Bayesian(inference(of( biogeographical(histories(for ......Figure 2: Cartoon of the computation of the distance-dependent dispersal rate-modiﬁer, ⌘(·). Here, we are interested

Bayesian inference of biogeographical histories for hundreds of discrete areas

Michael Landis Nick Matzke Brian Moore

John Huelsenbeck

Evolu>on 06/23/13

[email protected]

Biogeography

“Every species has come into existence coincident both in space and 5me with a pre-‐exis5ng closely allied species.”

AR Wallace, 1855

Con>nental-‐scale biogeography

Octodon degus Photo by José Cañas

(Mol Phylo Evol 2012)

Cuniculus taczanowskiiCuniculus pacaGalea musteloidesCavia apereaCavia porcellusCavia tschudiiMicrocavia australisDolichotis patagonumKerodon rupestrisHydrochoerus hydrochaerisDasyprocta leporinaMyoprocta acouchyCoendou bicolorErethizon dorsatumSphiggurus melanurusChinchilla lanigeraLagidium viscaciaLagostomus maximusDinomys branickiiEuryzygomatomys spinosusClyomys laticepsTrinomys setosusTrinomys paratusTrinomys eliasiTrinomys yonenagaeTrinomys iheringiTrinomys dimidiatusCapromys piloridesMyocastor coypusThrichomys apereoidesHoplomys gymnurusProechimys quadruplicatusProechimys simonsiProechimys longicaudatusProechimys robertiKannabateomys amblyonyxDactylomys boliviensisDactylomys dactylinusLonchothrix emiliaeMesomys hispidusMesomys occultusEchimys chrysurusToromys grandisPhyllomys blainvilliiMakalata didelphoidesPhyllomys brasiliensisMakalata macruraIsothrix barbarabrownaeIsothrix bistriataIsothrix sinnamariensisCtenomys steinbachiCtenomys boliviensisCtenomys haigiTympanoctomys barreraePipanacoctomys aureusOctomys mimaxSpalacopus cyanusAconaemys fuscusAconaemys sageiAconaemys porteriOctodon degusOctodon lunatusOctodon bridgesiOctodontomys gliroidesAbrocoma bennettiiAbrocoma cinerea

A B C D E F G H I

A

B

C

D

E

FG

HI

Supplemental Figure 1

a) b)

Global-‐scale biogeography

For 8 areas

For 80 areas

For 800 areas

For 8 zillion areas

Cuniculus taczanowskiiCuniculus pacaGalea musteloidesCavia apereaCavia porcellusCavia tschudiiMicrocavia australisDolichotis patagonumKerodon rupestrisHydrochoerus hydrochaerisDasyprocta leporinaMyoprocta acouchyCoendou bicolorErethizon dorsatumSphiggurus melanurusChinchilla lanigeraLagidium viscaciaLagostomus maximusDinomys branickiiEuryzygomatomys spinosusClyomys laticepsTrinomys setosusTrinomys paratusTrinomys eliasiTrinomys yonenagaeTrinomys iheringiTrinomys dimidiatusCapromys piloridesMyocastor coypusThrichomys apereoidesHoplomys gymnurusProechimys quadruplicatusProechimys simonsiProechimys longicaudatusProechimys robertiKannabateomys amblyonyxDactylomys boliviensisDactylomys dactylinusLonchothrix emiliaeMesomys hispidusMesomys occultusEchimys chrysurusToromys grandisPhyllomys blainvilliiMakalata didelphoidesPhyllomys brasiliensisMakalata macruraIsothrix barbarabrownaeIsothrix bistriataIsothrix sinnamariensisCtenomys steinbachiCtenomys boliviensisCtenomys haigiTympanoctomys barreraePipanacoctomys aureusOctomys mimaxSpalacopus cyanusAconaemys fuscusAconaemys sageiAconaemys porteriOctodon degusOctodon lunatusOctodon bridgesiOctodontomys gliroidesAbrocoma bennettiiAbrocoma cinerea

A B C D E F G H I

A

B

C

D

E

FG

HI

Supplemental Figure 1

a) b)

13,264 occurrences available (GBIF)

86 occurrences used (Upham & PaYerson, 2012)

Why 9 areas?

Transi>on between two ranges

Ancestral Observed & extant

Founda>onal work: Ree et al. (Evolu5on 2005) Ree & Smith (Syst Biol 2008)

Range

Character

>me

Transi>on probability

QInstantaneous rate matrix

Matrix exponen>a>on accounts for all intermediate events.

For few areas, no problem 3 areas

For more areas, explodes Q

3 areas 10 areas

210 ⇥ 210 = 1024⇥ 1024

Matrix exponen>a>on too slow for more than ten areas.

Download BayArea: bayarea.googlecode.com

Landis et al. (Syst Biol, in press)

BayArea: Method for more areas


Inspired by Robinson et al. (Mol Biol Evol 2003)



Inspired by Robinson et al. (Mol Biol Evol 2003) Key concepts: 1.  Sample biogeographic histories, H



Inspired by Robinson et al. (Mol Biol Evol 2003) Key concepts: 1.  Sample biogeographic histories, 2.  Compute likelihood, L�,H

H



Inspired by Robinson et al. (Mol Biol Evol 2003) Key concepts: 1.  Sample biogeographic histories, 2.  Compute likelihood, 3.  Approximate using

Markov chain Monte Carlo (MCMC)

L�,H

P (�, H | D)

H


1. Sample biogeographic histories, H


Nielsen (Syst Biol 2002)

L�,H2. Compute likelihood,

Range evolu>on events from range : sum of rates leaving prob any event at >me prob next event is

= product of event types & >mes over tree L�,H

ri/rre�rt

j


r =X

rj

P (�, H | D)

L�,Hhigh L�,Hlow

3. Approximate using MCMC P (�, H | D)


Can we infer distance effects?

Distance-‐dependent dispersal model Redistributes the rate of area gain…

Simula>on: 600 areas, 50 replicates, 8 distances


½ ¼ 0 1 2 3 4 6

Nearby

Collapses to “independence” model

Anywhere

BayArea recovers true parameters


00.25

0.51

23

46

0.0035 0.0040 0.0045 0.0050 0.0055 0.0060

Data sim

ulated under distance power, β

Mean posterior of rate of gain, λ1

00.25

0.51

23

46

0.035 0.040 0.045 0.050 0.055 0.060

Data sim


Mean posterior of rate of loss, λ0

Distance effe

cts 0

0.250.5

12

34

6

0 2 4 6

Data sim


Mean posterior of distance power, β

0 ¼ ½ 1 2 3 4 6

Rate of area loss Rate of area gain Distance effects

Bayes factors iden>fy distance effects

0

25

50

75

100

0 0.25 0.5 1 2 3 4 6

Simulation dataset per

% o

f sim

ulat

ions

favo

ring

MD

BFD0 support for MD

Favors M0

Insubstantial

Substantial

Strong

Very strong

Decisive

0

25

50

75

100

0 0.25 0.5 1 2 3 4 6

Simulation dataset per

% o

f sim

ulat

ions

favo

ring

MD

BFD0 support for MD

Favors M0

Insubstantial

Substantial

Strong

Very strong

Decisive

0 ¼ ½ 1 2 3 4 6

100%

0%

50%

25%

75%


Bayes factors support for distance model

% of sim

ula>

ons sup

ported

Malesian Rhododendron Vireya


Re-‐analysis of Webb & Ree (2012) work 65 species, 20 areas




Wallace’s Line

Known dispersal barrier Vireya crossing?




Wallace’s Line

Known dispersal barrier Vireya crossing?

Data from

Brown et al. (J Biogeogr 2012) Webb & Ree (Chapter 8 in Bio5c Evolu5on and Environmental Change in Southeast Asia 2012)

Distance maYers for Vireya dispersal

0.10 0.15 0.20 0.25

Rate of area loss, λ0

0.005 0.015 0.025

Rate of area gain, λ1

-4 -2 0 2 4

Distance power, βRate of area loss Rate of area gain Distance


Prior

Posterior

1.00.50.0Node maps:Posterior probabilityof presence per area

Branches:% of inferred rangeeast of Wallace’s Line

0.0 0.5 1.0

W E

W

E

B)

Wallace’s Line: 3+ crossings West East

Wallace’s Line & Lydekker’s Line: 1 crossing West East

East of Wallace’s Line West of Wallace’s Line

Posterior of ancestral ranges

Phylowood: biogeographic anima>ons

Future direc>ons Rate-‐modifiers for other traits/features Incorpora>ng on paleo-‐etc.-‐ical data Occupancy models to handle “false absences” Specia>on models (allopatry vs sympatry) Adding to RevBayes (easy to develop models)

Summary Allows hundreds of areas for analysis Joint posterior of parameters and ancestral ranges Simple distance-‐dependent dispersal model Efficient model tes>ng framework Open-‐source soqware available

Thanks! Ques>ons?

Contact

[email protected] twiYer.com/landismj

BayArea Biogeography for many areas Nick Matzke bayarea.google.code.com Brian Moore John Huelsenbeck

Phylowood Biogeographic anima>ons

Trevor Bedford mlandis.github.com/phylowood Helpful folks

Bas>en Boussau Tracy Heath Josh Schraiber Sebas>an Höhna

Extra slides

Malesian paleogeography ES42CH10-Lohman ARI 26 September 2011 14:33

Land

Deep seaTrenches

Shallow seaLakes

Volcanoes

Carbonateplatforms

Highlands

110˚E 120˚E 130˚E100˚E 110˚E 120˚E 130˚E100˚E

110˚E 120˚E 130˚E100˚E

10˚S

20˚S

0˚

10˚S

20˚S20˚S

0˚

10˚S

20˚S

0˚

110˚E 120˚E 130˚E100˚E

10˚S

20˚S

0˚

10˚S

20˚S

0˚

10˚S

0˚

a b

c d

e f

60 Mya Paleocene

40 Mya Late Eocene

30 MyaMiddle Oligocene

20 MyaEarly Miocene

10 MyaLate Miocene

5 MyaEarly Pliocene

Figure 2Six Cenozoic reconstructions of land and sea in the Indo-Australian Archipelago.

www.annualreviews.org • Indo-Australian Biogeography 207

Ann

u. R

ev. E

col.

Evol

. Sys

t. 20

11.4

2:20

5-22

6. D

ownl

oade

d fr

om w

ww

.ann

ualre

view

s.org

by U

nive

rsity

of C

alifo

rnia

- B

erke

ley

on 1

2/06

/12.

For

per

sona

l use

onl

y.

Lohman et al. (2011)

We assume constant geography, but…

Vireya results Assume Vireya root age is 55 Mya Ancestral range posterior

Joint WL and LL crossing once ~40 Mya All other WL crossings < 15 Mya

Plausible biogeographical scenario

Single long distance dispersal event around 40 Mya As Sundi and Sahul Shelf converge, repeated short dispersals

Dispersal-‐ex>nc>on model

R(a)Yi,Yj

=

8>>><

>>>:

�0 if Yj,a = 0

�1 if Yj,a = 1

0 if Yi and Yj di↵er at more than one area

0 if Yj = (0, 0, . . . , 0)

iid, Jukes-‐Cantor, forbids ex>nc>on

Rate-‐modified dispersal model

only a single area can be gained or lost. In other words, each row of Q contains up to N positive,

non-zero entries, which correspond to the rates at which any one of the N areas switches between

absent and present (i.e., the N 0 ! 1 and 1 ! 0 positive entries of the row). Additionally, each

row contains a single, negative diagonal entry, which accounts for the time during which no change

in geographic range occurs, defined as Qii = � Pi 6=j Qij , and ensures that each row of of Q sums

to zero. The remaining entries in Q have a value of zero, as they entail an instantaneous change in

geographic distribution involving two or more areas.

We define a distance-dependent dispersal model, MD, where the rate of gaining a particular area

(0 ! 1) depends on the relative proximity of available areas to those currently occupied by a lineage;

that is, the rate of colonizing a nearby area just outside the perimeter of the current geographic

range should be greater than that of colonizing a relatively remote area. The precise nature of the

relationship between geographic distance and dispersal probability might be specified in numerous

ways (see, e.g., Wallace 1887; MacArthur and Wilson 1967; Hanski 1998). Our distance-dependent

model specifies a simple relationship in which the probability of dispersal between two areas is

inversely related to the geographic distance between them.

Let R(a)Yi,Yj

be the rate of change from the geographic range Yi to the geographic range Yj , where

Yi and Yj di↵er only at the single area index a (again, reflecting the fact that this is a one-change-

at-a-time model). Also, let �0 2 ✓ and �1 2 ✓ be the respective rates at which an individual area

is lost or gained within a geographic range, and ⌘(Yi, Yj , a, �) be a dispersal-rate modifier that

accounts for correlative distance e↵ects. We define the instantaneous dispersal rate as

R(a)Yi,Yj

=

8>>>>>>>>>><

>>>>>>>>>>:

�0 if Yj,a = 0

�1⌘(Yi, Yj , a, �) if Yj,a = 1

0 if Yi and Yj di↵er at more than one area

0 if Yj = (0, 0, . . . , 0)

(1)

and the distance-dependent dispersal rate modifier as

⌘(Yi, Yj , a, �) =NX

n=1

1{Yi,n=1}d(Gn, Ga)��

⇥PN

m=1 1{Yj,m=0}PN

m=1 1{Yj,m=0}

⇣PNn=1 1{Yi,n=1}d(Gn, Gm)��

⌘ (2)

7

Per-‐area rate of gain depends on current biogeographical range.

L�,HCompute likelihood,

0111 2 3

0011 2 3

1011 2 3

1011 2 3

Distance-‐dependent dispersal model

Figure 2: Cartoon of the computation of the distance-dependent dispersal rate-modifier, ⌘(·). Here,

we are interested in computing the rate of Yi = (1, 1, 0, 0) transitioning to Yj = (1, 1, 0, 1). The

first term computes the sum of inverse distances raised to the power � between the area of interest

(i.e., 4) and all currently occupied areas (i.e., areas 1 and 2). The second term then normalizes this

quantity by dividing by the sum of inverse distances raised to the power � between all occupied-

unoccupied area-pairs (i.e., the denominator), then multiplying by number of currently unoccupied

areas (i.e., 2, the numerator).

0 01 2

3 40 0

1 2

3 4

0 01 2

3 4

⌘(Yi = (1, 1, 0, 0) ! Yj = (1, 1, 0, 1), a = 4, �) =

d(G1, G4)�� + d(G2, G4)

��

| {z }

⇥ 2

d(G1, G3)�� + d(G2, G3)

��

| {z } + d(G1, G4)�� + d(G2, G4)

��

| {z }

1 1

1 1 1 1

28

Rate-‐modifier

Normaliza>on

BayArea recovers rate of area gain

0 0.25 0.5 1 2 3 4 6

0.0035

0.0040

0.0045

0.0050

0.0055

0.0060

Data simulated under distance power, β

Mea

n po

ster

ior o

f rat

e of

gai

n, λ1

Posterior rate of area gain

True rate

True distance effects


0 0.25 0.5 1 2 3 4 6

0.035

0.040

0.045

0.050

0.055

0.060


Mea

n po

ster

ior o

f rat

e of

loss

, λ0

BayArea recovers rate of area loss

True distance effects

Posterior rate of area loss

True rate


BayArea recovers distance effects

0 0.25 0.5 1 2 3 4 6

02

46


Mea

n po

ster

ior o

f dis

tanc

e po

wer

, β

True value Po

sterior o

f distance effe

cts

True distance effects Landis et al. (Syst Biol, in press)

Documents

Bayesian(inference(of( biogeographical(histories(for ......Figure 2: Cartoon of the computation of the distance-dependent dispersal rate-modiﬁer, ⌘(·). Here, we are interested