Trophic Ecosystem Models

1. Trophic Ecosystem Models

2. Overview • Logistic growth model • Lotkavolterra predation models • Competition models • Multispecies production models • MSVPA • Size structured models LeMans • EcopathEcosim • Atlantis

3. Logistic growth Verhulst 1838

4. Lotka and Volterra Volterra, V., “Variazioni e fluttuazioni del numero d’individui in specie animali conviventi”, Mem. Acad. Lincei Roma, 2, 31–113, (1926) Lotka, A.J., Elements of Physical Biology, Williams and Wilkins, (1925)

5. Lotka (1925) Volterra (1926) W prey numbers L predator numbers r W intrinsic rate of increase e predator predation efficiency m predator natural mortality a predator assimilation efficiency

6. Biological unrealism of Lotka Volterra • No prey self limitation • No predator self limitation • No limit on prey consumption per predator • Known as functional response

7. Dynamic behavior These models are either unstable or cyclic

8. Adding some biological realism

9. Functional Responses (C.S. “Buzz”) Holling

10. The type II functional response (the disk equation) Na number attacked N number there (density) a’ area searched pc probability of successfully detecting and attacking b handling time

11. Multiprey functional response

12. Dynamic behavior in time

13. Predator prey phase diagram

14. Predator or Prey self limitation • Do we allow for self limitation, or assume that food (in the form of prey eaten) is the only limiting factor?

15. LotkaVolterra competition equations

16. Multispecies Production Models • Biomass dynamics models with trophic interactions • Captures predation effects • Problems: what you eat and who eats you changes through the life history – size or age usually needed to capture this • Switch to simple example in EXCEL

17. A simple 4 trophic level modelphytoplankton, zooplankton, grazer, piscivore • Phytoplankton bottom up driven • Predation equations for other species

18. Tkill’=Pred*

19. Mpredation = Tkill/Prey Mother = other natural mortality F = fishing mortality Survival = exp(-(Mpredation+Mother+F)) Preyt+1=Preyt*Survival+PreyConsumed*EcotrophicEfficiency

20. MSVPA • Multi species virtual population analysis • Uses the VPA equation to calculate how much must have been eaten by other species

21. VPA Back-calculation - I The “terminal” numbers-at-age determine the whole N matrix Oldest-age Ns Most-recent- year Ns (year ymax) Terminal numbers-at-age

22. VPA Back-calculation - II Given Ny+1,a+1 and Cy,a, Fy,a and Ny,a are calculated as follows: + Find Fy,a from the catch equation, i.e. by solving (using bisection or Newtons method): + Find Ny,a from Ny+1,a+1 and Fy,a :

23. How MSVPA differs from VPA • Instead of assuming M constant, M depends on how much other species at of prey species • This requires diet composition • Thousands and thousands of stomachs need to be examined!

24. Simulating MSVPA using MSFOR • What do you assume about diet composition? • Does it change with relative abundance? • Do you allow for a functional response? • What about a spawner recruit relationship?

25. Size structured models LeMans • Number of individuals by species and size class Nij • Growth parameters to calculate proportion growing between size classes each time interval ϕij proportion moving from i to j • Mortality has three components • Predation accounted for in model M2 • Other natural mortality M1 • Fishing mortality F

26. LeMans sequence

27. Limitations in LeMans • No relation between food availability and growth (or consumption) and survival or recruitment • Thus we can’t use it to examine impact on top predators of reducing their prey • Or bottom up forcing • BUT we can look at impacts of reducing predators on prey species

28. Ecopath and Ecosim • Switch to Walters Slide show

29. Atlantis • Wait for lecture from Isaac