Metapopulation models and movement Continued

1
Meta-population models and movement (Continued)

• Fish 458 Lecture 19

2
Modeling Approaches

• Discrete areas / grids with exchange (be careful
  of discrete assumptions).
• Advection / diffusion models.
• Occupancy models (a population is either extant
  or extinct at a given site).
• Individual-based models.

3
Advection-Diffusion Modeling(diffusion)

• Consider a site x at time t. Let C(x,t) denote
  the concentration at the point (x,t).
• Now consider the total amount of material between
  x and x?x.
• General law the rate of change of the amount in
  this interval equals the net rate at which
  material flows across its boundary (in the
  positive direction, J(x,t)) plus the net creation
  of material in the interval, Q(x,t)

4
Advection-Diffusion Modeling(diffusion)

• By the integral mean value theorem
• Dividing by ?x and taking the limit ?x?0 gives
• Discrete interpretation the change in the
  concentration at a site over time is determined
  by the net amount entering the site plus net
  production at the site

5
Advection-Diffusion Modeling(diffusion)

• If animals move randomly, the net movement will
  be from areas of high concentration to those of
  low concentration.
• The simplest way to model this is though Ficks
  Law
• Substituting into the previous model gives

6
Advection-Diffusion Modeling(back to the
logistic model)

• Now, let us assume that the diffusion rate is a
  constant, m, and the net production at x0 is
• This leads to
• But for ?x ?t 1 this is the discrete logistic
  model with migration!

7
Time for an Example!

• Consider a channel (filled with water and of
  infinite length), assume that some pollutant is
  dropped in the centre of the channel.
• How does the density of pollutant as a function
  of distance change with time.
• This is typical diffusion problem. To solve it,
  we discretize the diffusion equation and run it
  forwards.

8
Pollutants in Channels-II

• Note that you need to be very careful when
  choosing the step sizes (?x and ?t). I used ?x1,
  ?t1 and D0.1.

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Pollutants in Channels-III
The pollutant diffuses outward from the point
source
This eventually convergences to a normal
distribution
Distance
10
Drunks and Diffusion

• Consider a drunk walking down a north-south road.
  At each time-step, he (she) moves north or south
  with equal probability.
• We can consider the probability distribution for
  where he (she) is in the road a concentration
  and apply the diffusion model.
• Recall a random walk leads to a diffusion
  process.

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Drunks and Diffusion(reflective boundary)
There are bouncers at either end of the road and
if our drunk gets to them, they put him back in
the road!
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Drunks and Diffusion(reflective boundary)
There is a bouncer at one end of the road but the
sea at the other (our drunk must be sailor!)
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Advection-Diffusion Modeling(Multiple dimensions)

• The standard diffusion model can be extended
  (constant D) into multiple dimensions

14
Estimating Movement Rates using Tagging Data

• The model and likelihood

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Needs and Assumptions

• Needs
• Tagging in all areas.
• Tag return rates known from other sources.
• Survival rate known from other sources (either
  from an assessment or the exploitation rate is
  assumed proportional to fishing effort).
• Assumptions
• Usual assumptions of tagging analyses.
• Movement rate is independent of density / time /
  age.
• Survival, movement are independent of age / sex,
  etc.

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An Example

• Three areas with true migration matrix

M 0.2yr-1 1000 released in each area. Effort
known exactly No non-reporting Recaptures are
Poisson.
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An Example
The fits are very good unrealistically so!
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Reminder

• When fitting tagging data, it is often worthwhile
  assuming a negative binomial likelihood as
  tagging data are frequently overdispersed
  (variance larger than the mean with respect to
  the Poisson).

19
Readings

• Quinn and Deriso (1999) Chapter 10.
• Hilborn (1990).