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Project Salmon

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?

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Project Salmon

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  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. • Xnis 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.

  4. 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. Y(t) *Xn to te t

  5. 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)

  6. 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.

  7. 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

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

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

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

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

  12. What we saw... CHAOS!!!!!

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

  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. Modified Model • 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) ] * 

  17. Modified Model (fishing)

  18. Modified Model 2 • 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...

  19. Super-duper Combo Model • 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) ] * 

  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.

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