Post on 17-Dec-2015
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
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
Disturbance level:LowMediumHigh
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
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
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
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
tySaturated
Linear
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
tySaturated
Linear
Disturbance level:LowMediumHigh
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
tySaturated
Linear
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
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
(???)