General Ecology: Lecture 14
November 2, 2005
I.
II.
Predation defined
Models of predation
A.
Lotka-Volterra equation for predation
1.
Assumptions of the model
a)
Exponential growth
b)
Predators move randomly among prey.
c)
The proportion of encounters that result in prey capture and
consumption are the same at all predator and prey densities

NOTE: This is the same as saying that there is a linear
response between number of predators present and number
prey captured (text shows as an additional assumptions)
d)
No time lag for prey handling and consumption
2.
Prey equation
a)
Continuous time equation: dV/dt = rV-a´CV

V = prey (I’m using this because it is what is used in the
spreadsheet exercises, and is less ambiguous than a P)

r = intrinsic rate of growth (as in exponential growth equation)

a´ = attack/kill rate per consumer per prey

C = number of consumers

Discrete time equation: Vt+1 = RVt – a´CtVt.




With a little rearrangement, get V t+1 -Vt = RVt – aCtVt, or
ΔV = RVt- a´CtVt in one generation (= Δt)
Equation assumes the passing of one generation (= Δt), so
ΔV/Δt = RVt- a´CtVt; this is basically the same as the continuous
time equation.
b)
3.
Solve for equilibrium: dV/dt = 0

r/a´ = C. This is the isocline for prey growth [Fig. 15.1a]
Predator equation
a)
Continuous time equation: dC/dt = fa´CV –qC

f is the efficiency at which prey is turned into offspring,
multiplied by a´CV, the total number of prey
captured/consumed (see prey equation)

q is the per capita death rate

b)
B.
Discrete time equation is similar: C t = C t+1 + f a´C t V t -qC t
Solve for equilibrium: dC/dt = 0

V = q/ fa´ [Fig. 15.1b]
4.
Put the two equations together [Fig. 15.1c]
a)
Model predicts the cycling of predator and prey [Fig. 15.1c]

Need to think about individual vectors for the predator and the
prey here!

What this would look like through time [Fig. 15.1d]
Rosenzweig-MacArthur model [Fig. 15.3]
1.
NOTE: We will not go through the mathematics of this model; rather, you
should focus on what you can learn from the graphs.
2.
This model is similar to the Lotka-Volterra model, except that the prey
population grows according to the logistic growth model
a)
III.
Hump shape of dV/dt = 0 (prey zero-growth) isocline)reflects the fact
that the lowest growth rates for the prey would be at its lowest and
highest densities

At low prey densities, each act of predation has a greater
impact on the total population

At high prey densities, density-dependent factors (including
predation) decrease population growth rate.

Similar to dN/dt of logistic growth equation graphed against N.
b)
The predator population still grows exponentially and thus dC/dt = 0
is a vertical line.
3.
The outcome of the predator-prey interaction depends upon where the prey
and predator isoclines intersect. You should work through the vectors to
make sure you understand what leads to each of these three outcomes:
a)
Stable oscillation
b)
Dampened oscillation
c)
Increasing oscillations and an unstable system. Can eventually lead to
extinction of one or both species.

How does predator efficiency compare in graphs b and c, and
what is the consequence?
d)
Prey refuge

What would this correspond to in the real world of predators
and prey?
Experimental predator-prey systems
A.
Gause’s studies of Didinium (a ciliate) populations preying on Paramecium
caudatum [Fig. 15.4]
1.
Basic experiment in a closed system [Fig. 15.4a]
a)
Description and outcome
2.
Experiment with sediment as a refuge [Fig. 15.4b]
a)
Description and outcome
b)
Harvest refugia
3.
Experiment with continued immigration of both predator and prey [Fig.
15.4c]
a)
Description and outcome.
b)
Why significant
4.
Gause’s key conclusion: immigration and emigration are necessary to
maintain a balanced predator-prey relationship
a)
Note that these are not in any of the models.
B.
Huffaker’s studies of mite populations
1.
Basics of experiments (study organisms and set-up)
2.
His goal: create a closed system where predator and prey oscillated
3.
Variations and results
4.
Huffaker’s key conclusions: Like Gause, he concluded that immigration
was required…
C.
A few laboratory experiments have been successful, in that predator and prey
populations exhibit stable oscillations without immigration.
1.
Not the norm, though.
Study questions
1.
What are the basic assumptions of the Lotka-Volterra model for predation? How do the
assumptions of the Rosenzweig-MacArthur model differ?
2.
Are the Lotka-Volterra equations based on exponential growth or logistic growth? What
about the Rosenzweig-MacArthur model? (HINT: Think carefully about the R-M
model—answer not straightforward!)
3.
Be able to recognize and write out the Lotka-Volterra equations for both the prey and the
predator. We will focus on the continuous time model. You should be able to define all
the terms in both equations and state what each equation means in words.
4.
Make a sketch of the Lotka-Volterra predator-prey graph. Be sure you:
a.
label your axes,
b.
Identify and label both the predator and prey isoclines. Know what these lines
actually mean (state in words).
c.
Identify and label the x and y-intercepts.

Solve for from the basic equations. Show your work!
5.
What pattern of predator and prey populations does the Lotka-Volterra model predict? Be
able to draw prey, predator and “net” vectors in each quadrant.
6.
Why is the zero-growth isocline for the prey drawn as a hump in the RosenweigMacArther model?
7.
Given any of the graphs shown in Fig. 15.3 (that illustrate the Rosenzweig-MacArthur
model), be able to
a.
b.
c.
d.
e.
f.
Label the axes
Label the lines
Draw the proper vectors in each of the regions
Describe and explain the outcome of each situation (i.e. will the population oscillate, reach
a stable equilibrium, etc… and why?)
Be able to match each graph on the right with the phase-plane diagram on the left.
Be sure you understand the difference between Fig. 15.3 b and 15.3 c. In which is the
predator more efficient? In practical terms, why might that be ultimately problematic for
the predator as well as the prey?
8.
Discuss the three different outcomes obtained by Gause in his predator-prey experiments
with ciliates (Fig. 15.4). Be sure you know the differences in the experimental conditions
that led to the different outcomes.
9.
What general conclusion did Gause reach after he completed his experiments? Are these
conclusions consistent with the Lotka-Volterra models? Do they match any of the models?
10.
What is a “harvest refugium” Describe the simple experiment Gause did to show the
potential value of harvest refugia to prey species. What is the ecological value of harvest
refugia in the real world?
11.
What did Huffaker think was necessary for creating a stable, oscillating predator-prey
system in the lab? Was he successful? Explain. What general conclusion did Huffaker
reach?