Polar bears in the Hudson Bay region are being threatened by climate change-induced warming. As temperatures rise, sea ice melts earlier in the summer, reducing the time polar bears have to feed. This leads to lower body weights, fewer cubs, and higher cub mortality. The author builds a mathematical model to analyze how temperature increases could affect the polar bear population over time. Simulation results show that if temperatures reach a critical threshold, the polar bear population will decline to extinction. Maintaining sea ice and preventing temperature increases above this critical point are necessary for polar bear survival.

1. Polar Bears in the Warming Arctic
Modeling Climate Change and the Hudson Bay Polar Bears
Kirsten Bell - 6852082
Math 3820 - Winter 2013

2. ∗ Climate change = global temperature rising
∗ Changes are most extreme at the earth poles
∗ Average temperatures in the arctic region are rising twice as
fast as they are elsewhere in the world.[1]
∗ In the artic, the average summer temperature has increased
around 20
C over the last half century[11]
∗ Melting sea ice amplifies warming
∗ Ice melts  open ocean dark water and land reveled  soaks up
more sunlight  accelerates warming
∗ If this trend continues, summers in the Arctic could become ice-
free by the end of the century.[1]
Introduction
2

3. With these increasing temperatures, species that rely on the
fragile ecosystem of the artic are being put in danger. Polar
Bears, being at the top of the food chain, are amongst the
most affected by these changes.
3
[1]

4. ∗ Polar bears rely on sea ice to survive
∗ Cycle of feasting and fasting
∗ They feed heavily in August and September and then reduce their
metabolic rate to fast through the winter.
∗ Most affected by a shorter feasting time due to loss of sea ice are the
pregnant females.
∗ After feasting they stay in their dens from October to March or April
∗ Without a high enough level of body fat, the bear will have fewer cubs or
none at all, and the cubs will weigh less.[4]
∗ Overall, a loss in sea ice causes:
∗ Reduced access to food
∗ Drop in body condition
∗ Lower cub survival rates
∗ Increase in drowning
∗ Increase in cannibalism
∗ Loss of access to denning areas[4]
∗ For every 2°F of warming, a 15% decrease of annually averaged sea ice
and a 25% decrease in September Arctic sea ice are expected.[2]
4

5. ∗ There are 19 populations of polar bears throughout the arctic. In
my model I am looking at the polar bears of the West Hudson
region. This region includes the bears in Churchill, MB.[5]
∗ These bears can
provide a preview as to
what is to come for the
entire population of
species as
temperatures continue
to rise.
5
[5]

6. ∗ How will increasing temperatures due to climate
change affect the population of polar bears in the
West Hudson region?
∗ Real World Problem = Mechanistic approach
∗Future population = present population + change
∗Differential Equations
6
Question

7. Modeling
∗ Assumptions:
∗ Average Summer temperatures in the West Hudson region is uniform
throughout the region
∗ If the temperatures fall below T0 the ice freezes up and the bears will
not be able to hunt enough to sustain themselves through the winter
∗ Polar Bears have a fixed birth and death rate
∗ Polar bears depend on sea ice for survival
∗ B(t) = population of polar bears in the West Hudson region at time t
T(t) = summer temperature in the West Hudson region at time t
7

8. ∗ Parameters
∗ r=growth constant of polar bears
∗ b=constant birth rate of polar bears
∗ d=constant death rate of polar bears
∗ k=growth constant of temperature
∗ T0=minimum summer temperature required for bears to survive
∗ T1=maximum summer temperature required for bears to survive
∗ Tm=maximum temperature the artic can withstand and maintain itself as
an ecosystem
8

9. ∗ Setting dB/dt=0 and dT/dt=0 gives the 4 equilibria to be:
∗ Phase Plane Analysis gave nullclines as:
∗ B-nullclines:
∗ T-nullclines:
Mathematical Analysis
9

10. 10
b/d
Tm
B
T
(0,0)
( r/dTm(Tm-T0)(T1-Tm)+b/d , Tm )
= B nullclines
= T nullclines

11. ∗ Linearization
∗ The equilibria (x*,y*) is locally asymptotically stable if:
∗ det(J(x*,y*))>0
∗ Tr(J(x*,y*))<0
∗ λ1<0, λ2<0, real
Mathematical Analysis
11

12. 1. (0,0) : det(J)>0
tr(J)>0
λ1, λ2>0
 unstable node
2. (b/d,0) : det(J)<0
tr(J)>0
λ1, λ2 opposite signs
 unstable saddle point
 (0 , Tm) : det(J)>0
tr(J)>0
λ1, λ2>0
unstable saddle point
2. (r/dTm(Tm-T0)(T1-Tm)+b/d , Tm) : det(J)>0
tr(J)>0
λ1, λ2>0
LAS if, for Q = rTm(Tm-T0)(T1-Tm),
∗ kTmQ<bkTm  Q<b
∗ -Q<b+kTm  Q>-b-kTm
∗ -Q<b  Q>-b 12

13. ∗ If temperatures reach Tm, polar bears will not survive
∗ Unless if, for Q = rTm(Tm-T0)(T1-Tm):
∗ Q<|b|, and
∗ Q>-b-kTm.
∗ The other 3 equilibria are unstable nodes and saddle
points
Mathematical Analysis
13

14. ∗ Parameters:
r = 20[6]
k = 0.05[7]
d = 50 births/year[8]
b = 200 deaths/year[8]
T0 = -5 degrees Celsius[9]
T1 = 20 degrees Celsius[9]
Tm = 40 degrees Celsius[9]
∗ Initial Conditions:
B(0) = 1000 Bears[8]
T(0) = 12o
C[10]
∗ Simulations:
Numerical Results
14

15. Numerical
15

16. Numerical
16

17. Conclusion
17
∗ If temperatures continues to rise in the artic, the polar
bears will not survive.

18. 1. http://www.nrdc.org/globalwarming/qthinice.asp
2. http://www.epa.gov/climatechange/science/future.html
3. http://globalwarming.markey.house.gov/impactzones/arctic.html
4. http://www.polarbearsinternational.org/about-polar-bears
5. http://wwf.panda.org/what_we_do/where_we_work/arctic/wildlife/polar_bear/population/
6. http://www.wwf.ca/conservation/species/polar_bear_factsheet.cfm
7. http://arctic-news.blogspot.ca/2013/04/how-much-will-temperatures-rise.html
8. http://www.polarbearalley.com/blog/index.php/2013/02/27/polar-bear-blog-state-of-the-western-h
/
9. http://nsidc.org/cryosphere/seaice/
10. http://www.climate.weatheroffice.gc.ca/climateData/dailydata_e.html?
StationID=48969&Month=4&Day=16&Year=2013&timeframe=2
11. http://acia.cicero.uio.no/factsheets/1_arctic_climate_trends.pdf
References
18