This model demonstrates the basic concept of an idealized migration model through a habitat zone. In this context, the habitat zone is a "lake" and the migratory subjects are juvenile salmon called smolts. Individuals enter the habitat zone -- the square "lake" in this example -- from its tributaries (here represented by a single channel "in" ) and leave the habitat zone. This is also an illustration of a larger concept -- that of a compartmental model in which some substance enters the compartment from external sources and exits through other portals.
powered by NetLogo
view/download model file: MigrationConcept.nlogo
The setup command initializes the system and the plots, after which "go" repeatedly generates smolts entering the lake at the expected rate shown on the slider and exiting at the rate E indicated by the plots and monitors.
One the Setup command has been executed, simply press go to set the simulation in motion. The simulation advances in ticks, and the slider ticks-per-unit-time allows the number of ticks corresponding to a unit of time to be adjusted. The flow rate determines how much the water flow influences the smolts in the lake. At a rate of 1.00, the smolts almost certainly follow the flow, whereas at 0 the smolts in the lake move randomly independently of the flow.
The basic idea that this simple model seeks to illustrate is that the population reaches an equilibrium when the rates I and E are approximately the same. Also, it is important to notice that the relative rate of emigration is more or less constant throughout the simulation. This means that this type of density-dependent migration can be modeled by
a differential equation of the form dN/dt = I - m N, and thus that the equilibrium is approximately I / m.
Try varying the expected number of arrivals per unit time. Also, try changing the relative rate of flow of the smolts (relative to the flow of the lake, that is).
This model created by Dr. Jeff Knisley as part of the Symbiosis project module on migratory models in differential calculus. It is one of the simulations located at http://math.etsu.edu/symbiosis/migrations.
globals [ from0to1 MaxDist Current-Time Previous-Time AllTime N0P N1P m0 m1
Arrivals-per-unit-time Departures-per-unit-time IMI EMI Deaths-per-tick ]
patches-own [ iswater? inlake? pzone flow-heading flow-strength ] ; flow-strength is 0 to 1, percentage of main-channel flow speed
turtles-own [ zone ]
to setup
clear-all
set from0to1 0
set Previous-Time 0
set Current-Time 0
setup-patches
set Arrivals-per-unit-time [ 0 ]
set Departures-per-unit-time [ 0 ]
set N0P [ 0 ]
set-current-plot "N(t)"
plotxy 0 0
end
to setup-patches
;set source info and zone transition points
ask patches
[
ifelse ( pxcor >= min-pxcor and pycor > -4 and pycor < 4 )
[ set pcolor blue
set iswater? true
set inlake? false
set pzone 0
set flow-heading 90
set flow-strength 1
]
[ set pzone -1
set iswater? false
set inlake? false
]
]
make-compartment min-pxcor + 4 max-pxcor 1
end
to make-compartment [ L R zn ]
let U max-pycor
let B min-pycor
set MaxDist sqrt ( ( R - 4 - L ) ^ 2 + U ^ 2 )
ask patches [
if ( pxcor >= min-pxcor and pxcor >= L and pxcor < ( R - 4 ) and pycor < U and pycor > B )
[ set pcolor blue
set iswater? true
set inlake? true
set pzone zn
set flow-heading towardsxy ( R - 4 ) 0
set flow-strength 1 - sqrt ( ( pxcor - R ) ^ 2 + pycor ^ 2) / MaxDist ]
if ( pxcor <= R and pxcor >= ( R - 4 ) and pxcor <= max-pxcor and pycor > -4 and pycor < 4 )
[ set pcolor blue
set iswater? true
set inlake? false
set pzone 2
set flow-heading 90
set flow-strength 1 ]
]
end
to go
set Deaths-per-tick 0
ask turtles [
if ( random-float 1 < 1 / ticks-per-unit-time )
[ smoltMove
smoltLearn
]
]
smoltMake
smoltCount
end
to smoltMove
rt random 90
lt random 90
let p [ flow-strength ] of patch-here
if( [ pzone ] of patch-here >= 0 ) [ set p p * Relative-Flow-In-Lake ]
ifelse ( heading < 270 )
[ set heading ( ( 1 - p ) * heading + p * ( [ flow-heading ] of patch-here ) ) ]
[ set heading ( ( 1 - p ) * heading + p * ( [ flow-heading ] of patch-here + 360 ) ) ]
if not ( member? patch-ahead 1 neighbors with [ iswater? ] )
[ set heading towards one-of neighbors with [ iswater? ] ]
fd 1
set zone [pzone] of patch-here
if ( xcor >= max-pxcor )
[ die
set Deaths-per-tick ( Deaths-per-tick + 1 ) ]
end
to smoltMake
if( random-float 1 < Expected-Arrivals-per-unit-time / ticks-per-unit-time )
[ create-turtles 1
[
set shape "fish"
set color green
setxy min-pxcor 0
set heading 90
set zone 0
]
]
end
to smoltCount
tick
set Current-time ticks / ticks-per-unit-time
set Arrivals-per-unit-time lput count turtles with [ zone = 0 ] Arrivals-per-unit-time
set N0P lput count turtles with [ zone = 1 ] N0P
set Departures-per-unit-time lput ( Deaths-per-tick + count turtles with [ zone = 2 ] ) Departures-per-unit-time
if( Current-Time < Previous-Time )
[ clear-all-plots
reset-ticks
set Current-Time 0
]
set-current-plot "N(t)"
plotxy Current-Time last N0P
set-current-plot "Rates I (blue) and E (red)"
if( last N0P > 0 and Current-Time > 0 )
[ set IMI precision ( sum Arrivals-per-unit-time / ( length Arrivals-per-unit-time ) / 11 ) 5
set EMI precision ( sum Departures-per-unit-time / ( length Departures-per-unit-time ) / ( 7 + 3 * Relative-Flow-In-Lake ) ) 5
set m0 precision ( EMI / last N0P ) 5
set-current-plot-pen "I"
plotxy Current-Time IMI
set-current-plot-pen "E"
plotxy Current-Time EMI
]
if( length N0P > ticks-per-unit-time )
[ set N0P remove-item 1 N0P
set Departures-per-unit-time remove-item 1 Departures-per-unit-time
set Arrivals-per-unit-time remove-item 1 Arrivals-per-unit-time
]
set Previous-Time Current-Time
end
to smoltLearn
; Zone number, simple flow concept means no going to earlier zone /font>
end