Polar Bear Math Modeling Presentation

Polar Bears in the Warming Arctic

Modeling Climate Change and the Hudson Bay Polar Bears

Kirsten Bell - 6852082Math 3820 - Winter 2013

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 20C 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

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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.

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[1]

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]

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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.

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[5]

How will increasing temperatures due to climate change affect the population of polar bears in the West Hudson region?

Real World Problem = Mechanistic approachFuture population = present population + changeDifferential Equations

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Question

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 t7

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

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Setting dB/dt=0 and dT/dt=0 gives the 4 equilibria to be:

Phase Plane Analysis gave nullclines as:

B-nullclines:

T-nullclines:

and the resulting graph:

Mathematical Analysis

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b/d

Tm

B

T

(0,0)

( r/dTm(Tm-T0)(T1-Tm)+b/d , Tm

)

= B nullclines = T nullclines

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

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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

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

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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) = 12oC[10]

Simulations:

Numerical Results

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Numerical

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Numerical

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Conclusion

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

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.html4. 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.cfm7. http://arctic-news.blogspot.ca/2013/04/how-much-will-temperatures-

rise.html8. http://www.polarbearalley.com/blog/index.php/2013/02/27/polar-bear-

blog-state-of-the-western-hudson-bay-polar-bears/

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=211. http://acia.cicero.uio.no/factsheets/1_arctic_climate_trends.pdf

References

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