trophic ecosystem models
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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
predatoreach by year per killed andfound andfor searchedprey theof proportion theis predation thesurvivingfraction theminus one is kill The
)exp(1on assimilati is survival is - dynamics (L)Predator
kill isK -- dynamics (W)Prey
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1
1
h
hLWKas
aKsLL
Kk
WrWWW
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Functional Responses (C.S. “Buzz”) Holling
The type II functional response (the disk equation)
NphaNpaTNc
cTa '1
'
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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iciiTai Npah
NpaTN'1
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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
-
5,000
10,000
15,000
20,000
25,000
30,000
- 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
ma
max
max
max
max max m
x
ax max
2,2
1
3,4
2,4
1,4
,1 ,2 ,3 ,
3
4
,
y
y
y
y y
y
y
y
y
NNNN
N N
N
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
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