104
6. Spatial Fisheries Model
Next, we define a movement rate M, for fish in each region . This will be the
base rate at which fish move out of region i (i = 1, ... 4). Given the movement rate M; random variations around that number, and a total fish population N, in region i, we can calculate the total number of fish moving into
a region by summing all the immigrants from other regions to region i and
subtracting all the emigrants from region i. For example, net movement between region 1 and the rest of the system is
M[Region 1] =«[Region 2]*M
j[Region
2]*RAND[P21l!RAND N2
+ «[Region 3]*M j[Region 3]*RAND[P31]/RAND N3
+ «[Region 4]*M j[Region 4]*RAND[P41l!RAND N4
- «[Region l]*M
1]*RAND[P12]/RAND N1
j[Region
- «[Region l]*M;fRegion 1]*RAND[P13l!RAND N1
1]*RAND[P14]/RAND Nl.
(11)
In equation (11) the random numbers defined for movement from each region to any other region are normalized by the sum of the random numbers
used to define movement from any specific region to the other three
regions .
Set the flow of net migration , M, as an array and specify net migration for
the remaining regions analogously to equation (11). Total net migration is
TOTAL M =ARRAYSUM(M[*]).
(12)
Similarly,
ARRAYSUM(<< DOn*])
(13)
is the total change in N, that occurs in the system at a point in time.
The complete model is shown in Figure 6.4. Before you run that model ,
set M, =0 for all i =1, . . . 4. This will enable you to compare the model of
Figure 6.4 with the one of the previous section and find any errors you may
have made in adopting that model. Then set M, =.8 for all i = 1, . . . 4, and
ALPHA =.64, and all other values as in the previous section.
As you can see from Figures 6.5 and 6.6, for our choice of parameters the
system more or less constantly oscillates with the same amplitude and periodicity-it does not settle down to some long-term steady-state, nor doe s it
show explosive behavior. Be aware that this statement is not a very precise
one-its validity for any specific model run depends on the choice of random numbers. Recognize also that even though for the aggregate system
there is (roughl y) some uniform behavior, that observation does not hold at
the regional level, as Figure 6.7 indicates . This must be the case because, at
the aggregate level, net-migration must be zero (there are no fish leaving or
entering our system). At the regional level, in contrast, migration does
occur (Figure 6.8) and is likely different from region to region .
6. Spatial Fisheries Model
Next, we define a movement rate M, for fish in each region . This will be the
base rate at which fish move out of region i (i = 1, ... 4). Given the movement rate M; random variations around that number, and a total fish population N, in region i, we can calculate the total number of fish moving into
a region by summing all the immigrants from other regions to region i and
subtracting all the emigrants from region i. For example, net movement between region 1 and the rest of the system is
M[Region 1] =«[Region 2]*M
j[Region
2]*RAND[P21l!RAND N2
+ «[Region 3]*M j[Region 3]*RAND[P31]/RAND N3
+ «[Region 4]*M j[Region 4]*RAND[P41l!RAND N4
- «[Region l]*M
1]*RAND[P12]/RAND N1
j[Region
- «[Region l]*M;fRegion 1]*RAND[P13l!RAND N1
1]*RAND[P14]/RAND Nl.
(11)
In equation (11) the random numbers defined for movement from each region to any other region are normalized by the sum of the random numbers
used to define movement from any specific region to the other three
regions .
Set the flow of net migration , M, as an array and specify net migration for
the remaining regions analogously to equation (11). Total net migration is
TOTAL M =ARRAYSUM(M[*]).
(12)
Similarly,
ARRAYSUM(<< DOn*])
(13)
is the total change in N, that occurs in the system at a point in time.
The complete model is shown in Figure 6.4. Before you run that model ,
set M, =0 for all i =1, . . . 4. This will enable you to compare the model of
Figure 6.4 with the one of the previous section and find any errors you may
have made in adopting that model. Then set M, =.8 for all i = 1, . . . 4, and
ALPHA =.64, and all other values as in the previous section.
As you can see from Figures 6.5 and 6.6, for our choice of parameters the
system more or less constantly oscillates with the same amplitude and periodicity-it does not settle down to some long-term steady-state, nor doe s it
show explosive behavior. Be aware that this statement is not a very precise
one-its validity for any specific model run depends on the choice of random numbers. Recognize also that even though for the aggregate system
there is (roughl y) some uniform behavior, that observation does not hold at
the regional level, as Figure 6.7 indicates . This must be the case because, at
the aggregate level, net-migration must be zero (there are no fish leaving or
entering our system). At the regional level, in contrast, migration does
occur (Figure 6.8) and is likely different from region to region .