Download - Trophic Ecosystem Models
![Page 1: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/1.jpg)
Trophic Ecosystem Models
![Page 2: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/2.jpg)
Overview
• Logistic growth model• Lotka volterra predation models• Competition models• Multispecies production models• MSVPA• Size structured models LeMans• Ecopath Ecosim• Atlantis
![Page 3: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/3.jpg)
Logistic growth Verhulst 1838
![Page 4: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/4.jpg)
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)
![Page 5: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/5.jpg)
5
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
![Page 6: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/6.jpg)
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
![Page 7: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/7.jpg)
7
-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
![Page 8: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/8.jpg)
8
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
1
1
1
h
hLWKas
aKsLL
Kk
WrWWW
ttt
ttt
tt
ttt
![Page 9: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/9.jpg)
Functional Responses (C.S. “Buzz”) Holling
![Page 10: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/10.jpg)
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
![Page 11: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/11.jpg)
11
Multiprey functional response
jjcjjj
iciiTai Npah
NpaTN'1
'
![Page 12: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/12.jpg)
12
Dynamic behavior in time
-
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
![Page 13: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/13.jpg)
13
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
![Page 14: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/14.jpg)
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?
![Page 15: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/15.jpg)
Lotka Volterra competition equations
![Page 16: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/16.jpg)
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
![Page 17: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/17.jpg)
A simple 4 trophic level modelphytoplankton, zooplankton, grazer, piscivore
• Phytoplankton bottom up driven• Predation equations for other species
![Page 18: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/18.jpg)
Tkill’=Pred*
![Page 19: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/19.jpg)
Mpredation = Tkill/PreyMother = other natural mortalityF = fishing mortalitySurvival = exp(-(Mpredation+Mother+F))Preyt+1=Preyt*Survival+PreyConsumed*EcotrophicEfficiency
![Page 20: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/20.jpg)
MSVPA
• Multi species virtual population analysis• Uses the VPA equation to calculate how
much must have been eaten by other species
![Page 21: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/21.jpg)
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)
![Page 22: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/22.jpg)
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
![Page 23: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/23.jpg)
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!
![Page 24: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/24.jpg)
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?
![Page 25: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/25.jpg)
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
![Page 26: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/26.jpg)
LeMans sequence
![Page 27: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/27.jpg)
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
![Page 28: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/28.jpg)
Ecopath and Ecosim
• Switch to Walters Slide show
![Page 29: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/29.jpg)
Atlantis
• Wait for lecture from Isaac
![Page 30: Trophic Ecosystem Models](https://reader035.vdocuments.site/reader035/viewer/2022062218/56816616550346895dd96403/html5/thumbnails/30.jpg)