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Metapopulation and Intertrophic Dynamics
From single species population dynamics
(and how to harvest them) to complex
multi-species (pred-prey) dynamics in time
and space.
900
750
600
450
300
1975 1980 1985 1990 1995 2000
Metapopulation and Intertrophic Dynamics
abiotic factors?
Herons, UK
(density independence)
stability
fluctuations
biotic factors?
(density dependence)
BHT: fig. 10.17
Metapopulation and Intertrophic Dynamics
A+B+C
SdrJylland
Density
A
DK
1950
1975
C
B
2000 1968 1969 1970 1971 1972
Population-level analysis!
Then again … where is the population-level?
Metapopulation and Intertrophic Dynamics
Dispersal – an important population process
Searocket
(Cakile edentula)
BHT: fig. 15.19
Metapopulation and Intertrophic Dynamics
(1) Metapopulations: living in a patchy environment
(2) Intertrophic dynamics: squeezed from above and below
Metapopulation Dynamics
Do animals occupy all suitable habitats within their geographic range?
39 sites
water vole
Slope, vegetation,
heterogeneity
human disturbance
10 core, 15 peripheral & 14 no-visit
Lawton & Woodroffe 1991
Metapopulation Dynamics
Do animals occupy all suitable habitats within their geographic range?
PCA performed
Increase in % grass
core sites
water vole
no-visit sites
reduced colonization rates
Increasing bank angle and structural heterogeneity
55% with suitable habitats ...
...30% lack voles because...
Know your species...!
Lawton & Woodroffe 1991
predation
Metapopulation Dynamics
...and know your landscape!
Hanski & Gilpin 1997
Metapopulation Dynamics
Metapopulation theory
The MacArthur-Wilson Equilibrium theory
Equilibrium “population” of species
(extinction - recolonization)
Metapopulation Dynamics
Metapopulation theory
Metapopulation
Mainland-Island model
(Single-species version of the M-W multi-species model)
Metapopulation Dynamics
Metapopulation theory
Metapopulation
Mainland-Island model
(Single-species version of the M-W multi-species model)
Levins’s metapopulation model
(no mainland; equally large habitat patches)
Metapopulation Dynamics
Metapopulation theory
Levins’s model (equal patch size)
P : fraction of patches occupied
(1-P) : fraction not occupied
m : recolonization rate
e : extinction rate
dP
 mP(1  P)  eP
dt
recolonization – increases with BOTH the no of empty
patches (1-P) AND with the no of occupied patches (P).
extinction – increases with the no of patches prone to
extinction (P).
Metapopulation Dynamics
Metapopulation theory
Levins’s model
P : fraction of patches occupied
(1-P) : not occupied
m : recolonization rate
e : extinction rate
dP
 mP(1  P)  eP
dt
 mP  mP 2  eP
 (m  e) P  mP 2
P


 ( m  e) P 1 

 1  e / m  
Metapopulation Dynamics
Metapopulation theory
Levins’s model
P : fraction of patches occupied
(1-P) : not occupied
m : recolonization rate
e : extinction rate
dP
P


 ( m  e ) P 1 

dt
 1  e / m  
0.4
P
1-e/m
0.3
0.2
0.1
dN
N
time
0
 rN 1  
0
5
10
15
20
dt
 K
Given that (m – e) > 0, the metapop will grow until equlibrium:
dP/dt = 0 => P* = 1 – e/m
(trivial: P* = 0)
Metapopulation Dynamics
Metapopulation theory
NOTE: the metapop persists, stably, as a result of the balance between m
and e despite unstable local populations!
Melitaea cinxia
local patches
the metapop
persists:
ln(1991) = ln(1993)
Hanski et al. 1995
Metapopulation Dynamics
Metapopulation theory
Mainland-Island model
dP
 m(1  P)  eP
dt
Levins
0.4
0.3
M-I
0.2
0.1
0
0
5
10
Levins’s metapopulation model
dP
 mP(1  P)  eP
dt
15
20
Metapopulation Dynamics
Metapopulation theory
Mainland-Island model
Variable patch size
Levins’s metapopulation model
Metapopulation Dynamics
Metapopulation theory
Mainland-Island model
a=0
Variable patch size model
a=
Levins’s metapopulation model
dP
P 
 (1  a ) 
 a
 mP(1  P)

  eP

dt
 (a  P) 
 (1  a ) (a  P) 
Increasing
a, the freq of larger patches decreases
Metapopulation Dynamics
Metapopulation theory
Levins’s model:
dP
 mP(1  P)  eP
dt
dP/dt = 0 => P1* = 1 – e/m
P2* = 0
Melitaea cinxia
Value of
across metapops
Hanski et al. 1995
a!
Hanski & Gyllenberg (1993) Two
general metapopulation models and
the core-satellite species hypothesis.
American Naturalist 142, 17-41
Intertrophic Dynamics
(2) Intertrophic dynamics: squeezed from above and below
(i) Predation on prey are biased
Thomson’s Gazelle
BHT: fig. 8.9
Intertrophic Dynamics
(2) Intertrophic dynamics: squeezed from above and below
(ii) Predators AND prey are also
”squeezed from the side”
mates
territories
There is density dependence (crowding), which may influence or be
influenced buy predation!
Intertrophic Dynamics
Demonstrating the effect of predation is NOT straight forward
Hokkaido
Long winter
DD intense
multiannual cyclic
seasonal fluctuations
Short winter
DD weak
Intertrophic Dynamics
American mat-biol
The Lotka - Volterra model
dN
 rN  a' PN
dt
Alfred J Lotka
(1880-1949)
dP
 fa' PN  qP
dt
Italian mat-phys
% pred fish
40
Vito Volterra
(1860-1940)
0
1912
1916
1920
1924
Intertrophic Dynamics
q: mortality
a': hunting eff.
Predator (P)
The Lotka-Volterra model
dP
 fa’PN - qP
dt
per predator
f: ability to convert
food to offspring
- a’PN
+ fa’PN
r: intrinsic rate of
increase
dN
 rN - a’PN
dt
Prey (N)
Intertrophic Dynamics
isoclines, dN/dt = dP/dt = 0
The Lotka-Volterra model
dP *
 fa’P*N* - qP* = 0 (Predator isocline)
dt
dN *
 rN*- a’P*N* = 0 (Prey isocline)
dt
predator
mortality
=>
offpring/prey
fa’P*N* = qP*
rN* = a’P*N*
=>
N* = q/fa’
P* = r/a’
hunting
effeciency
prey
reproduction
Intertrophic Dynamics
isoclines, dN/dt = dP/dt = 0
The Lotka-Volterra model
P
P*
N
N*
predator
mortality
offpring/prey
P
N* = q/fa’
P*
P* = r/a’
N*
N
hunting
effeciency
prey
reproduction
Intertrophic Dynamics
The Lotka-Volterra model
BHT: fig. 10.2
P
P*
Predator isocline:
Prey isocline:
N*
N
N* = q/fa’
P* = r/a’
Intertrophic Dynamics
Crowding in the Lotka-Volterra model
P
Crowding in predators:
Hunting effeciency
(a’ ) decreases with
increasing P
P*
N
N*
Predator isocline: N* = q/fa’
Prey isocline: P* = r/a’
Intertrophic Dynamics
Crowding in the Lotka-Volterra model
P
Crowding in predators:
Hunting effeciency
(a’ ) decreases with
increasing P
P*
N
N*
Crowding in prey:
Reproduction rate (r )
decreases with increasing N
Predator isocline: N* = q/fa’
Prey isocline: P* = r/a’
Intertrophic Dynamics
Crowding in the Lotka-Volterra model
P
Crowding in predators:
Hunting effeciency
(a’ ) decreases with
increasing P
P*
K
N
N
N*
Crowding in prey:
Reproduction rate (r )
decreases with increasing N
Predator isocline: N* = q/fa’
Prey isocline: P* = r/a’
Intertrophic Dynamics
Crowding in the Lotka-Volterra model
Combining DD in predator and prey
BHT: fig. 10.7
Predator isocline
Prey isocline
Less effecient predator
Predator isocline
Prey isocline
Strong DD in predator
Prey isocline
Predator isocline
The greater the distance from Eq,
the quicker the return to Eq!
Predator isocline: N* = q/fa’
Prey isocline: P* = r/a’
Intertrophic Dynamics
Functional response and prey-switching
Switch of prey
P
eat another
prey
eat this prey
N (this prey)
P
K
N
N (this prey)
Intertrophic Dynamics
Functional response and prey-switching
Switch of prey
P
eat another
prey
eat this prey
N (this prey)
At low N there’s no
effect of predator
P
K
N
N (this prey)
P
N (this prey)
Intertrophic Dynamics
Functional response and prey-switching
Switch of prey
P
eat another
prey
eat this prey
N (this prey)
At low N there’s no
effect of predator
P
P
Independent of prey
(DD still in work)
Degree of DD
determines level
K
N
N (this prey)
N (this prey)
Intertrophic Dynamics
Functional response and prey-switching
BHT: fig. 10.9
Predator isocline
Predator isocline (high DD)
Prey isocline
Stable pattern with prey density below
carrying capacity
Intertrophic Dynamics
Functional response and prey-switching
Many other combinations! Despite initial settings
they all become stable!
Combining DD in predator and prey
Predator isocline
Prey isocline
Less effecient predator
Predator isocline
Prey isocline
Strong DD in predator
Prey isocline
Predator isocline
BHT: fig. 10.7
Intertrophic Dynamics
Crowding in practice
Indian Meal moth
Log density
Heterogeneous media
Structural simple media
time
BHT: fig. 10.4
Intertrophic Dynamics
Crowding in practice
Indian Meal moth
Intrinsic and extrinsic causes of
population cycles (fluctuations)
Heterogeneous media
Structural simple media
Intertrophic Dynamics
Population cycles and their analysis
Intertrophic Dynamics
Lynx – hare interactions
• pattern: the distinct 10-year cycle (hunting data!)
• processes?: obscure!
(4) sunspots
• hypotheses: (1) vegetation-hare (2) hare-lynx
(3) vegetation-hare-lynx
+
lynx
Sunspot
Intertrophic Dynamics
Lynx – hare interactions
• pattern: the distinct 10-year cycle (hunting data!)
• processes?: obscure!
(4) sunspots
• hypotheses: (1) vegetation-hare (2) hare-lynx
(3) vegetation-hare-lynx
-
lynx
Sunspot
Intertrophic Dynamics
Lynx – hare interactions
• pattern: the distinct 10-year cycle (hunting data!)
• processes?: obscure!
(4) sunspots
• hypotheses: (1) vegetation-hare (2) hare-lynx
(3) vegetation-hare-lynx
+
lynx
Sunspot
Intertrophic Dynamics
Lynx – hare interactions: The Kluane Project
Factorial design large-scale
experiment:
(1) control blocks
(2) ad lib supplemental food blocks
(3) predator exclusion blocks
(4) 2+3 blocks
• monitored everything over 15
years (species composition, population
dynamics, life histories ...)
Intertrophic Dynamics
Lynx – hare interactions: The Kluane Project
Hare density
(-pred, + food)
(-pred)
(+food)
(control)
year
• Increased cycle period ...
… but neither food addition
and predator exclosure
prevented hares from cycling Why?
10-fold
• Non-additive
response
Vegetation-hare-predator
Intertrophic Dynamics
Lynx – hare interactions: A spatial perspective
NAO
Open forest
Closed forest
Forest/Grassland
Continental
Atlantic
Pacific
Intertrophic Dynamics
Lynx – hare interactions: the lynx perspective
Kluane indicates that harepredator interactions are
central.
Nt = f(Nt-1,Nt-2,..., Nt-11)!...
… dynamics non-linear!
density
High dependence (80%) on hare density ...
hare
Nt =
lynx
year
f(Nt-1,Nt-2)
increase
f(Nt-1,Nt-2)
decrease
Intertrophic Dynamics
A geographical gradient in rodent fluctuations: a statistical modelling approach
Clethrionomys
Lemmus
Ottar Bjørnstad
27 populations
Microtus
Effect of predators?
Bjørnstad et al. 1995
Hanski et al. 1991
BHT: fig. 15.13
Intertrophic Dynamics
A geographical gradient in rodent fluctuations: a statistical modelling approach
Two hypotheses
Clethrionomys
(1) The specialist predator hypothesis
Lemmus
(predator numerically linked to prey, that is
through reproduction; variations come
from variations in predator efficiency)
delayed effect on prey
Microtus
(2) The generalist predator hypothesis
(more generalist predators in south than
north)
direct effect
on prey
Analysis of prey
population dynamics:
AR(2): Nt = f(Nt-1,Nt-2)
AR(1): Nt = f(Nt-1) efficiency
Bjørnstad et al. 1995
Hanski et al. 1991
BHT: fig. 15.16
no of pred
Intertrophic Dynamics
A geographical gradient in rodent fluctuations: a statistical modelling approach
Ottar analysed 19 time series (>15 years) using
autoregression (AR):
Clethrionomys
Lemmus
17 (89%) time series best described by:
AR(2): Nt = f(Nt-1,Nt-2)
Increasing no of gen pred increases
the direct negative effect on prey
Nt-2
Bjørnstad et al. 1995
Microtus
Nt-1
The generalist predator hypothesis
Metapopulation and Intertrophic Dynamics
Combining metapopulation and predator-prey theory
BHT section 10.5.5
Comins et al. (1992) The spatial dynamics of hostparasitoid systems. Journal of Animal Ecology 61, 735-748
Fagprojekter
(1) Harvesting natural populations
Toke
Niels
(2) Cohort variation and life histories
Mads
(3) Climate and density dependence in
population dynamics
Max 3-4 pax/group
Max 10-15 pp + figs/tabs