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CEE262C Lecture 4: Phase line analysis and NPZ models

CEE262C Lecture 4: Phase line analysis and NPZ models. Overview. Phase line analysis Logistic equation Allee effect model Q-logistic model Nutrient-phytoplankton-zooplankton (NPZ) models

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CEE262C Lecture 4: Phase line analysis and NPZ models

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  1. CEE262C Lecture 4: Phase line analysis and NPZ models Overview • Phase line analysis • Logistic equation • Allee effect model • Q-logistic model • Nutrient-phytoplankton-zooplankton (NPZ) models References: Mooney & Swift, Ch 5.2-5.3; Paper: Truscott, J. E., and J. Brindley (1994), "Ocean Plankton Populations as Excitable Media", Bull. Math. Biol.56, 981-998. CEE262C Lecture 4: Phase line analysis and NPZ models

  2. Levels of ODE models: Linear O(x1) CEE262C Lecture 4: Phase line analysis and NPZ models

  3. Logistic equation O(x2) CEE262C Lecture 4: Phase line analysis and NPZ models

  4. Improved predator-prey Noncompetitive parts CEE262C Lecture 4: Phase line analysis and NPZ models

  5. q-logistic model CEE262C Lecture 4: Phase line analysis and NPZ models

  6. Phase directions for the logistic equation Do small populations always recover as this model implies? CEE262C Lecture 4: Phase line analysis and NPZ models

  7. Allee-effect equation O(x3) CEE262C Lecture 4: Phase line analysis and NPZ models

  8. Phase directions for the Allee-effect equation CEE262C Lecture 4: Phase line analysis and NPZ models

  9. Allee-effect equation K=1,T=0.25,r=0.75 K=1,T=0.15,r=0.5 Carrying capacity (K) K=1,T=0.25,r=0.5 Threshold populations (T) CEE262C Lecture 4: Phase line analysis and NPZ models

  10. Example: Fixed harvesting CEE262C Lecture 4: Phase line analysis and NPZ models

  11. CEE262C Lecture 4: Phase line analysis and NPZ models

  12. Example: Harvesting effort CEE262C Lecture 4: Phase line analysis and NPZ models

  13. CEE262C Lecture 4: Phase line analysis and NPZ models

  14. Higher-order functions: Modeling phytoplankton blooms CEE262C Lecture 4: Phase line analysis and NPZ models

  15. Phytoplankton blooms or "red tides":model requirements: • Existence of two states • equilibrium P-Z level – stable, can remain throughout the red-tide season • outbreak P-Z level – quasi-stable, can remain for weeks or months • Rapid outbreaks followed by slow relaxation: implies 2 time scales • Trigger mechanism • Cyclic nature – both triggering and restoring mechanism CEE262C Lecture 4: Phase line analysis and NPZ models

  16. Modified Predator-Prey CEE262C Lecture 4: Phase line analysis and NPZ models

  17. Model for the grazing rate G(P,Z) CEE262C Lecture 4: Phase line analysis and NPZ models

  18. All F(P)1 as P ; Which function to use? Functional forms of G(P,Z)=RmZ F(P) CEE262C Lecture 4: Phase line analysis and NPZ models

  19. Behavior of Nonzero fixed points: 1 Stable at P=K 1 unstable Nonzero fixed points: 1 Stable at P=K 1 unstable Nonzero fixed points: 2 Stable! 1 unstable CEE262C Lecture 4: Phase line analysis and NPZ models

  20. The phytoplankton bloom model CEE262C Lecture 4: Phase line analysis and NPZ models

  21. Behavior of P2/(a2+P2) a is the "Predation response concentration" Predators respond and start grazing when prey population is roughly P=a. Low a: Predators respond quickly to low P High a: Predators respond more slowly and wait for P to increase. CEE262C Lecture 4: Phase line analysis and NPZ models

  22. Nondimensionalize CEE262C Lecture 4: Phase line analysis and NPZ models

  23. Nullclines and fixed points CEE262C Lecture 4: Phase line analysis and NPZ models

  24. CEE262C Lecture 4: Phase line analysis and NPZ models

  25. Excitability based on initial conditions npzdemo.m CEE262C Lecture 4: Phase line analysis and NPZ models

  26. CEE262C Lecture 4: Phase line analysis and NPZ models

  27. Excitability of the red tide depends on F(P) CEE262C Lecture 4: Phase line analysis and NPZ models

  28. CEE262C Lecture 4: Phase line analysis and NPZ models

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