Download Trophic Ecosystem Models

Trophic Ecosystem Models
Overview
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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
Volterra, V., “Variazioni e
Lotka, A.J., Elements of Physical fluttuazioni del numero d’individui
Biology, Williams and Wilkins,
in specie animali conviventi”,
(1925)
Mem. Acad. Lincei Roma, 2, 31–
113, (1926)
Lotka (1925) Volterra (1926)
dW
 rW  eWL
dt
dL
 mL  eaWL
dt
W prey numbers
L predator numbers
r W intrinsic rate of
increase
e predator predation
efficiency
m predator natural mortality
a predator assimilation
efficiency
5
Biological unrealism of Lotka
Volterra
• No prey self limitation
• No predator self limitation
• No limit on prey consumption per predator
– Known as functional response
6
Dynamic behavior
90,000,000
7,000,000
80,000,000
70,000,000
6,000,000
5,000,000
60,000,000
50,000,000
40,000,000
30,000,000
4,000,000
Wild
3,000,000
Lions
These models are
either unstable or
cyclic
2,000,000
20,000,000
10,000,000
1,000,000
-
0
50
100
150
200
250
300
Time
7
Adding some biological realism
 Wt 
Wt 1  Wt  rWt 1    K t
k 

Prey (W) dynamics - - K is kill
Lt 1  Lt s  K t a
Predator (L) dynamics - s is survival a is assimilation
K t  Wt 1  exp(hLt ) 
The kill is one minus the fraction surviving the predation
h is the proportion of the prey searched for and found
and killed per year by each predator
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Functional Responses
(C.S. “Buzz”) Holling
The type II functional response
(the disk equation)
TT a ' pc N
Na 
1  ha ' pc N
Na number attacked
N number there (density)
a’ area searched
pc probability of successfully detecting and attacking
b handling time
Multiprey functional response
TT ai ' pci N i
N ai 
1   h j a j ' pcj N j
j
11
Dynamic behavior in time
1,200,000
18,000
16,000
1,000,000
14,000
800,000
12,000
600,000
400,000
10,000
Wild
8,000
Lions
6,000
4,000
200,000
2,000
-
0
50
100
150
200
250
300
12
Predator prey phase diagram
30,000
25,000
Lions
20,000
15,000
10,000
5,000
-
500,000
1,000,000
1,500,000
2,000,000
Wildebeest
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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
𝑎𝑃𝑟𝑒𝑦
𝑁𝑎 =
𝑏 + 𝑃𝑟𝑒𝑢
Tkill’=Pred*𝑁𝑎
𝑇𝑘𝑖𝑙𝑙′
𝑇𝑘𝑖𝑙𝑙 = 𝑃𝑟𝑒𝑦 1 − 𝑒𝑥𝑝 −
𝑃𝑟𝑒𝑦
Mpredation = Tkill/Prey
Mother = other natural mortality
F = fishing mortality
Survival = 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
The “terminal” numbers-at-age determine
the whole N matrix
N ymax 2,2
N ymax ,1
Most-recentyear Ns (year ymax)
N ymax ,2
N ymax 1,3
N ymax ,3
Oldest-age Ns
N ymax 3,4
N ymax 2,4
N ymax 1,4
N ymax ,4
Terminal numbers-at-age
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):
Fy ,a
( M  Fy , a )
C y ,a 
N y 1,a 1 (e
 1)
M  Fy ,a
+ Find Ny,a from Ny+1,a+1 and Fy,a :
N y ,a  N y 1,a 1 e
M  Fy ,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 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