1. Predation and Competition
Abdessamad Tridane
MTBI summer 2008
MTBI summer 2008

2. Two-species interactions
Neutral
Mutualism -
Competition
Amensalism
Commensalism +
Herbivory
Parasitoidism
Parasitism
Predation
Response
of Sp B
of Sp A
Types

3. Predation

4. Types of predators Carnivores – kill the prey during attack
Herbivores – remove parts of many prey,
rarely lethal.
Parasites – consume parts of one or few prey,
Parasitoids – kill one prey during prolonged
attack.

5. Adaptations to avoid being eaten:
How has predation influenced evolution?
Adaptations to avoid being eaten:
spines (cactii, porcupines)‏
hard shells (clams, turtles)‏
toxins (milkweeds, some newts)‏
bad taste (monarch butterflies)‏
camouflage
aposematic colors
mimicry

6. Camouflage – blending in

7. Aposematic colors – warning

8. Mimicry – look like something that is dangerous or tastes bad
MTBI summer 2008

9. Mimicry – look like something that is dangerous
or tastes bad
Mullerian mimicry – convergence of several unpalatable species
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10. Mimicry – look like something that is dangerous
or tastes bad
Batesian mimicry – palatable species mimics an
unpalatable species
model
mimic
mimics
model
MTBI summer 2008

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13. A verbal model of predator-prey cycles:
Predators eat prey and reduce their numbers
Predators go hungry and decline in number
With fewer predators, prey survive better and increase
Increasing prey populations allow predators to increase
And repeat…
MTBI summer 2008

14. Why don’t predators increase at the same time as the prey?
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15. The Lotka-Volterra Model: Assumptions
Prey grow exponentially in the absence of predators.
Predation is directly proportional to the product of prey and predator abundances (random encounters).
Predator populations grow based on the number of prey. Death rates are independent of prey abundance.
MTBI summer 2008

16. An introduction to prey-predator Models
Lotka-Volterra model
Lotka-Volterra model with prey logistic growth
Holling type II model

17. Generic Model f(x) prey growth term g(y) predator mortality term
h(x,y) predation term
e prey into predator biomass conversion coefficient
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18. Lotka-Volterra Model r prey growth rate : Malthus law
m predator mortality rate : natural mortality
Mass action law
a and b predation coefficients : b=ea
e prey into predator biomass conversion coefficient
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19. Number of predators depends on the prey population.
isocline
Number of
Predators (y)‏
Predators
decrease
Predators
increase
m/b
Number of prey (x)‏
MTBI summer 2008

20. Number of prey depends on the predator population.
Prey decrease
Prey
Isocline
Number of
Predators (y)‏
r/a
Prey increase
m/b
Number of prey (x)‏
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21. Lotka-Volterra nullclines
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24. Direction field for Lotka-Volterra model
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25. Local stability analysis
Jacobian at positive equilibrium
detJ*>0 and trJ*=0 (center)‏
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26. Linear 2D systems (hyperbolic)‏
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27. Local stability analysis
Proof of existence of center trajectories (linearization theorem)‏
Existence of a first integral H(x,y) :
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28. Lotka-Volterra model
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29. Lotka-Volterra model
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30. Hare-Lynx data (Canada)‏
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31. Logistic growth (sheep in Australia)‏
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32. Freshmen and donuts: an example
There is a room with 100 donuts – what does a typical male freshmen do?
First – eat several donuts. (A male freshman can eat 10 donuts)‏
Second – rapidly tell friends
But not too many!
Third – Room reaches carrying capacity at 10 male freshmen.
So K=10 for male freshmen.

33. Lotka-Volterra Model with prey logistic growth
MTBI summer 2008

34. Nullclines for the Lotka-Volterra model with prey logistic growth
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35. Lotka-Volterra Model with prey logistic growth
Equilibrium points : (0,0) (K,0) (x*,y*)‏
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36. Local stability analysis
Jacobian at positive equilibrium
detJ*>0 and trJ*<0 (stable)‏
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37. Condition for local asymptotic stability
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38. Lotka-Volterra model with prey logistic growth : coexistence
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39. Lotka-Volterra with prey logistic growth : predator extinction
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40. Transcritical bifurcation
(K,0) stable and (x*,y*) unstable
and negative
(K,0) and (x*,y*) same
(K,0) unstable and (x*,y*) stable
and positive
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41. Loss of periodic solutions
coexistence
Predator extinction
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42. Competition
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43. How do species interact?
Competition
Predation
Herbivory
Parasitism
Disease
Mutualism
MTBI summer 2008

44. Interspecific Competition
When two species use the same limited resource to the detriment of both species.
Assessment-some general features of interspecific competition
Competitive exclusion or coexistence
Tilman’s model of competition for specific resources (ZINGIs)‏
Coexistence: reducing competition by dividing resources
MTBI summer 2008

45. Assessment mechanisms
consumptive or exploitative — using resources (most common)‏
preemptive — using space, based on presence
overgrowth — exploitative PLUS preemptive
chemical — antibiotics or allelopathy
territorial — like preemptive, but behavior
encounter — chance interactions
MTBI summer 2008

46. Modeling coexistence? Can we model the growth of 2 species?
Remember logistic model?
What is K?
Now we add another factor that can limit the abundance of a species.
Another species.
MTBI summer 2008

47. Freshmen and donuts: an example
What happens if a male and female discover the room at the same time?
First – eat several donuts. (A female freshman can eat 5 donuts)‏
Second – rapidly tell friends
But not too many!
Third – Room reaches carrying capacity at ? males and ? females.
What is the carrying capacity?
It depends…
MTBI summer 2008

48. Lotka-Volterra Need a way to combine the two equations.
If species are competing, the number of species A decreases if number of species B increases.
Such that:
Where alpha is the competition coefficient
Lotka-Volterra: A logistic model of interspecific competition of intuitive factors.
MTBI summer 2008

49. Freshman Example In a room we have 100 donuts.
Need 10 donuts for each male freshmen.
So K1 = 10
Need only 5 donuts for each female freshmen.
So K2 = 20
If room is at K1 and 1 male leaves, how many females can come in?
So, , where α = 0.5
And, , where B = 2
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50. Possible outcomes when put two species together.
Species A excludes Species B
Species B excludes Species A
Coexistence
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51. Changes in population 1:
Yellow: both increase
White: both decrease
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52. Changes in population 2:
Yellow: both increase
White: both decrease
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53. Yellow: both increase White: both decrease Green: Sp 1 increase
Brown: Sp 2 increase
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54. Tilman’s model Problems with Lotka-Voltera model? No mechanism
Logistic-competition theory is based on the dynamics of the consumer populations involved, i.e., it does not explicitly consider changes in resources utilized by the competitors. Tilman (1982) treated the regulation of population size from the standpoint of resource dynamics, i.e., supply and consumption.
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56. 1 – no species can survive 2 – Only A can live
3 – Species A out competes B
4 – Stable coexistence
5 – Species B out competes A
6 – Only B can live
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57. MTBI summer 2008

58. Will be posted on my website http://math.asu.edu/~tridane
Homework
Will be posted on my website