Modeling the Re-invasion of Sea Otters along the Coast of

California

M.J. Krkosek

J.S. Lauzon

A Little History…

• Thought extinct in the early 1900’s

• Small population of 50 found in 1914

• Began to re-invade former range

Life History• Occupies rocky coastline

habitat less than 40 meters deep

• Live in “rafts” of 4 to 40 individuals

• Females have one pup a year, in the spring

• Pups wean after 4 months and take 5 years to reach reproductive maturity

• Feed primarily on sea urchins

Scale and Independence of Data

Lubina & Levin’s Model

Re-invasion Data

Krkosek & Lauzon’s

Model

Independent Life-history & Dispersal data

Re-invasion Data

Models of California Sea Otters Re-invasion

Lubina and Levin’s Model

t( )n ,x t ( )f , , ,n x y t D

2

x2( )n ,x t

V 2 rD

• One-dimensional Diffusion Model

• Rate of Invasion

Growth function Diffusion

r: intrinsic rate of increase

D:diffusion coefficient

•Assumes random movement

•Rate of invasion is constant

Our Model

• Based on life history and spatial use data independent of invasion

• Begin with a stage structured growth model

• Eigenvalue = intrinsic growth rate

• Eigenvector = stable stage distribution

( )Yt x

( )Jt x

( )At x

( )Y t 1 x

( )J t 1 x

( )A t 1 x

0 0 R

SYJ SJJ 0

0 SJA SAA

Leslie Matrix

Incorporate Spatial Dynamics

• Stage-structured integro-difference equations

( )J t 1 x SYJ ( )Yt x SJJ d

( )kJ

x y ( )Jt x y

( )A t 1 x SAA d

( )kA

x y ( )At x y SJA d

( )kJ

x y ( )Jt x y

( )Y t 1 x R ( )At x

Dispersal Kernel

What is a Dispersal Kernel?

• Frequency distribution of dispersal distances• Types: Normal, leptokurtic, …

( )k x1

42 e

( ) x( )k x

1

2

2 e

1

2

x2

2

2

How do you parameterize a kernel?

2 = 2D,

where D = M2/(4t)

and M = mean displacement per time = find by least squares fit to a

density distribution of annual displacement

( )k x1

2

2 e

1

2

x2

2

2

( )k x1

42 e

( ) x

Results: How does the kernel affect the rate of spread?

Gaussian• Linear spread

Fat Tailed Kernels

• exponential spread

A closer look at exponential spread

= 3.4598

= 3.7 (south)

= 5.2 (north)

km

Results: Population growth

Exponential growth Logistic growth

( )Y t 1 x R ( )At x ( )Y t 1 x R ( )At x

1

( )Nt x 2

K2

Carrying capacity and spatial distribution

Exponential Logistic

So how do we compare?Lubina & Levin

• r = 0.056

• D = 13.5 – 104 km2/yr

• Spread = piecewise linear

Krkosek & Lauzon

• r = 0.0631

• DJ = 76.75 km2/yr

• DA = 15.55 km2/yr

• Spread = exponential

• Growth = logistic

Population size vs time Population

spread vs time

Conclusion

• We have shown that independent small scale data on sea otter life history and spatial movement can predict large scale patterns in population growth and dispersal.

• Due to their discrete nature, life history characteristics of sea otters are better described by integro-difference equations than by diffusion equations.

• Integro-difference equations appear to accurately predict growth and dispersal patterns of re-invading sea otters along the California coast.

ReferencesEberhardt, L.L., 1995. Using the Lotka-Leslie model for sea otters. Journal of Wildlife Management, 59(2): 222-227. Kot, M., M.A. Lewis, and P. van den Driessche, 1996. Dispersal data and the spread of invading organisms. Ecology,

77(7): 2027-2042. Laidre, K.L., R.J. Jameson, and D.P. DeMaster, 2001. An estimation of carrying capacity for sea otters along the

California coast. Marine Mammal Science, 17(2): 294-309. Lewis, M.A., 1997. Variability, patchiness, and jump dispersal in the spread of an invading population. in: Spatial

Ecology: The Role of Space in Population Dynamics and Interspecific Interactions (D. Tilman and P. Kareiva, eds). Princeton University Press, Princeton, New Jersey.

 Lubina, J.A. and S.A. Levin, 1988. The spread of a reinvading species: Range expansion in the California sea otter. American Naturalist, 131(4): 526-543.

 Monnett, C., and L.M. Rotterman, 2000. Survival rates of sea otter pups in Alaska and California. Marine Mammal Science, 16(4): 794-810.

 Monson, D.H. , and A.R. DeGange, 1995. Reproduction, preweaning survival, and survival of adult sea otters at Kodiak Island, Alaska. Canadian Journal of Zoology, 73(6):1161-1169.

 Ralls, K, T.C. Eagle, and D.B. Siniff, 1996. Movement and spatial use patterns of California sea otters. Canadian Journal of Zoology, 74(10): 1841-1849.

 Riedman, M.L., J.A. Estes, M.M Staedler, A.A. Giles, and D.R. Carlson, 1994. Breeding patterns and reproductive success of California sea otters. Journal of Wildlife Management, 58(3): 391-399.

 Udevitz, M.S., and B.E. Ballachey, 1998. Estimating survival rates with age-structure data. Journal of Wildlife Management, 62(2): 779-792.