1. California Sea Otters Diana White and Michelle Browne

3. Background  California Sea Otters were thought to be extinct before the year 1914. After this year a population was found off the coast of Point Sur.  Information about their spread along the coast and their population numbers has been taken between the years of 1914 and 1986.

4.  Their population has increased slowly since 1914, but, in the mid 70’s to early 80’s a decrease in the population has been observed.  Reasons for this are not clear but are believed to be caused by human disturbance, competition with fisheries and pollution.

5. Population Dynamics

6. The Logistic equation  What model will fit this data?  Try the logistic model!!!

7. Solution  The solution to the logistic equation gives There are three parameters that need to be There 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.

8.  N(t)=Population of the sea otters at time t  K=Carrying capacity  α=Maximum rate of growth of population  C=constant parameter

9. Estimating Parameters  K is determined from looking at the plot of population as a function of time  C is determined using the initial condition N(0)=50.  α is estimated by plotting a range of solutions for varying α’s. K=1800 C=0.0191 α=0.0875

10. Population as a Function on Time

11. Best Fit

12. Spatial Ecology  How do the otters move?  At what speed do they invade the northern and southern regions?  Do they prefer one area more than the other?

13. Invasion in the North Distance as a function of time

14. Invasion in South Distance as a function of time

15. How fast do they move? C n =2.0986 km/year C s =4.0772 km/year - It is obvious that the otters are moving more to the southern regions. - It is not clear why this happens. We can only guess that it is due to a more favorable Climate.

16. Bias Behavior of the Otter’s Movement  We will let V=drift (bias)  V can be calculated by the following equation:

17. Spatial Model  Fisher’s model can be used to describe spread within the otter population. This model does not involve a drift term(bias). The following model involves such a term.

18. U(x,t) = Population as a function of space and time D = Diffusion Coefficient V = Bias term α = Maximum rate of population growth K = Carrying capacity (Maximum sustainable population

19. Traveling Waves  A method that can be used to study the invasion of the otters is to study the traveling wave solutions. u(x,t)=Ф(x-ct) Ф(-∞)=1, Ф(+∞)=1 u(x,t)=Ф(x-ct) Ф(-∞)=1, Ф(+∞)=1 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 right (south)

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

21. Method of Separation of Variables

22. Equilibrium points: (0,0),(k,0) Solving for (0,0) Using linearization techniques we find the following two eigenvalues. When C>V>0 there are stable solutions When C >2(αD) ½ +V there is a stable node When C <2(αD) ½ +V there is a stable spiral (k,0) is always a saddle point

23.  Solutions can not be negative for populations so C < 2(αD) ½ +V is not biologically relevant.  The minimal speed for which a wave front can exist is C * = 2(αD) ½ ± V

24. Range as a Function of Time

25.  We can calculate the diffusion coefficient from the total range of the otters movement north and south.  Range = (C s + C n )t = (2C * )t C * is the average speed at which the otters move. C * is the average speed at which the otters move. C * = Slope/2 C * = Slope/2

26. Derivation of the Diffusion Coefficient C * = 2(αD) 1/2 =slope/2 Slope = total range/time = 3.9199 km/year D = (3.9199 2 /16α) D = 10.97

27. Could we have used a better model? Continuous model vs Discrete model - Otters do not continuously reproduce. They reproduce approximately once a year. - Overshoot - Invasion of otters does not occur at a constant speed. Can not use a diffusion model.