The Markovian Patch- Occupancy (MPO) framework in Omri Allouche Prof. Ronen Kadmon Sep 2008...

The Markovian Patch-Occupancy (MPO) framework

in

Omri Allouche

Prof. Ronen KadmonSep 2008

Community Ecology

Photos taken from Life, the science of biology, WH Freeman

Semi-universal patterns

Disturbance level:LowMediumHigh

Habitat diversity

Habitat loss

ProductivityDisturbanceProductivity

Isolation

Regional diversity

Area

a b c

d e

f g h

Dis

turb

ance

Spec

ies

div

ersi

tySp

ecie

s d

iver

sity

Spec

ies

div

ersi

ty

Saturated

Linear


Habitat diversity

Habitat loss


Isolation

Regional diversity

Area

a b c

d e

f g h

Dis

turb

ance

Spec

ies

div

ersi

tySp

ecie

s d

iver

sity

Spec

ies

div

ersi

ty

Saturated

Linear



Habitat diversity

Habitat loss


Isolation

Regional diversity

Area

a b c

d e

f g h

Dis

turb

ance

Spec

ies

div

ersi

tySp

ecie

s d

iver

sity

Spec

ies

div

ersi

ty

Saturated

Linear

Habitat diversity

Habitat loss


Isolation

Regional diversity

Area

a b c

d e

f g h

Dis

turb

ance

Spec

ies

div

ersi

tySp

ecie

s d

iver

sity

Spec

ies

div

ersi

ty

Habitat diversityHabitat diversity

Habitat lossHabitat loss

ProductivityProductivityDisturbanceDisturbanceProductivityProductivity

IsolationIsolation

Regional diversityRegional diversity

AreaArea

a b c

d e

f g h

Dis

turb

ance

Spec

ies

div

ersi

tySp

ecie

s d

iver

sity

Spec

ies

div

ersi

ty

Saturated

Linear

The AimDevelop a general framework for modeling

ecological communities:

• Elementary community dynamics

• Simplified representation of communities

• Analytic solution

• Able to qualitatively produce known patterns of species-diversity

• Useful for the study of complex ecological phenomena

The importance of Demography

• Species richness is determined by species extinctions and colonizations

• Species extinctions and colonizations result from actions of individuals

• The basic elements of ecological communities are individuals

• Individuals go through demographic processes of birth, death and migration

The MPO Framework

• Individual-based

• Island receiving immigrants from a mainland

• Multiple species

• Basic demographic processes:

Local Reproduction

Mortality

Immigration + Emigration

General Framework for Modeling Ecological

Communities

• In a small-enough time interval only a single

event can take place

• A Markov chain:

• Implicit space - Global dispersal

• We wish to find the steady state probability of

1( ,..., )SN N N

1 1 0 0 1 1{ | ,..., } { | }n n n n n n n nP X i X i X i P X i X i

The MPO Framework

dt

Mortality

Emigrationk

Nr

Decrease in the number of

individuals of species k

Possible events:

Local reproduction

Immigration from the mainland

kN

g

Increase in the number of

individuals of species k

1

( ) M

k k

Sk k k k

k kN e N e N Nk

dP NP N e r P N e g P N g r

dt

1 if =

0 otherwisek l

k le

The Master Equation:

The transition rates are highly flexible, and can

incorporate complex ecological phenomena.

We only require that:

This holds for many interesting cases, and can

be easily checked for any transition rates.

i k i k k

i

i k

k i i kN e N e e N e N e e

k i kN N e N N e

g r g r

g r g r

Analytic Solution

The steady-state probability

of state

is given by:

Analytic Solution

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

Abundance of species 1 (N1)

Abu

ndan

ce o

f sp

eci

es

2 (N

2)

1

( ) ( ) ( )MPON

P N X N X N

{1,..., 1}

{1,..., 1}

1

1 0 ( 1)

( )kM

k k

k k

kNSN m e

kk m N m e

gX N

r

((0,0,..., 0)) 1X {1,..., }

if k n

0 otherwisek

n

NN

1( ,..., )MSN N N

{1,..., 1}

{1,..., 1} {1,..., 1}

{1,..., 1}{1,..., 1}

{1,..., 1}

1( 1)

1 0( 1)

( 1)

( )

( )

i k i

iMk k k i k i

i ik k k

i i

iN e m e

k iNSN N e N e m eMPO k

iki k m N m eMPO N N e

iN m e

g

g rP N egrP N

r

{1,..., } {1,..., 1} {1,..., 1}

{1,..., } {1,..., 1} {1,..., 1}

1( 1)

1 0 ( 1)

iMk i k i i i

k k i i i k i

k i iNSN N e m e N m e

k i ii k mN e N m e N e m e

g g r

r g r

( ) ( )a b b aP a P P b P

““detailed balance”detailed balance” ( ) ( )k

k kMPO MPO kN N e

P N g P N e r

{1,..., 1} {1,..., 1} {1,..., 1} {1,..., 1}

{1,..., 1} {1,..., 1} {1,..., 1} {1,...,

1( 1) ( 1)

1 0 ( 1)

iMi i k i i i i i i k

i i i i k i i i

i i k kNSN m e e N m e N m e N m e e

i i ki k m N m e N m e e N m e N

g r g r

g r g r

1}

{1,..., 1} {1,..., 1} {1,..., }

{1,..., 1} {1,..., 1} {1,..., }

1

1 0 ( 1)

1

iM

i k

Mi i i i k k k

i i i i k k k

NS

ki k m m e e

k k kkSN N e N e N eN

k k k ki k N N N e e N N e

g r rg

g r g r

Analytic Solution

Steady-state distribution:

Abundance distribution:

:

( ) ( )k

localk PO

N N n

P n P N

1

(1 (0))MS

localk

k

SR P

Species Richness:

1

( ) ( ) ( )MPON

P N X N X N

{1,..., 1}

{1,..., 1}

1

1 0 ( 1)

( )kM

k k

k k

kNSN m e

kk m N m e

gX N

r

:

( ) ( )kk

PON N J

p J P N

Community size distribution:

• Island of area A

• Multiple species

• Rates of:

Birth bk

Death dk

Immigration ik

• Relative regional abundance

• Individuals only establish in vacant sites

Multispecies community with competition for space

k k

A Jb N dt

A

k kd N dt

( )regk ki P A J dt

Local reproduction:

Mortality:

Immigration from the regional pool:

kk kN

r d N

kN

regk k k k

g

A Jb N i P A

A

1

1( )

!

kM

k

regNS k k NJ kJ

k k k

PA J bX N

A d N

The solution:

1

0

( )y

yi

x x i

kk

k

iA

b

0 20 40 60 80 1000

20

40

60

80

100

Habitat loss)%(

Spe

cies

ric

hnes

s

1.535

10

Reproduction

AD = The number of destroyed sites

Habitat Loss“The greatest existing

threat to biodiversity“

( )k regk k k

DkN

A Jg b N i P A

A

A

kk kN

r d N

kk

k

iA

b

1

1( ) ,

!

kM

k

regNS k k ND J kJ

k k k

PA A J bX N

A d N

1. An island consisting of

A sites, divided among

H habitats

2. Each species is able to

establish and persist in

only one habitat

3. Individuals disperse

and immigrate to

random sites

Habitat Heterogeneity and niche partitioning

Habitat Heterogeneity

2 4 6 8 100

20

40

60

80

100

120

140

Heterogeneity )H(

Sp

ecie

s ri

chne

ss

Reproduction = 1.5

3

510

Kadmon and Allouche 2007 Am Nat

( ) k kk regk k k k

H H

N

Ag b N

Ji

AP A

kk kN

r d N

kHA

kHJ

total area

1 1

1( ) 1

!

kM

k

h

regNSH k k Nkh hJ J

h k k k

PbX N A J

A d N

kk

k

iA

b

and H is the total number of habitats

community size

in habitat of species k

Habitat PreferenceOffspring arrive in suitable habitats

more than in unsuitable habitatsdue to active site selection or environmental

autocorrelation and limited dispersal

(( )

)k

k

k

k

k

k

H Hk regk k k kN

H

k H

k H H

A Jg b N i

v A

v A AP

A AA

kk kN

r d N

1 1

( ) 1( 1) !

kkM

k

h

k

N regNSH k k Nk kh h J

h k k H k k

Pb vX N A J

d A v A N

kk

k

iA

b

Non-random Dispersal

Offspring tend to arrive to vacant sites more than to occupied sites

( )

( )( )k reg

k k k kN

v A J

v A J Jg b N i P A

kk kN

r d J

1

1( )

!

kM

k

regNSk k NJJ k

k k kJ

PA J bX N

A d N

1

v

v

k

kk

iA

b

Productivity

R = Productivity0.2 0.4 0.6 0.8 1

40

60

80

100

Productivity (R)S

pec

ies

rich

nes

s

b=1.5

5

10

20

( )k regk k k kN

A Jg b N i P A

AR

kk kN

r d J

1

1( )

!

kM

k

regNS k k NJ kJ

k k k

PA J bX N

A d N

1

0

( )y

yi

x x i

kk

k

iA

b

Increased reproduction due to more resources

Reduced reproduction of small populationsdue to mate finding, predation, aggregation, environmental modification…

( )k regk

kk kN

kk

k

N

N

A Jg b N i P A

A

kk kN

r d N

12

0

1

1( )

!

k

kM

k

Nreg reg

N k k k k kSJ k m

Jk k k k N

m P m PA J b

X Nd NA

k the level of Allee effect for species k ,kk

k

iA

b

Allee Effect

( )k regk k k kN

A Jg b N i P A

A

( ) kk k

kk

kkN

Nd

Kr bd N

1

1( )

! 1

kM

k

k

regNSk k NJ k

Jk k k

kk N

PA J bX N

A dN

, k k kk k

k k

i b dA

b K

Population-Level Density Dependence

Increased mortality of large populations

due to parasites and diseases, predation, environmental modification…

Community-Level Carrying Capacity

k regk k k kN

g b N i P A

kk kN

r dJ

NK

1

( )! !

kM

k

regNSJ k k Nk

k k k

PbKX N

J d N

k

kk

iA

b

Increased mortality of large communitiesdue to parasites and diseases, predation, environmental modification…

5.14

$@#$!

2

ˆ( ) lim( )

_

! 4

! ! 2

kk regk k k kN x

D

Ng b N i P A

Bugs Bunny

A A J n b b acdt

A r n r a

kk kN

r d N

The Meaning of Life

General Framework for Neutral Models

Hubbell’s mainland-island model (Hubbell 2001)

( )k regk kN

A Jg bN iP A

A

kkN

r dN

b dA

i dA ( 1)

iAm

iA b A

where:

Independent neutral species (Volkov et al. 2003, 2005 Nature, He 2005 Func Ecol., Etienne et al. 2007 JTB)

k regk kN

g bN iP

kkN

r dN

Community-Level Density-Dependence (Haegeman & Etienne 2008 JTB)

( ) ( )k regk kN

bg N PJ i J

( )kkN

r d J N

The MPO Framework - Summary

• General framework for modeling ecological

communities

• Individual-based

• Basic demographic processes

• Demographic differences among species

• Analytically tractable

• A general framework for neutral null models

• Extends patch-occupancy theory

The MPO Framework

• Highly flexible

• Useful for the study of complex ecological phenomena


Habitat diversity

Habitat loss


Isolation

Regional diversity

Area

a b c

d e

f g h

Dis

turb

ance

Spec

ies

div

ersi

tySp

ecie

s d

iver

sity

Spec

ies

div

ersi

tySaturated

Linear


Habitat diversity

Habitat loss


Isolation

Regional diversity

Area

a b c

d e

f g h

Dis

turb

ance

Spec

ies

div

ersi

tySp

ecie

s d

iver

sity

Spec

ies

div

ersi

tySaturated

Linear



Habitat diversity

Habitat loss


Isolation

Regional diversity

Area

a b c

d e

f g h

Dis

turb

ance

Spec

ies

div

ersi

tySp

ecie

s d

iver

sity

Spec

ies

div

ersi

tySaturated

Linear

Habitat diversity

Habitat loss


Isolation

Regional diversity

Area

a b c

d e

f g h

Dis

turb

ance

Spec

ies

div

ersi

tySp

ecie

s d

iver

sity

Spec

ies

div

ersi

ty

Habitat diversityHabitat diversity

Habitat lossHabitat loss

ProductivityProductivityDisturbanceDisturbanceProductivityProductivity

IsolationIsolation

Regional diversityRegional diversity

AreaArea

a b c

d e

f g h

Dis

turb

ance

Spec

ies

div

ersi

tySp

ecie

s d

iver

sity

Spec

ies

div

ersi

tySaturated

Linear

• Able to qualitatively produce leading patterns of species- diversity

• Useful for other fields

(???)

The Markovian Patch- Occupancy (MPO) framework in Omri Allouche Prof. Ronen Kadmon Sep 2008...

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