Spatial synchrony and extinction risk
in metapopulations:
a spatial “hydra effect”
Jeremy Fox
University of Calgary
dynamicecology.wordpress.com
David Vasseur
Yale University
The “hydra effect”
Metapopulation persistence time
The usual story: intermediate dispersal rates
maximize metapopulation persistence
Indep. patches
(async.)
Coloniz.-extinction
(async.)
“One big patch”
(sync.)
Big patch persistent
Big patch
extinction-prone
Zero/low
Intermediate
Dispersal rate
High
Intermediate dispersal rates maximize
metapopulation persistence
Yaari et al. 2012
Intermediate dispersal maximizes metapopulation persistence
Huffaker 1958
Holyoak and Lawler 1996:
Protist microcosms: a model system for spatial synchrony
Euplotes
patella
Tetrahymena
pyriformis
Prey density (ml-1)
Cyclic dynamics are easily synchronized
(“phase locked”) by dispersal
1500
0
0
72 0
Day
72
• Dispersal rates <0.5%/prey generation can give synchrony
Vasseur & Fox 2009; Fox et al. 2011, unpublished
Spatial synchrony in nature
Measles
Lynx
Gypsy moth
Lemming abundance index
10
Collared lemming
0
1994
1995
1996
1997
1998
1999
Wren
2000
Year
Blasius et al. 1999, Johnson et al. 2006, Rohani et al. 1999, Paradis et al. 2000, Krebs et al. 2002
A puzzle: How are asynchronous colonization-extinction
dynamics possible?
An answer: A spatial hydra effect
Local extinctions are desynchronizing
• Anything that reduces synchrony promotes recolonization,
and thus persistence
• Empirical examples of colonization-extinction
dynamics involve extinction-prone subpopulations
• Empirical examples of synchrony at low dispersal
rates involve persistent subpopulations
An illustration of the spatial hydra effect
• Nicholson-Bailey host-parasitoid model with demogr. stochas.
(Yaari et al. 2012)
• 4 patches
• Global density-independent dispersal of both spp.
after births & deaths
• At end of timestep: random subpop. destruction
800
600
400
200
0
abundance
Host subpopulation
n.h[, 1]
Subpopulation dynamics under low dispersal,
no subpop. destruction
0
10
20
Index
Timestep
30
40
1500
1000
500
0
abundance
Host subpopulation
n.h[, 1]
Subpopulation dynamics under intermediate dispersal,
no subpop. destruction
0
50
100
Index
Timestep
150
4000
3000
2000
1000
0
n.h[, 1]
abundance
Host subpopulation
Subpopulation dynamics under high dispersal,
no subpop. destruction
0
10
20
Index
Timestep
30
40
600
500
400
300
200
100
0
abundance
Host subpopulation
n.h[, 1]
Subpopulation dynamics under high dispersal
with random subpopulation destruction
0
10
20
30
40
Index
Timestep
50
60
Metapopulation
persistence time (mean)
A spatial hydra effect
90
Subpopulation
destruction rate
0
0.025
0.5
0.075
0.1
0
0.0001
0.001
0.01
0.1
Dispersal rate (log scale)
1
Conclusions and future directions
• Hydras are real
Really exists.
• Effect can vary in strength, be swamped by other effects
-Matter & Roland 2010 Proc Roy Soc B
• Biological details only matter via effects on colonization and
extinction rates
Mean metapop. persist. time
Weak spatial hydra effect
800
Stochastic Ricker
Stochastic logistic map
Destruct. rate
0
0.025
0.05
0.075
0.1
0
0
1 0
Dispersal rate
1
Low rates of “stepping stone” dispersal phase lock
entire metapopulations
Mean prey synchrony ±SE
1.8
Moran Disp.
n
n
y
n
n
y
y
y
0.9
0
1
2
3
4
Spatial lag
5
Fox et al. 2011 Ecol. Lett.
0.6
0.4
0.2
0.0
Prey synchrony
0.8
1.0
Even low dispersal rates can rapidly synchronize
cycling populations
0
2
4
6
8
10
12
Dispersal rate (% per event)
Fox et al. unpublished