Transcript Lecture 2 - DKE Personal & Projects Websites

Modeling Nature
February 2009
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Modeling Nature
LECTURE 2: Predator-prey models
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and more general information …
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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)
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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 overruled by these new descriptions !!!
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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.
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Complete time table
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•
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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
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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.
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General Information
More questions?
Ask after the Lecture or your personal tutor
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Lecture 2:
PREDATOR-PREY
SYSTEMS
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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
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Recall the Logistic Model
Pn1  Pn 1  Pn 
Large P slows down P
Logistic model a.k.a. the Verhulst model
• Pn is the fraction of the maximum population
size 1
•  is a parameter that describes the
strength of the coupling
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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
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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.
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percentages of predator
species (sharks, skates, rays, ..)
Percentage of predators in Fiume catch
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10
year
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% predators
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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
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Lotka–Volterra equation :
predator
prey
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.
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Lotka–Volterra equation :
Two species
species #1: population size: x
species #2: population size: y
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Lotka–Volterra equation :
Remember Verhulst-equation:
Predator (x) and prey (y) model:
xn+1 = xn(α – βyn)
yn+1 = yn(γ – δxn)
: y is the limitation for x
: x is the limitation for y
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Lotka–Volterra equation :
Equivalent formulation:
rate of change: dx/dt = in/de-crease
per unit time
(e.g. 2000 hares per year)
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Behaviour of the Volterra’s model
Oscillatory behaviour
Limit cycle
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Effect of changing the parameters
(1)
Behaviour is qualitatively the same. Only the amplitude changes.
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Effect of changing the parameters
(2)
Behaviour is qualitatively different. A fixed point instead of a limit cycle.
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Different modes…
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Predator-prey interaction in vivo
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 the24
six-spotted mite (Eotetranychus sexmaculatus) as the prey
Online simulation of PP model
• http://www.xjtek.com/models/?archive=ecosystem_dynamics/predat
or_prey/model.jar,xjanylogic6engine.jar&root=predator_prey.Simulati
on$Applet&width=800&height=650&version=6
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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
–…
Exponential growth
Limited growth
Exponential decay
Oscillation
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time
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.
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Lynx and hares
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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
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behavior of both populations
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?
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“equilibrium” states
• Complex systems are assumed to
converge towards an equilibrium state.
Equilibrium state: two (or more) opposite processes
take place at equal rates
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unstable
stable
Adaptations
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Evolutionary arms race
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This is the basis for evolution
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This is the basis for evolution
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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.
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Sequoias
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Predator versus Prey?
Fire is dangerous when caused by
surrounding bushes
Fire is needed to clean area and
to open the seeds of the Sequoia
• 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
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predator dies out too
More complicated interactions [2]
Predators, Preys and Hurricanes
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More complicated interactions [3]
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)40
The role of biodiversity in global change
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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)
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Relation to theTasks
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Predation, competition, and interaction
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END of LECTURE 2
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