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)

Volterra, V., “Variazioni e fluttuazioni del numero d’individui in specie animali conviventi”, Mem. Acad. Lincei Roma, 2, 31–113, (1926)

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Lotka (1925) Volterra (1926)

eaWLmLdtdL

eWLrWdt

dW

W prey numbers

L predator numbers

r W intrinsic rate of increase

e predator predation efficiency

m predator natural mortality

a predator assimilation efficiency

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Biological unrealism of Lotka Volterra

• No prey self limitation• No predator self limitation• No limit on prey consumption per predator

– Known as functional response

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-10,000,00020,000,00030,000,00040,000,00050,000,00060,000,00070,000,00080,000,00090,000,000

0 50 100 150 200 250 300

Time

-

1,000,000

2,000,000

3,000,000

4,000,000

5,000,000

6,000,000

7,000,000

Wild

Lions

Dynamic behavior

These models are either unstable or cyclic

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Adding some biological realism

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

The type II functional response (the disk equation)

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Na number attackedN number there (density)a’ area searchedpc probability of successfully detecting and attackingb handling time

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Multiprey functional response

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Dynamic behavior in time

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200,000

400,000

600,000

800,000

1,000,000

1,200,000

0 50 100 150 200 250 300-

2,000

4,000

6,000

8,000

10,000

12,000

14,000

16,000

18,000

Wild

Lions

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Predator prey phase diagram

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5,000

10,000

15,000

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- 500,000 1,000,000 1,500,000 2,000,000

Wildebeest

Lions

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 modelphytoplankton, zooplankton, grazer, piscivore

• Phytoplankton bottom up driven• Predation equations for other species

Tkill’=Pred*

Mpredation = Tkill/PreyMother = other natural mortalityF = fishing mortalitySurvival = exp(-(Mpredation+Mother+F))Preyt+1=Preyt*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

max

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N N

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N N

Terminal numbers-at-age

The “terminal” numbers-at-age determine the whole N matrix Oldest-age Ns

Most-recent-year Ns (year ymax)

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 :

,( ),, 1, 1

,

( 1)y aM Fy ay a y a

y a

FC N e

M F

,, 1, 1

y aM Fy a y aN N e

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 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

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