California Sea OttersCalifornia Sea Otters

Diana White and Michelle BrowneDiana White and Michelle Browne

BackgroundBackground

California Sea Otters were thought to be California Sea Otters were thought to be extinct before the year 1914. After this extinct before the year 1914. After this year a population was found off the coast year a population was found off the coast of Point Sur.of Point Sur.

Information about their spread along the Information about their spread along the coast and their population numbers has coast and their population numbers has been taken between the years of 1914 and been taken between the years of 1914 and 1986.1986.

Their population has increased slowly Their population has increased slowly since 1914, but, in the mid 70’s to early since 1914, but, in the mid 70’s to early 80’s a decrease in the population has 80’s a decrease in the population has been observed.been observed.

Reasons for this are not clear but are Reasons for this are not clear but are believed to be caused by human believed to be caused by human disturbance, competition with fisheries and disturbance, competition with fisheries and pollution.pollution.

Population DynamicsPopulation Dynamics

The Logistic equationThe Logistic equation

What model will fit this data?What model will fit this data? Try the logistic model!!!Try the logistic model!!!

)1( KN

dtdN N

SolutionSolution

The solution to the logistic equation givesThe solution to the logistic equation gives

There are three parameters that need to beThere are three parameters that need to be

evaluated in order to get a best fit with the data.evaluated in order to get a best fit with the data.

kceKttN )(1

)(

N(t)=Population of the sea otters at time tN(t)=Population of the sea otters at time t K=Carrying capacityK=Carrying capacity αα=Maximum rate of growth of population=Maximum rate of growth of population C=constant parameterC=constant parameter

Estimating ParametersEstimating Parameters

K is determined from looking at the plot of K is determined from looking at the plot of population as a function of timepopulation as a function of time

C is determined using the initial condition C is determined using the initial condition N(0)=50.N(0)=50.

αα is estimated by plotting a range of is estimated by plotting a range of solutions for varying solutions for varying αα’s.’s.

K=1800 C=0.0191 K=1800 C=0.0191 αα=0.0875=0.0875

Population as a Function on TimePopulation as a Function on Time

Best FitBest Fit

Spatial EcologySpatial Ecology

How do the otters move?How do the otters move? At what speed do they invade the northern At what speed do they invade the northern

and southern regions?and southern regions? Do they prefer one area more than the Do they prefer one area more than the

other?other?

Invasion in the NorthInvasion in the North

Distance as a function of time

Invasion in SouthInvasion in South

Distance as a function of time

How fast do they move?How fast do they move?

CCnn =2.0986 km/year =2.0986 km/yearCCss =4.0772 km/year =4.0772 km/year- It is obvious that the otters are moving more- It is obvious that the otters are moving moreto the southern regions. to the southern regions. - It is not clear why this happens. We can only- It is not clear why this happens. We can onlyguess that it is due to a more favorableguess that it is due to a more favorableClimate.Climate.

t

dc

Bias Behavior of the Otter’s Bias Behavior of the Otter’s MovementMovement

We will let V=drift (bias)We will let V=drift (bias) V can be calculated by the following equation:V can be calculated by the following equation:

9893.02

||

ns CC

V

Spatial ModelSpatial Model

Fisher’s model can be used to describe Fisher’s model can be used to describe spread within the otter population.spread within the otter population.

This model does not involve a drift term(bias).This model does not involve a drift term(bias).

The following model involves such a term.The following model involves such a term.

xxxt Vuk

uuuDu )1(

)1(k

uuDuu xxt

U(x,t) = Population as a function of space U(x,t) = Population as a function of space and timeand time

D = Diffusion Coefficient D = Diffusion Coefficient

V = Bias term V = Bias term

αα = Maximum rate of population growth = Maximum rate of population growth

K = Carrying capacity (Maximum K = Carrying capacity (Maximum sustainable populationsustainable population

Traveling WavesTraveling Waves

A method that can be used to study the A method that can be used to study the invasion of the otters is to study the invasion of the otters is to study the traveling wave solutions.traveling wave solutions.

u(x,t)=u(x,t)=ФФ(x-ct) (x-ct) ФФ(-∞)=1, (-∞)=1, ФФ(+∞)=1(+∞)=1

C is the speed at which the otters invade.C is the speed at which the otters invade.

If C<0 the otters will travel left (north)If C<0 the otters will travel left (north)

If C>0 the otters will travel right (south)If C>0 the otters will travel right (south)

SolutionSolution

By using the method of separation of By using the method of separation of variables we can determine the minimum variables we can determine the minimum speed of the traveling wave, therefore speed of the traveling wave, therefore determining the speed at which the otters determining the speed at which the otters invade new locations.invade new locations.

Method of Separation of VariablesMethod of Separation of Variables

),(2

2txu

x

ctxut

),(

),( txux

)1()(

)1(

)1(

kDD

V

D

cD

V

kDd

c

kVDc

Equilibrium points: (0,0),(k,0)Equilibrium points: (0,0),(k,0)

Solving for (0,0)Solving for (0,0)

Using linearization techniques we find theUsing linearization techniques we find the

following two eigenvalues.following two eigenvalues.

When CWhen C>V>0 there are stable solutions>V>0 there are stable solutions

When When CC >2( >2(ααD) D) ½ ½ +V there is a stable node+V there is a stable node

When When CC <2( <2(ααD) D) ½ ½ +V there is a stable spiral+V there is a stable spiral

(k,0) is always a saddle point(k,0) is always a saddle point

D

DVccV

2

4)()( 22,1

Solutions can not be negative for Solutions can not be negative for populations so Cpopulations so C < 2( < 2(ααD) D) ½ ½ +V is not +V is not biologically relevant.biologically relevant.

The minimal speed for which a wave front The minimal speed for which a wave front can exist is Ccan exist is C** = 2( = 2(ααD) D) ½ ½ ± V ± V

Range as a Function of TimeRange as a Function of Time

We can calculate the diffusion coefficient We can calculate the diffusion coefficient from the total range of the otters from the total range of the otters movement north and south. movement north and south.

Range = (CRange = (Cs s + C+ Cnn)t = (2C)t = (2C**)t)t

CC* * is the average speed at which the otters is the average speed at which the otters move.move.

CC* * = = Slope/2Slope/2

Derivation of the Diffusion Derivation of the Diffusion CoefficientCoefficient

CC* * = 2(= 2(ααD)D)1/21/2 =slope/2 =slope/2

Slope = total range/time = 3.9199 km/yearSlope = total range/time = 3.9199 km/year

D = (3.9199D = (3.919922/16/16αα))

D = 10.97D = 10.97

Could we have used a better Could we have used a better model?model?

Continuous model vs Discrete modelContinuous model vs Discrete model- Otters do not continuously reproduce. Otters do not continuously reproduce.

They reproduce approximately once a They reproduce approximately once a year. year.

- OvershootOvershoot- Invasion of otters does not occur at a Invasion of otters does not occur at a

constant speed. Can not use a diffusion constant speed. Can not use a diffusion model. model.