Multispecies Models (Introduction to predator-prey dynamics)

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Multispecies Models(Introduction to
predator-prey dynamics)

• Fish 458, Lecture 26

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Overview

• All of the models examined so far ignore
  multispecies considerations.
• We can divide multispecies considerations into
  biological and technological interactions.

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Biological and Technological Interactions

• Technological Interactions linkage among species
  occurs because of their co-occurrence in catches.
• Biological Interactions linkage among species
  occurs because one eats the other or they compete
  for the same prey.
• This lecture and the next lecture will focus on
  biological interactions.

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Biological Interactions

• We will develop our models of biological
  interactions using lumped differential equations
  (i.e. we are modelling the rate of change of
  population size / biomass).
• Multi-species / eco-system models are, however,
  extremely complicated and we will quickly have to
  resort to numerical methods to make use of them.

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Example 1 Foxes and Rabbits

• In the absence of foxes, rabbits increase
  uncontrolled while in the absence of rabbits,
  foxes die due to starvation
• Now let the foxes prey on the rabbits and see
  what happens

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Example 1 Foxes and Rabbits
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How Did We Do That?

• Simple method
• Keep h very small. However, this simple approach
  can be very inaccurate.
• I used the Runge-Kutta method it is much more
  accurate (and pretty fast).

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Understanding Predator-prey Dynamics

• The properties of the predator-prey system can be
  worked out from the form of the differential
  equation the phase diagram.
• The population trajectories will often be
  strongly impacted by the initial conditions.

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Constructing a Phase Diagram

1. Find any equilibrium points, i.e, values of y
   such that
2. Draw the isoclines lines for which the
   derivative is zero for one of the variables.
3. Draw arrows on each isocline indicating the rate
   of change of all other variables.

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Back to Foxes and Rabbits

• The equilibrium point is (F1,R1).
• The isoclines are defined by

Isoclines
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Adding in Rates of Change
Rabbits unchanging Foxes increasing
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But Rabbit Populations dont Grow Forever!

• We will extend the model by allowing for some
  density-dependence in the growth rate for the
  rabbit population, i.e.

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This stablizes the population
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Computing the Phase Diagram

• We proceed as before
• Compute the equilibrium point (R1F0.8).
• Compute the iscolines

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The Phase Diagram-I
The point (R1F0.8) is a stable equilibrium
t0
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The Phase Diagram-II
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Feeding Functional Relationships - I

• The current model assumes that the amount
  consumed per capita is related linearly to the
  amount of the prey.
• This may be realistic at low prey population size
  but there must be predator saturation.
• We model this effect using feeding functional
  relationships

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Feeding Functional Relationships - II
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Feeding Functional Relationships - III
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Multispecies Models

• Advantages
• Predator-prey dynamics are clearly realistic!
• Managers are often interested in ecosystem
  considerations.

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Multispecies Models

• Disadvantages
• It is very difficult to select functional forms /
  the number of species.
• The number of parameters in a multispecies model
  can be enormous (000s).
• The results of multispecies models are often
  sensitive to their specifications.
• The methods required to conduct the numerical
  integrations can be complicated and, if not done
  correctly, numerical integration impacts the
  results markedly.

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Readings

• Press et al. (1988), Chapter 16.
• Starfield and Bleloch, Chapter 6.