0,00

1 0,19 19 0,19 2 0,38 0,82 0,31

2 0,69 13 0,13 3 0,39 0,67 0,26

3 0,62 8 0,08 4 0,33 0,55 0,18

4 0,73 6 0,06 4 0,24 0,45 0,11

5 0,60 4 0,04 3 0,11 0,37 0,04

6 0,54 2 0,02 2 0,04 0,30 0,01

Influence of growth and survival costs of

reproduction on Atlantic cod, Gadus morhua,

population growth rate

Jeffrey A. Hutchings

Abstract: A stochastic, age-structured life history model was used to examine how age at maturity (θ), pre- (Zimm) andpostreproductive (Zmat) mortality, and postreproductive growth rate can affect maximum reproductive rates of fish atlow population size. Simulations suggest that annual (r) and per-generation (R0) metrics of population growth forNewfoundland’s northern Grand Bank Atlantic cod, Gadus morhua, are primarily influenced by changes to mortalityprior to and following reproduction. At observed weights at age and Zmat = 0.2, r ranged between 0.135 and 0.164 forcod maturing at between 4 and 7 years. Incremental increases in either Zimm or Zmat of 0.1 were associated with 0.03–0.05 reductions in r. To effect similar reductions, individual growth rate would have to decline by approximately onehalf. At observed weights at age, increases in Zmat from 0.20 to 0.45 increased the probability of negative per-generation growth from 3 to 26% for cod maturing at 4 years and from 6 to 46% for cod maturing at 7 years. Thus,even in the absence of fishing mortality, little or no population growth by Atlantic cod may not be unexpected in thepresence of environmental stochasticity, particularly when accompanied by increases in mortality and decliningindividual growth.

Résumé : Un modèle stochastique du cycle vital selon l’âge a été utilisé pour étudier de quelle manière l’âge à lamaturité (θ), la mortalité avant la reproduction (Zimm) et après la reproduction (Zmat), et le taux de croissance après lareproduction peuvent influer sur le taux maximal de reproduction des poissons d’une population peu abondante. Selonles simulations, les mesures annuelles (r) et les mesures par génération (R0) de la croissance de la population de lamorue (Gadus morhua) de la partie nord du Grand Banc de Terre-Neuve sont surtout influencées par des modificationsde la mortalité avant et après la reproduction. Pour les poids selon l’âge observés et une valeur de Zmat = 0,2, r étaitcompris entre 0,135 et 0,164 dans le cas de la morue qui atteint la maturité entre 4 et 7 ans. Les augmentations paréchelon de 0,1 de Zimm ou de Zmat étaient associées à des baisses du facteur r comprises entre 0,03 et 0,05. Pourobtenir des baisses similaires, le taux de croissance individuel devrait chuter d’environ la moitié. Pour des valeursobservées de poids selon l’âge, des hausses de Zmat variant de 0,20 à 0,45 augmentaient la probabilité d’une croissancepar génération négative de 3 à 26 % chez la morue qui atteint la maturité à l’âge de 4 ans, et de 6 à 46 %, chez cellequi atteint la maturité à l’âge de 7 ans. En conséquence, même en l’absence de mortalité due à la pêche, unecroissance faible, voire nulle, de la population de morue en présence de la stochasticité environnementale n’est pas àexclure, en particulier lorsqu’elle est accompagnée d’une augmentation de la mortalité et d’une baisse de la croissanceindividuelle.

[Traduit par la Rédaction] Hutchings 1623

Introduction

The parameter of greatest import in population dynamics,evolutionary ecology, and evolutionary biology is rate ofincrease. The fitness of a genotype, for example, is definedas the rate at which it transmits its genes to future genera-tions relative to that of other genotypes in the same popula-tion (Stearns 1992). From an evolutionary perspective, Bell(1997) has argued that rate of self-replication is the onlyattribute that can be selected directly. And the ability of apopulation to persist in the face of demographic and envi-ronmental stochasticity in survival and fecundity appears to

be best reflected by that population’s rate of increase (Lande1993). This point is of particular importance to commer-cially harvested populations for which spatial and temporalvariation in natural mortality is compounded by spatial andtemporal variation in mortality due to harvesting.From a conservation perspective, quantifying rate of in-

crease is perhaps most important when a population hasbeen reduced to a small fraction of the size at which it waspresumed stable. For commercially harvested fish popula-tions, the ability of a stock to “recover,” i.e., return to someusually arbitrary level of abundance after population col-lapse, and the time required for such recovery to occur areboth functions of that stock’s rate of increase at low abun-dance.The most widely accepted metrics of maximum popula-

tion growth rate are the instantaneous, or intrinsic, rate ofnatural increase, given as r, and the net reproductive rate,R0 (Roff 1992; Stearns 1992; Charlesworth 1994). The

Can. J. Fish. Aquat. Sci. 56: 1612–1623 (1999) © 1999 NRC Canada

1612

Received March 11, 1998. Accepted May 19, 1999.J14488

J.A. Hutchings. Department of Biology, DalhousieUniversity, Halifax, NS B3H 4J1, Canada.e-mail: [email protected]

J:\cjfas\cjfas56\CJFAS-09\F99-088.vpTuesday, August 24, 1999 1:39:20 PM

Color profile: DisabledComposite Default screen

Influence of growth and survival costs of

reproduction on Atlantic cod, Gadus morhua,

population growth rate

Jeffrey A. Hutchings

Abstract: A stochastic, age-structured life history model was used to examine how age at maturity (θ), pre- (Zimm) andpostreproductive (Zmat) mortality, and postreproductive growth rate can affect maximum reproductive rates of fish atlow population size. Simulations suggest that annual (r) and per-generation (R0) metrics of population growth forNewfoundland’s northern Grand Bank Atlantic cod, Gadus morhua, are primarily influenced by changes to mortalityprior to and following reproduction. At observed weights at age and Zmat = 0.2, r ranged between 0.135 and 0.164 forcod maturing at between 4 and 7 years. Incremental increases in either Zimm or Zmat of 0.1 were associated with 0.03–0.05 reductions in r. To effect similar reductions, individual growth rate would have to decline by approximately onehalf. At observed weights at age, increases in Zmat from 0.20 to 0.45 increased the probability of negative per-generation growth from 3 to 26% for cod maturing at 4 years and from 6 to 46% for cod maturing at 7 years. Thus,even in the absence of fishing mortality, little or no population growth by Atlantic cod may not be unexpected in thepresence of environmental stochasticity, particularly when accompanied by increases in mortality and decliningindividual growth.

Résumé : Un modèle stochastique du cycle vital selon l’âge a été utilisé pour étudier de quelle manière l’âge à lamaturité (θ), la mortalité avant la reproduction (Zimm) et après la reproduction (Zmat), et le taux de croissance après lareproduction peuvent influer sur le taux maximal de reproduction des poissons d’une population peu abondante. Selonles simulations, les mesures annuelles (r) et les mesures par génération (R0) de la croissance de la population de lamorue (Gadus morhua) de la partie nord du Grand Banc de Terre-Neuve sont surtout influencées par des modificationsde la mortalité avant et après la reproduction. Pour les poids selon l’âge observés et une valeur de Zmat = 0,2, r étaitcompris entre 0,135 et 0,164 dans le cas de la morue qui atteint la maturité entre 4 et 7 ans. Les augmentations paréchelon de 0,1 de Zimm ou de Zmat étaient associées à des baisses du facteur r comprises entre 0,03 et 0,05. Pourobtenir des baisses similaires, le taux de croissance individuel devrait chuter d’environ la moitié. Pour des valeursobservées de poids selon l’âge, des hausses de Zmat variant de 0,20 à 0,45 augmentaient la probabilité d’une croissancepar génération négative de 3 à 26 % chez la morue qui atteint la maturité à l’âge de 4 ans, et de 6 à 46 %, chez cellequi atteint la maturité à l’âge de 7 ans. En conséquence, même en l’absence de mortalité due à la pêche, unecroissance faible, voire nulle, de la population de morue en présence de la stochasticité environnementale n’est pas àexclure, en particulier lorsqu’elle est accompagnée d’une augmentation de la mortalité et d’une baisse de la croissanceindividuelle.

[Traduit par la Rédaction] Hutchings 1623

Introduction

The parameter of greatest import in population dynamics,evolutionary ecology, and evolutionary biology is rate ofincrease. The fitness of a genotype, for example, is definedas the rate at which it transmits its genes to future genera-tions relative to that of other genotypes in the same popula-tion (Stearns 1992). From an evolutionary perspective, Bell(1997) has argued that rate of self-replication is the onlyattribute that can be selected directly. And the ability of apopulation to persist in the face of demographic and envi-ronmental stochasticity in survival and fecundity appears to

be best reflected by that population’s rate of increase (Lande1993). This point is of particular importance to commer-cially harvested populations for which spatial and temporalvariation in natural mortality is compounded by spatial andtemporal variation in mortality due to harvesting.From a conservation perspective, quantifying rate of in-

crease is perhaps most important when a population hasbeen reduced to a small fraction of the size at which it waspresumed stable. For commercially harvested fish popula-tions, the ability of a stock to “recover,” i.e., return to someusually arbitrary level of abundance after population col-lapse, and the time required for such recovery to occur areboth functions of that stock’s rate of increase at low abun-dance.The most widely accepted metrics of maximum popula-

tion growth rate are the instantaneous, or intrinsic, rate ofnatural increase, given as r, and the net reproductive rate,R0 (Roff 1992; Stearns 1992; Charlesworth 1994). The

Can. J. Fish. Aquat. Sci. 56: 1612–1623 (1999) © 1999 NRC Canada

1612

Received March 11, 1998. Accepted May 19, 1999.J14488

J.A. Hutchings. Department of Biology, DalhousieUniversity, Halifax, NS B3H 4J1, Canada.e-mail: [email protected]

J:\cjfas\cjfas56\CJFAS-09\F99-088.vpTuesday, August 24, 1999 1:39:20 PM

Color profile: DisabledComposite Default screen

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