Ronald Westra

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• Ronald Westra
• Department of Mathematics
• Faculty of Humanities Sciences
• Universiteit Maastricht

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• 2. PREDATOR-PREY
• SYSTEMS

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Overview

• From one to two equations
• Volterras model of predator-prey (PP) systems
• Why are PP models useful?
• Examples from nature
• Sequoias
• Predators, Preys, and Hurricanes
• Three-species interactions
• Biodiversity
• Relation to future tasks

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Recall the Logistic Model
Large P slows down P
Logistic model a.k.a. the Verhulst model

• Pn is the fraction of the maximum population size
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• ? is a parameter

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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, ..)
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Volterras 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

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Behaviour of the Volterras model
Limit cycle
Oscillatory behaviour
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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 the
six-spotted mite (Eotetranychus sexmaculatus) as
the prey
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3D Java simulation of PP model

• http//www.stensland.net/java/erin.html

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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

time
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More complicated interactions

• Clinton established the Giant Sequoia National
  Monument to protect the forest from culling,
  logging and clearing.
• But many believe that Clintons 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.
• Smokey Bear did too good a job, said Matt
  Mathes, a Forest Service spokesman. It was a
  well-meaning policy with unintentional
  consequences.

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Sequias
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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

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Predators, Preys and Hurricanes
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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)
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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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equilibrium states

• Complex systems are assumed to converge towards
  an equilibrium state.Equilibrium state two (or
  more) opposite processes take place at equal rates

VIDEO
stable
unstable
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Predator-Prey ModelsForaging according to the
random walk model
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Two examples from research

• Modelling foraging
• Decaying step-lengths in foraging
• Modelling semantic network dynamics
• Growth of knowledge

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Foraging patterns in nature
Random walk
Levy flight
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Distribution of step lengths l
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Slope of the log plot equals ?
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Universal foraging behaviour

• Foraging behaviour in sparse food environments is
  characterised by Lévy-flights with ? ? 2 is
  performed by
• Albatrosses
• foraging bumblebees
• Deer
• Amoebas
• In dense food environments ? gt 3 (random walk)

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END LECTURE 2