Predation

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Predation

• Community Level Effects

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Predation

• Functional Responses
• As prey density increases, each predator can
  consume more prey

• Numerical Responses
• As prey density increases, predators increase in
  number, and that larger number of predators
  consumes more prey.

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Predation

• As an example, consider the work of Holling in
  1959.
• Studied 3 species of small mammal
• Peromyscus leucopus
• Blarina brevicauda
• Sorex cinereus

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Predation

• These 3 species all consume saw-fly larvae, and
  each does it in such a way that it is possible to
  determine who consumed what.
• Holling sampled his study site to determine prey
  (saw-fly larva) density, and estimated small
  mammal density.

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Predation

• Notice, the number of Blarina and Peromyscus was
  basically constant regardless of pupae density.
• Blarina and Peromyscus appear to have a
  relatively small numerical response to increased
  prey density.
• Sorex appears to have a relatively large
  numerical response.

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Predation

• We could also look at number of pupae consumed by
  each species relative to density of pupae.
• This gives us an index of the functional response.

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Predation

• Here we note that the relationships are opposite
  of what we saw before.
• Blarina has a large functional response,
  Peromyscus is intermediate, and Sorex has the
  smallest functional response.
• What does this tell you about these predators?

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Predation

• We can combine these relationships into a single
  graph, by looking at precent predation relative
  to prey density.
• This is done by multiplying number of pupae eaten
  by number of predators present, and dividing by
  density of the pupae (proportion of pupae eaten).

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Predation

• Notice that the proportion of prey eaten by each
  species peaks at a different pupal density.
  Therefore, as sawfly density increases, it
  encounters first significant Blarina predation,
  then Sorex predation, then Peromyscus predation.
• This is very different than if all peaked at the
  same density.

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Predation

• The functional response is a consequence of the
  coevolutionary interaction between predator and
  prey, and the reproductive biology of the
  predator.
• Holling predicted 3 forms of the functional
  response.

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Functional Responses

• Type I a linear response between number of prey
  consumed and prey density.
• Type II prey consumption is asymptotic.
• Type III prey consumption is logistic.

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Functional Response

• The asymptotic behavior of Type I and Type II
  functional responses is a consequence of
  satiation of the predator, or increased handling
  time as prey are consumed at a high rate.
• This is important in terms of the effect the
  predator has on the prey population.

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Functional Response

• Look at the proportion of prey eaten over the
  range of prey densities, for each kind of
  functional response.

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Functional Response

• The Type I response is density independent.
• Type II response is density dependent, with, but
  it is decreasing with increased prey density.
• Type III is density dependent, but in different
  directions depending on prey density.

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Functional Response

• Only the Type III functional response is density
  dependent in a way that promotes population
  regulation.
• The Type III functional response is the one most
  likely to regulate prey populations.

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Functional Response

• What causes a Type III functional response?
• Factors that cause low hunting efficiency at low
  prey density.
• Failure to develop an appropriate search image
  without positive reinforcement.
• Presence of prey refugia at low densities.

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Functional Response

• Clearly, it is highly unlikely that a predator
  will cause the extinction of its prey.
• Just as in parasites, killing your host will be
  selected against.

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Predator Prey Coexistence.

• Under what conditions will we see stable
  coexistence of a predator and its prey?
• This is very similar to what we did with 2
  competing species.

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Predator Prey Coexistence

• Our basic strategy is to
• 1) write simple differential equations describing
  the growth of the 2 populations
• 2) define equilibrium as the point where the
  populations do not change.
• 3) do a phase plane analysis using the isoclines
  for the 2 species.

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Predator Prey Coexistence

• Model for the predator population

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Predator Prey Coexistence

• Here, dP/dt is the growth rate of the predator.
• a production efficiency of the predator
  (proportion of energy assimilated by predator
  that is converted into new predator biomass.
• pingestion efficiency of the predator
  (proportion of available prey actually consumed).

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Predator Prey Coexistence

• H density of the prey.
• d death rate of the predator. Notice, in the
  absence of prey, the predator population must
  certainly decrease.

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Predator Prey Coexistence

• For the prey population, we have

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Predator Prey Coexistence

• dH/dt growth rate of the prey population.
• p, H, and P are as in the previous equation.
• r is the birth rate of the prey.

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Predator Prey Coexistence

• Note, the births of prey are decreased by deaths
  (pHP).
• Note also, that encounters between predator and
  prey is the product of their numbers. This is a
  brownian motion idea.

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Predator Prey Coexistence

• At equilibrium,

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Equilibrium
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Equilibrium

• Predator isocline

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Equilibrium

• Prey isocline

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Equilibrium

• As in the case of competing species, these
  differential equations have no explicit
  solutions. So we simply plot the isoclines.
• This produces the following graph

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Equilibrium

• The behavior of this system is very intuitive,
  and very pleasing.
• It produces exactly the type of behavior we see
  in the moose and wolves of Isle Royale.
• We see this behavior in lynx/showshoe hare
  systems. From another view, we could plot it as

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Equilibrium

• The lags make sense.
• It takes time for the predator population to
  catch up with they prey.
• Predators do not produce new predators
  instantaneously. Nor do they stop reproducing
  instantaneously.

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Equilibrium

• We can make the model more realistic.
• We know that there will be a carrying capacity
  for the prey population, and probably for the
  predator as well.
• There will aslo be an Allee effect some minimum
  population size necessary to sustain the
  population.

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Equilibrium

• New the system becomes much more interesting.
  The exact position of the predator isocline will
  be very important.
• The results will be different for systems in
  which the predator isocline is to the left or
  right of the hump in the prey isocline.

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Unstable equilibrium

• In this first scenario, once the system is
  perturbed from the equilibrium point, the systems
  spirals out of bounds and extinction results.
  Why?

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Unstable equilibrium

• Here, the hump is to the right of the predator
  isocline, and we get an unstable system. Why?
• The predator population is capable of growing
  even at very low prey density - the predator is
  an efficient humter. As the predator becomes
  less efficient, the predator isocline shifts to
  the right.

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Stable equilibrium

• When the hump is to the left, the region in
  which the predator population does not grow is
  larger.
• Very high prey densities are necessary for the
  predator to increase. This might be the result
  of crypsis, or inefficient foraging by the
  predator.

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An interesting twist

• There was a wonderful paper by Rosenzweig in
  Science quite a few years ago, titled the
  paradox of the plankton.
• In the paper, Rosenzweig showed that zoo plankton
  did not follow the predictions.

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A twist

• The predator isocline was to the left, but the
  system persisted and did not cycle to extinction.
  It did cycle, but not to extinction.
• What happened?

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A twist

• If prey have a refugia, or an escape from
  predation, the isocline will look different

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Equilibrium

• The phytoplankton had a refugia, and consequently
  the zooplankton were unable to exploit the entire
  prey population. The result was a cyclical
  system as shown.

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An important point

• What are the assumptions of these models?
• First, there is a type I functional response.
• Second, there are no stochastic effects.
• Nevertheless, the models give us a basic
  understanding of how the system work.