1. Project Salmon

2. 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?

3. Assumptions One cycle is equal to the birth of larvae to their adulthood. X n 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 = t e - t o. All adult salmon die at the end of each cycle.

4. Life cycle # of salmon larvae born is proportional (  ) to number of adult salmon at beginning of each cycle. Namely = (  *X n ) Adult Salmon cannibalize a proportion (  ) of the larvae population during time t e - t o. There is a proportion (  ) of juvenile salmon that survive at sea. (Some just don’t make it) Surviving juveniles become new adult salmon population. toto tete *Xn*Xn Y (t) t

5. Model Start with initial # of larvae (  *X n ) @ y( t o ) for each cycle. Larvae population then changes with time: dy = -  *X n* y(t) ← during  t = t e - t o. dt dy = (-  *X n ) * dt ← (rearrange and integrate) y(t)

6. Model (cont’d) ln (y (t) ) = (-  *X n )*(t e – t o ) ← [ solve for y(t) ] y(t) = exp ( -  *Xn*(te – to) ) X n+1 = [  * X n * exp(-  *(t e –t o )*X n ) ] *  Remember that X n+1 is the salmon population after each cycle.

7. Modeling process SO: all information is collected into one equation. Convenient!!  X n+1 =  *  * X n * exp(-  *(t o –t e )*X n ) 3 <  *  < 20 ↑  * , larger pop. next cycle ↓  * , smaller pop. next cycle. 1 <  *(t o –t e )< 10 ↑  *dt, more larvae were eaten ↓  *dt, less larvae were eaten

8. What could happen... Because we could have an infinite number combinations – let’s looks at specific results.

9. Stability: X o =3,  *(te–to) =1,  *  =7 What we saw...

10. 2-cycle: X o =1,  *(te–to) =1,  *  =10 What we saw...

11. 4-cycle X o =1,  *(te–to) =1,  *  =13 What we saw...

12. CHAOS!!!!!

13. Stability X*  |-----stable------ |---- Cyclical -----| Stable: 3 ≤  ≤ 7 2 cycle: 7 <  ≤ 12 4 cycle: 12 <  ≤ 14 8 cycle:  > ~15 CHAOS!!:  = ???? Stability ? ? ? ? ?

14. 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.

15. 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.

16. Let f = ratio of fish caught 0 ≤ f ≤ 1 If X n < 2 f = 0 NEW MODEL BECOMES: X n+1 = (1-f)* [  * X n * exp(-  *(t e –t o )*X n ) ] *  Modified Model

17. Modified Model (fishing)

18. 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(-  *(t e –t o )*Xn) ] *  similar results as fishing are expected but... Modified Model 2

19. 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

20. What we saw... Super-duper combo model

21. 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.

22. 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.

23. 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.