Project Salmon

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.