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Predation and Competition
Abdessamad Tridane
MTBI summer 2008 1
Two-species interactions
Types Response Response
of Sp A of Sp B
Competition - -
Predation + -
Parasitism + -
Parasitoidism + -
Herbivory + -
Neutral
Mutualism
Commensalism
Amensalism
2
Predation
3
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,
rarely lethal.
Parasitoids – kill one prey during prolonged
attack.
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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
5
Camouflage – blending in
6
Aposematic colors – warning
7
Mimicry – look like something that is dangerous
or tastes bad
Mimicry – look like something that is dangerous
or tastes bad
Mullerian mimicry – convergence of several unpalatable
species
Mimicry – look like something that is dangerous
or tastes bad
Batesian mimicry – palatable species mimics an
unpalatable species
model
mimic
mimics
model
11
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A verbal model of predator-prey
cycles:
1. Predators eat prey and reduce their
numbers
2. Predators go hungry and decline in
number
3. With fewer predators, prey survive
better and increase
4. Increasing prey populations allow
predators to increase
Andrepeat…
13
Whydon’tpredatorsincreaseatthesame
time as the prey?
14
The Lotka-Volterra Model:
Assumptions
1. Prey grow exponentially in the
absence of predators.
2. Predation is directly proportional to the
product of prey and predator
abundances (random encounters).
3. Predator populations grow based on
the number of prey. Death rates are
independent of prey abundance.
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An introduction to prey-predator Models
• Lotka-Volterra model
• Lotka-Volterra model with prey logistic growth
• Holling type II model
16
Generic Model
• f(x) prey growth term
• g(y) predator mortality term
• h(x,y) predation term
• e prey into predator biomass conversion coefficient
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
Number of predators depends on the prey population.
Predator
isocline
m
x=
b
Number of r
Predators (y) y=
a
Predators Predators
decrease increase
m/b
Number of prey (x)
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umber of prey depends on the predator population.
m
Prey decrease x=
Prey
Isocline
b
Number of r
r/a
Predators (y) y=
a
Prey increase
m/b
Number of prey (x)
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Lotka-Volterra nullclines
22
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Direction field for Lotka-Volterra model
Local stability analysis
• Jacobian at positive equilibrium
• detJ*>0 and trJ*=0 (center)
• Linear 2D systems (hyperbolic)
Local stability analysis
• Proof of existence of center trajectories (linearization theorem)
• Existence of a first integral H(x,y) :
Lotka-Volterra model
Lotka-Volterra model
Hare-Lynx data (Canada)
• Logistic growth (sheep in Australia)
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. dN KN
rN
• So K=10 for male freshmen. dt
K
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• Lotka-Volterra Model with prey logistic growth
Nullclines for the Lotka-Volterra model
with prey logistic growth
• Lotka-Volterra Model with prey logistic growth
• Equilibrium points : (0,0) (K,0) (x*,y*)
Local stability analysis
• Jacobian at positive equilibrium
• detJ*>0 and trJ*<0 (stable)
• Condition for local asymptotic stability
Lotka-Volterra model with prey
logistic growth : coexistence
Lotka-Volterra with prey logistic
growth : predator extinction
Transcritical bifurcation
Kx (K,0) stable and (x*,y*) unstable
and negative
(K,0) and (x*,y*) same
K=x
K x (K,0) unstable and (x*,y*) stable
and positive
• Loss of periodic solutions
x-y x-y
8 20
6,4 16
4,8 12
y
y
3,2 8
1,6 4
0 0
0 0,3 0,6 0,9 1,2 1,5 0 0,3 0,6 0,9 1,2 1,5
x x
coexistence Predator extinction
Competition
How do species interact?
• Competition
• Predation
• Herbivory
• Parasitism
• Disease
• Mutualism
Interspecific Competition
• 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 modelofcompetitionforspecific
resources (ZINGIs)
• Coexistence: reducing competition by dividing
resources
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
Modeling coexistence?
• Can we model the growth of 2 species?
• Remember logistic model?
dN KN
rN
dt K
• What is K?
• Now we add another factor that can limit the
abundance of a species.
– Another species.
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?
– Itdepends…
dN1 K1 N1 dN 2 K2 N2
r1 N1 r2 N 2
dt K1
dt K2
dN 2 K2 N2
Lotka-Volterra r2 N 2
dt K2
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: N = αN
1 2
Where alpha is the competition coefficient
Lotka-Volterra: A logistic model of interspecific
competition of intuitive factors.
dN1 K1 N1 αN 2
r1 N1
dt K1
Freshman Example
dN1 K1 N1
r1 N1
dt N1
• 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?
N 1 = αN2
• So, N 2 = βN 1 , where α = 0.5
• And, , where B = 2
dN1 K1 N1 αN 2 dN 2 K 2 N 2 βN1
r1 N1 r2 N 2
dt N1 dt K2
Possible outcomes when put two species together.
• Species A excludes Species B
• Species B excludes Species A
• Coexistence
Changes in population 1:
dN1 K1 N1 αN 2
r1 N1
dt K1
Yellow: both increase
White: both decrease
Changes in population 2:
dN 2 K 2 N 2 βN1
r2 N 2
dt K2
Yellow: both increase
White: both decrease
Yellow: both increase
White: both decrease
Green: Sp 1 increase
Brown: Sp 2 increase
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.
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
Homework
Will be posted on my website
http://math.asu.edu/~tridane
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