eulerian movement models example grid point model: an advection-diffusion-reaction model for the...

Eulerian Movement Models

• Example grid point model:An advection-diffusion-reaction model for the estimation of fish movement parameters from tagging data, with application to skipjack tuna (Katsuwonus pelamis). Sibert, J, J. Hampton, D. Fournier, and P. Bills (1999). CJFAS. 56: 925-938

• Example grid line (block transfer) model:Modeling effects of FADs and islands on movement of skipjack tuna (Katsuwonus pelamis): estimating parameters from tagging data. Kleiber, P. and J. Hampton. 1994. CJFAS. 51:2642-2653.

Estimating Skipjack Movement Parameters

The Data: ~90k releases and ~5k recaptures by pole-and-line and purse seine fishing fleets

Grid point model

The Universe:

Diffusion terms Advection terms

Grid points

Continuous form:

Discrete form:

Reaction term Parameters Alternating direction, implicit time steps

Movement parameter estimates

Observation model Poisson likelihood

Model Performance

Solomon Islands

Modeling effects of FADs and Islands on Skipjack Movement

• Many FADs

• Two fleets

• 7730 tag releases

• 914 returns

• 1/2 degree square grid

Grid line model

1, jiN

jiN ,1

1, jiN

jiN ,1

jiN ,

1, jim

jim ,

jim ,

jim ,1

jim ,

1, jim

jim ,1

jim ,

Movement by block transfer

transfer coefficients:

FAD effect

FAD

fact

or

no. FADs

Island effects

ba

Model Parameters

ADI -- Alternating Direction, Implicit time steps

Index time differently in alternating steps

step A: step B:

Example: step A (NS implicit). Collect ) terms on the left(i.e.i-1,j

i,j

i+1,j

add neighboring equations and arrange

ADI -- alternating direction, implicit time steps

i-1,j

i,j

i+1,j

makes tri-diagonal system

ADI -- alternating direction, implicit time steps

i-1,j

i,j

i+1,j

Objective function

Two alternatives:

• tag release sets combined

single model run includes all sets

Poisson likelihood

• individual tag release sets

separate model run for each set

Multinomial likelihood

Model Performance

Months at large Months at large

Ret

urns

per

mon

th

Mea

n di

stan

ce (n

m)

Return rate Mean distance moved

Results

• D = 3.2 nm2/mo

• FF = 0.36

• M = 0.15 mo-1

≈ 10 nm

radius of FAD actionparameter estimates

0.49

1.3 FADs

Diffusion bias