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Predator-Prey Cycles and
Effects of Climate Change
Hao Wang
Department of Mathematical and Statistical Sciences
Centre for Mathematical Biology
University of Alberta
CO2
Temperature
Global Climate Change
Hypothesis
Period
Amplitude
Incidence
Predator-Prey Cycles
Population Cycles
Large
groups
Taxon
Testing #
Periodic #
Fraction
Bird
139
18
13%
Mammals
328
109
33%
Fish
129
56
43%
Insects
79
13
16%
Crustaceans
12
6
50%
Gastropods
3
1
33%
Bivalves
3
1
33%
All
694
204
29%
(Bruce Kendall, John Prendergast and Ottar Bjornstad 1998, Ecology Letters, 1: 160-164)
Population Cycles in a Small
Mammal - Lemmings
brown lemming
in Alaska
collared lemming
in Greenland
Empirical Data
lemming (prey) density
stoat (predator) density
(Olivier Gilg, Ilkka Hanski et al 2003, Science 302:866-868)
Lemming Background
The population numbers of lemming fluctuate
greatly over a period of 3-4 years until now.
Ecologists suspect that such oscillations are
controlled by a trophic mechanism.
Their principal summer foods are moss and
vascular plants.
Lemmings are eaten by many animals, like
weasels, arctic fox and a bigger number dies
when they try to cross rivers, lakes or seas.
Trophic Mechanisms
Brown lemmings in Alaska: bottom-up
regulation (P.Turchin et al 2000, Nature).
Collared lemmings in NE Greenland: topdown regulation (Olivier Gilg et al 2003,
Science).
Top-down Regulation
The lemming cycle driven by predators
For collared lemmings in NE Greenland, the
major sources of mortality are predation by
stoats, arctic foxes, snowy owls and long-tailed
skuas. Only the stoat is a specialist.
Lemming-Stoat Delayed Model
lemming
2
dx
x
bx
(1
(x/K
))/(
1
K
)
y
2
2
Dx
dt
modified
logistic
lemming
growth
Holling
Type
III
functional
response
stoat
dy
x2(t
)
exp(
dj) 2 2
y
(t
)d
(x
)y
dt
Dx(t
)
lemming
stoat
growth
hmaturation
witdelay
depende
stoat
death
Hao Wang, John Nagy, Olivier Gilg, and Yang Kuang. The roles of predator maturation delay and functional
response in determining the periodicity of predator-prey cycles, Mathematical Biosciences, Vol. 221: 1-10
(2009)
Functional Response Test
(Olivier Gilg et al 2003, Science)
Predation by stoat is modeled with Holling Type III functional response, which was
used to incorporate a possible "refuge" for the lemming at very low densities. when
lemmings are so dispersed, then they must become very hard to locate for the stoat.
Lemming-Stoat Delayed Model
lemming
2
dx
x
bx
(1
(x/K
))/(
1
K
)
y
2
2
Dx
dt
modified
logistic
lemming
growth
Holling
Type
III
functional
response
stoat
dy
x2(t
)
exp(
dj) 2 2
y
(t
)d
(x
)y
dt
Dx(t
)
lemming
stoat
growth
hmaturation
witdelay
depende
stoat
death
Hao Wang, John Nagy, Olivier Gilg, and Yang Kuang. The roles of predator maturation delay and functional
response in determining the periodicity of predator-prey cycles, Mathematical Biosciences, Vol. 221: 1-10
(2009)
Stoat Maturation Delay
The stoat maturation delay is about 3
months.
The stoat juvenile/maturation death rate
is chosen to be the maximum stoat death
rate, 4/year.
Lemming-Stoat Delayed Model
lemming
2
dx
x
bx
(1
(x/K
))/(
1
K
)
y
2
2
Dx
dt
modified
logistic
lemming
growth
Holling
Type
III
functional
response
stoat
dy
x2(t
)
exp(
dj) 2 2
y
(t
)d
(x
)y
dt
Dx(t
)
lemming
stoat
growth
hmaturation
witdelay
depende
stoat
death
Hao Wang, John Nagy, Olivier Gilg, and Yang Kuang. The roles of predator maturation delay and functional
response in determining the periodicity of predator-prey cycles, Mathematical Biosciences, Vol. 221: 1-10
(2009)
Prey Dependent Death Rate
The stoat death rate depends on lemming density
d
(
x
)
d
(
d
d
)
(
1
/
2
arcta
b
(
x
N
))
/
)
h
h
l
1
c
This is tested by Olivier Gilg from field.
Lemming-Stoat Delayed Model
lemming
2
dx
x
bx
(1
(x/K
))/(
1
K
)
y
2
2
Dx
dt
modified
logistic
lemming
growth
Holling
Type
III
functional
response
stoat
dy
x2(t
)
exp(
dj) 2 2
y
(t
)d
(x
)y
dt
Dx(t
)
lemming
stoat
growth
hmaturation
witdelay
depende
stoat
death
Hao Wang, John Nagy, Olivier Gilg, and Yang Kuang. The roles of predator maturation delay and functional
response in determining the periodicity of predator-prey cycles, Mathematical Biosciences, Vol. 221: 1-10
(2009)
Modified Logistic Growth
for the lemming
Per capita growth rate
0.3
mammals
Population density x
Per
capita
growth
rate
b
(
1
(
x
/
K
)
)
（
/
1
K
)
(Richard M. Sibly et al and John D. Reynolds et al 2005,
Science)
Empirical Data Fitting
Sensitivity Analysis
Parameters
dj,,
, and
are
key
factors
for period.
theIncreasing
stoat
maturatio
death
rate
dj and
delay
,or
decreasing
predation
rate
and
conversion
rate
can
enlarge
the
period.
Compare the lemming cycle
to the snowshoe hare cycle
lynx
hare
Hare-Lynx Delayed System
In general view, the snowshoe hare cycle is also controlled
by predators like collared lemming in NE Greenland.
Differences: (i) Holling Type II functional response;
(ii) constant lynx death rate.
dx
x
bx
(
1
(
x
/K
))
/(
1
K
)
y
dt
D
x
dy
x
(
t
)
exp(
d
)
y
(
t
)
dy
j
dt
D
x
(
t
)
1
10-year period
1976
1996
Sensitivity Analysis
Goal: 4<10
lemming
Predation
rate
and
hareconversion rate are comparable.
lemming
hare
Lynx maturation death rate is less than that of stoat
(
d
)
(
d
)
opposite
directio
lemming
j
hare
whichjmakes
the period
of snowshoe hare cycle
smaller than the period of lemming cycle.
right
hare
Lynxlemming
maturation
delay
isdirection
1.5 years, much larger
than the stoat maturation delay, 3 months. This
makes ‘4<10’ possible.
Therefore the predator maturation delay is the key
factor to generate different periods (4-year and 10year) of lemming and hare cycles.
38-year moose cycle for moose-wolf interaction
Maturation Hypothesis
Climate change leads to variation
of predator maturation period.
Who knows the evolution direction?
Snow Melt Hypothesis
Climate change results in more frequent
freeze-thaw events leading to a more
shallow and icy snow pack, which will
probably act to expose small mammals
to predators……
(Callaghan et al 2004, Ambio;
Callaghan et al 1995, Phil. Trans. R. Soc. Lond. A)
Lemming-Stoat Delayed Model
predation rate increases with snow melt
lemming
2
dx
x
bx
(1
(x/K
))/(
1
K
)
y
2
2
Dx
dt
modified
logistic
lemming
growth
Holling
Type
III
functional
response
stoat
dy
x2(t
)
exp(
dj) 2 2
y
(t
)d
(x
)y
dt
Dx(t
)
lemming
stoat
growth
hmaturation
witdelay
depende
stoat
death
Hao Wang, John Nagy, Olivier Gilg, and Yang Kuang. The roles of predator maturation delay and functional
response in determining the periodicity of predator-prey cycles, Mathematical Biosciences, Vol. 221: 1-10
(2009)
Observation
For collared lemmings in NE Greenland,
period of the cycle is decreasing and
amplitude is almost unchanged. Climate
change has different effects on lemming
cycles in different locations because of
different mechanisms triggering them.