Distribution System Modeling and Analysis - The Nature of Loads
Modeling Nature February 2009
description
Transcript of Modeling Nature February 2009
![Page 1: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/1.jpg)
1
Modeling NatureFebruary 2009
![Page 2: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/2.jpg)
2
Modeling Nature
LECTURE 2: Predator-prey models
* and more general information …
![Page 3: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/3.jpg)
3
Task descriptions and required readings
• Each lecture has two associated tasks (a, b) • The task descriptions can be found on
ELEUM• Each task has a number of required readings
on the ELEUM website• Additional readings and Web pointers are also
available (not mandatory, but useful)
![Page 4: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/4.jpg)
4
Task Descriptions
• Updated task descriptions of are made available through ELEUM on a weekly basis
• Note that the task descriptions in the course manual (syllabus) become over-ruled by these new descriptions !!!
![Page 5: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/5.jpg)
5
Student research project
A TEAM consists of 2 to 3 students
A 2-3 PAGE PROPOSAL is submitted ultimately on Monday 2 March to the tutor of the team’s TG.
PRESENTATION: +/-10 minutes each presentation in the week of 23 March in the TGs.
REPORT: a short paper (2500 words) on the subject.
THE GRADE is for the TEAM and thus for all students in the TEAM. It consists of 50% for the report and 50% for the presentation.
![Page 6: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/6.jpg)
6
Complete time table
• 02/2 - Lecture 1 : Models, Growth & Decay• 09/2 - Lecture 2 : Predator-Prey Systems • 16/2 - Lecture 3 : Network Models• 02/3 - Lecture 4 : Chaos and Fractals;
first draft report and presentation – feedback from tutors
• 09/3 - Lecture 5 : Percolation and Phase Transitions • 16/3 - Lecture 6 : Self-Organization and Collective Phenomena
• 23/3 - Week 7 : Student presentations on chosen topic; report
• 30/3 - Week 8 : Final exam
![Page 7: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/7.jpg)
7
Final mark of the course
The final mark of the course consists of 40% of the project and 60% of the written exam.
Only students with sufficient attendance may attend the exam and present his/her project, when only 1 TG lacks for a valid pass, the student receives an additional task, otherwise the students fails the course.
![Page 8: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/8.jpg)
8
General Information
More questions?
Ask after the Lecture or your personal tutor
![Page 9: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/9.jpg)
9
Lecture 2:
PREDATOR-PREY SYSTEMS
![Page 10: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/10.jpg)
10
Overview
• From one to two equations• Volterra’s model of predator-prey (PP)
systems• Why are PP models useful?• Examples from nature• Relation to future tasks
![Page 11: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/11.jpg)
11
Recall the Logistic Model
• Pn is the fraction of the maximum population size 1
is a parameter that describes the strength of the coupling
nnn PPP 11 Logistic model a.k.a. the Verhulst model
Large P slows down P
![Page 12: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/12.jpg)
12
Interacting quantities
• The logistic model describes the dynamics (change) of a single quantity interacting with itself
• We now move to models describing two (or more) interacting quantities
![Page 13: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/13.jpg)
13
Fish statistics• Vito Volterra (1860-1940): a famous
Italian mathematician
• Father of Humberto D'Ancona, a biologist studying the populations of various species of fish in the Adriatic Sea
• The numbers of species sold on the fish markets of three ports: Fiume, Trieste, and Venice.
![Page 14: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/14.jpg)
14
percentages of predator species (sharks, skates, rays, ..)
Percentage of predators in Fiume catch
0
10
20
30
40
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
year
% p
red
ato
rs
![Page 15: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/15.jpg)
15
Volterra’s model
• Two (simplifying) assumptions– The predator species is totally dependent on the prey
species as its only food supply – The prey species has an unlimited food supply and no
threat to its growth other than the specific predator
predator prey
![Page 16: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/16.jpg)
16
predator prey
Lotka–Volterra equation :
The Lotka–Volterra equations are a pair of equations used to describe the dynamics of biological systems in which two species interact, one a predator and one its prey.
They were proposed independently by Alfred J. Lotka in 1925 and Vito Volterra in 1926.
![Page 17: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/17.jpg)
17
Lotka–Volterra equation :
Two species
species #1: population size: xspecies #2: population size: y
![Page 18: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/18.jpg)
18
Lotka–Volterra equation :
Remember Verhulst-equation:
Predator (x) and prey (y) model:
xn+1 = xn(α – βyn) : y is the limitation for x
yn+1 = yn(γ – δxn) : x is the limitation for y
![Page 19: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/19.jpg)
19
Lotka–Volterra equation :
Equivalent formulation:
rate of change: dx/dt = in/de-crease per unit time
(e.g. 2000 hares per year)
![Page 20: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/20.jpg)
20
Behaviour of the Volterra’s model
Limit cycleOscillatory behaviour
![Page 21: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/21.jpg)
21
Effect of changing the parameters (1)
Behaviour is qualitatively the same. Only the amplitude changes.
![Page 22: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/22.jpg)
22
Effect of changing the parameters (2)
Behaviour is qualitatively different. A fixed point instead of a limit cycle.
![Page 23: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/23.jpg)
23
Different modes…
![Page 24: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/24.jpg)
24
Huffaker (1958) reared two species of mites to demonstrate coupled oscillations of predator and prey densities in the laboratory. He used Typhlodromus occidentalis as the predator and the six-spotted mite (Eotetranychus sexmaculatus) as the prey
Predator-prey interaction in vivo
![Page 25: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/25.jpg)
25
Online simulation of PP model• http://www.xjtek.com/models/?archive=ecosystem_dynamics/predat
or_prey/model.jar,xjanylogic6engine.jar&root=predator_prey.Simulation$Applet&width=800&height=650&version=6
![Page 26: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/26.jpg)
26
Why are PP models useful?
• They model the simplest interaction among two systems and describe natural patterns
• Repetitive growth-decay patterns, e.g., – World population growth– Diseases– …
time
Exponential growth
Limited growth
Exponential decay
Oscillation
![Page 27: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/27.jpg)
27
Lynx and hares
Very few "pure" predator-prey interactions have been observed in nature, but there is a classical set of data on a pair of interacting populations that come close: the Canadian lynx and snowshoe hare pelt-trading records of the Hudson Bay Company over almost a century.
![Page 28: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/28.jpg)
28
Lynx and hares
![Page 29: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/29.jpg)
29
The Hudson Bay data give us a reasonable picture of predator-prey interaction over an extended period of time. The dominant feature of this picture is the oscillating behavior of both populations
![Page 30: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/30.jpg)
30
1. what is the period of oscillation of the lynx population?
2. what is the period of oscillation of the hare population?
3. do the peaks of the predator population match or slightly precede or slightly lag those of the prey population?
![Page 31: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/31.jpg)
31
“equilibrium” states
• Complex systems are assumed to converge towards an equilibrium state.
Equilibrium state: two (or more) opposite processes take place at equal rates
stableunstable
![Page 32: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/32.jpg)
32
Adaptations
![Page 33: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/33.jpg)
33
Evolutionary arms race
![Page 34: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/34.jpg)
34
This is the basis for evolution
![Page 35: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/35.jpg)
35
This is the basis for evolution
![Page 36: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/36.jpg)
36
More complicated interactions [1]
• Clinton established the Giant Sequoia National Monument to protect the forest from culling, logging and clearing.
– But many believe that Clinton’s measures added fuel to the fires.
– Tree-thinning is required to prevent large fires.– Fires are required to clear land and to promote new
growth.
![Page 37: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/37.jpg)
37
Sequoias
![Page 38: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/38.jpg)
38
Predator versus Prey?
• Fire acts as “prey” because it is needed for growth
• Fire acts as “predator” because it may set the tree on fire
• Tree acts as “prey” for the predator
• If trees die out, the predator dies out too
Fire is dangerous when caused by surrounding bushes
Fire is needed to clean area and to open the seeds of the Sequoia
![Page 39: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/39.jpg)
39
Predators, Preys and Hurricanes
More complicated interactions [2]
![Page 40: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/40.jpg)
40
Biodiversity
“Human alteration of the global environment has triggered the sixth major extinction event in the history of life and caused widespread changes in the global distribution of organisms. These changes in biodiversity alter ecosystem processes and change the resilience of ecosystems to environmental change. This has profound consequences for services that humans derive from ecosystems. The large ecological and societal consequences of changing biodiversity should be minimized to preserve options for future solutions to global environmental problems.”
F. Stuart Chapin III et al. (2000)
More complicated interactions [3]
![Page 41: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/41.jpg)
41
The role of biodiversity in global change
![Page 42: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/42.jpg)
42
Consequences of reduced biodiversity
"...decreasing biodiversity will tend to increase the overall mean interaction strength, on average, and thus increase the probability that ecosystems undergo destabilizing dynamics and collapses."
Kevin Shear McCann (2000)
![Page 43: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/43.jpg)
43
Relation to theTasks
![Page 44: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/44.jpg)
44
Predation, competition, and interaction
![Page 45: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/45.jpg)
45
![Page 46: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/46.jpg)
46
![Page 47: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/47.jpg)
47
![Page 48: Modeling Nature February 2009](https://reader036.fdocuments.net/reader036/viewer/2022062517/56813de1550346895da7b54c/html5/thumbnails/48.jpg)
48
END of LECTURE 2