Trophic Ecosystem Models

Overview Logistic growth model Lotka volterra predation models Competition models Multispecies production models MSVPA Size structured models LeMans Ecopath Ecosim Atlantis

Logistic growth Verhulst 1838

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

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

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

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

11 Multiprey functional response

12 Dynamic behavior in time

13 Predator prey phase diagram

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?

Lotka Volterra competition equations

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

A simple 4 trophic level model phytoplankton, zooplankton, grazer, piscivore Phytoplankton bottom up driven Predation equations for other species

M predation = T kill /Prey M other = other natural mortality F = fishing mortality Survival = exp(-(M predation +M other +F)) Prey t+1 =Prey t *Survival+PreyConsumed*EcotrophicEfficiency

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

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

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

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!

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?

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

LeMans sequence

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

Ecopath and Ecosim Switch to Walters Slide show

Atlantis Wait for lecture from Isaac