Problem:
• How does a salmon population change after consecutive cycles of larvae being born?
• How could the population be modeled?
• Are there equilibrium solutions, patterns and trends?
• What factors might affect the salmon population?
• How will these factors change the results?
Assumptions• One cycle is equal to the birth of larvae to their adulthood.
• Xn is the population of salmon after the n-th cycle in hundreds of millions. (discrete-time)
• y(t) is the population of larvae at a given time t. (continuous-time)
• All larvae are born in the river.
• Adult Salmon cannibalize a proportion () of the larvae population ONLY in the river during time.
t = te - to.
• All adult salmon die at the end of each cycle.
Life cycle
• # of salmon larvae born is proportional () to number of adult salmon at beginning of each cycle. Namely = (*Xn)
• Adult Salmon cannibalize a proportion () of the larvae population during time te - to.
• There is a proportion () of juvenile salmon that survive at sea. (Some just don’t make it)
• Surviving juveniles become new adult salmon population.
to te
*Xn
Y(t)
t
Model• Start with initial # of larvae (*Xn ) @ y(to) for each
cycle.
• Larvae population then changes with time:
• dy = -*Xn* y(t) ← during t = te - to.
dt
• dy = (-*Xn) * dt ← (rearrange and integrate)
y(t)
Model (cont’d)
• ln(y(t)) = (-*Xn)*(te – to) ← [ solve for y(t) ]
• y(t) = exp (-*Xn*(te – to))
• Xn+1 = [ * Xn * exp(- *(te–to)*Xn) ] *
• Remember that Xn+1 is the salmon population after each cycle.
Modeling process
• SO: all information is collected into one equation.• Convenient!! Xn+1 = * * Xn * exp(- *(to–te)*Xn)
• 3 < * < 20 ↑ *, larger pop. next cycle
↓ *, smaller pop. next cycle.• 1 < *(to–te)< 10 ↑ *dt, more larvae were eaten
↓ *dt, less larvae were eaten
What could happen...
• Because we could have an infinite number combinations – let’s looks at specific results.
Stability:
Xo=3, *(te–to) =1, * =7
Cobwebbing!
0
1
2
3
4
5
6
7
8
0 2 4 6 8
X(n)
X(n
+1) identity line
X(n+1) vs X(n)Cobwebbing!
What we saw...X(n+1) vs. n
0
0.5
1
1.5
2
2.5
3
3.5
-15 5 25 45 65 85 105
n (generations)
Po
pu
lati
on
(h
un
dre
d m
illio
ns)
c
2-cycle:
Xo=1, *(te–to) =1, * =10
X(n+1) vs. n
0
0.5
1
1.5
2
2.5
3
3.5
4
-15 5 25 45 65 85 105
n (generations)
Po
pu
lati
on
(h
un
dre
d m
illio
ns)
c
What we saw...
Cobwebbing!
012345678
0 2 4 6 8
X(n)
X(n
+1)
identity line
X(n+1) vs X(n)
Cobwebbing!c
4-cycle
Xo=1, *(te–to) =1, * =13
Cobwebbing!
0
1
2
3
4
5
6
7
8
0 2 4 6 8
X(n)
X(n
+1) identity line
X(n+1) vs X(n)Cobwebbing!
What we saw...
X(n+1) vs. n
0
1
2
3
4
5
6
-15 5 25 45 65 85 105
n (generations)
Po
pu
lati
on
(h
un
dre
d m
illio
ns)
c
What we saw...
X(n+1) vs. n
0
1
2
3
4
5
6
7
8
-15 5 25 45 65 85 105
n (generations)
Po
pu
lati
on
(h
un
dre
d m
illio
ns)
c
Cobwebbing!
012345678
0 2 4 6 8
X(n)
X(n
+1)
identity line
X(n+1) vs X(n)
Cobwebbing!c
CHAOS!!!!!
Stability
X*
|-----stable------|---- Cyclical-----|
Stable:3 ≤ ≤ 7
2 cycle:7 < ≤ 12
4 cycle:12 < ≤ 14
8 cycle: > ~15
CHAOS!!: = ????
Stability
?
?
?
?
?
Why are we getting cycles?!
• Consider a 2 cycle:• If lots eaten small salmon population next
cycle
small population means less cannibalism. More will survive.
large salmon population
4, 8, 16, etc. cycles are more complicated.
Modified Model
• Fishing affects the salmon population.
• Based on ocean fishing, limits are determined to ensure a minimum salmon “stock”, to prevent over-fishing.
• We assumed if the salmon population was below 2, no fishing was allowed.
• A proportion of the current salmon population would be fished, as opposed to a system of diff. equations.
• Let f = ratio of fish caught 0 ≤ f ≤ 1
• If Xn < 2
• f = 0
• NEW MODEL BECOMES:
Xn+1 = (1-f)*[ * Xn * exp(- *(te–to)*Xn) ] *
Modified Model
Modified Model (fishing)
Comparison of Fishing and Non-Fishing
0
0.5
1
1.5
2
2.5
3
3.5
1 10 19 28 37 46 55 64 73 82 91 100
n (generations)
po
pu
lati
on
s (h
un
dre
ds
of
mil
lio
ns)
X(n+1)F
X(n+1)
Cobwebbing!
0
1
2
3
4
5
6
7
8
9
0 2 4 6 8 10
Identity line
model
cobwebbing!
• Let p=ratio of fish killed by predators 0 ≤ p ≤ 1
• If Xn < 0.5• Then p = 0
• NEW MODEL BECOMES: Xn+1 = (1-p)*[ * Xn * exp(- *(te–to)*Xn) ] *
similar results as fishing are expected but...
Modified Model 2
• Let p=ratio of fish killed by predators 0 ≤ p ≤ 1
f = ratio of fish caught 0 ≤ f ≤ 1
• If Xn < 0.5 → p = 0 • If Xn < 2 → f = 0
• NEW MODEL BECOMES:
Xn+1 = (1-p-f)*[ * Xn * exp(- *(te–to)*Xn) ] *
Super-duper Combo Model
What we saw...
Comparing X(n+1)F+P and X(n+1)
0
1
2
3
4
5
6
7
0 20 40 60 80 100
n (generations)
po
pu
lati
on
(h
un
dre
ds
of
mil
lio
ns)
X(n+1)F+P
X(n+1)
Cobwebbing!!
0
1
2
3
4
5
6
7
8
0 2 4 6 8
X(n)
X(n
+1) identity line
model
Cobwebbing
Super-duper combo model
What does the new model do?
• Provides a slightly more realistic representation of salmon population over generations.
• Changes the stability and cyclical behavior of the original model.
Model Critique
• Predation depends on the animal-salmon interaction.– The Super-duper Combo Model poorly represents actual predation.
• Not all adult salmon die at sea. Some return to river to re-spawn. We assumed all die.
• Fishing and predation were dealt with as instantaneous effects on the model and should have been modeled as a system of differential equations.
• Infinite number of possibilities (depending on parameters) makes the model difficult to explore in great depth.
• A lot of macro work. Due to lack of programming knowledge, multiple macros had to be made.
• The effect of pollution could be a great MATH472 project.
Super summary
• Salmon population, under varying conditions, can result in a steady state, cyclic behavior or chaos from cycle to cycle.
• The salmon population was modeled using discrete and continuous time methods together.
• Factors such as fishing, predation, and pollution, amount born, eaten, and surviving at sea affected the salmon population.
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