1. FW364 Ecological Problem Solving
Class 20: Predation
November 12, 2013

2. Outline for Today Continuing with our shift in focus from
populations of single species to community interactions
Objectives from Last Class:
Introduced coupled predator-prey interactions
Introduced basic predator-prey equations
Objectives for Today:
Application of calculus to predator-prey dynamics
Continue building predator predator-prey equations
Look at a very important example of predator-prey interactions
Return the mid-term 2
Objectives for Next Two Classes:
Link predator-prey equations
Develop predator-prey models using different assumptions
No textbook chapters!

3. Recap from Last Class Vt+1 = Vt + Bv - Dv Pt+1 = Pt + Bp - Dp
Developed separate (not explicitly linked) predator and prey equations
Started with stock and flow models:
Prey
Births
Deaths
Predator
Births
Deaths
Revised familiar equation: Nt+1 = Nt + B - D
Victim, V:
Vt+1 = Vt + Bv - Dv
Predator, P:
Pt+1 = Pt + Bp - Dp

4. Which in a natural system look like:
Recap from Last Class
Victim, V:
Vt+1 = Vt + Bv - Dv
Predator, P:
Pt+1 = Pt + Bp - Dp
Dynamics go like this:
Prey go up…
Predators go up…
Prey go down…
Predators go down…
 predators go up
 prey go down
 predators go down
 prey go up
Which in a natural system look like:

5. Next Step Vt+1 = Vt + Bv - Dv Pt+1 = Pt + Bp - Dp Victim, V:
Predator, P:
Pt+1 = Pt + Bp - Dp
Next Step:
Link (explicitly) these two equations (Wednesday’s class)
We need shared variables/parameters in both equations
i.e., Some predator variables/parameters need to be in the prey equation
Some prey variables/parameters need to be in the predator equation

6. Next Step Vt+1 = Vt + Bv - Dv Pt+1 = Pt + Bp - Dp Victim, V:
Predator, P:
Pt+1 = Pt + Bp - Dp
Next Step:
BUT FIRST
To link equations,
we’ll need to investigate dynamics of predators and prey at the same time,
so we need to switch from discrete time to continuous time
i.e., we will use instantaneous rates
to study predator-prey interactions (and other two species interactions)

7. Recall difference between discrete and continuous growth:
Remember This?
Recall difference between discrete and continuous growth:
Discrete
Continuous
Population Size
Population Size
Nt+1 = Nt λ
dN/dt = r N
Nt = N0 λt
Nt = N0 ert

8. Let’s consider a driving example:
Calculus Refresher
Let’s consider a driving example:
Imagine we drive to MBC in Webberville
Say it takes us ½ hour to go 20 miles
What was our average speed (in mi/hr)?
 20 miles / ½ hr = 40 mi/hr
So, 40 mi/hr is our average speed,
but we were not going 40 mi/hr the whole trip
i.e., our instantaneous speed (i.e., speed on speedometer) varied
 Started slower than 40mi/hr in East Lansing
 Drove faster than 40mi/hr on I-96
 Drove slower than 40mi/hr when got off interstate in Webberville

9. Calculus Refresher Let’s look at a figure
Let’s consider a driving example:
Imagine we drive to MBC in Webberville
Say it takes us ½ hour to go 20 miles
What was our average speed (in mi/hr)?
 20 miles / ½ hr = 40 mi/hr
Average speed ~ finite rate
Like our discrete time rates (b’, d’, λ)
(average rate calculated between two end points)
Speed on speedometer ~ instantaneous rate
Like our continuous time rates (b, d, r)
(constantly changing rate)
Let’s look at a figure

10. Let’s consider a driving example:
Calculus Refresher
Let’s consider a driving example:

11. Let’s consider a driving example:
Calculus Refresher
Let’s consider a driving example:
Slope of the curve is equivalent to our speed
Slope =
Rise
Run ΔY ΔX =
Speed =
Distance
Time

12. Let’s consider a driving example:
Calculus Refresher
Let’s consider a driving example:
End slow:
Shallow slope
Fast on I-96:
Steep slope
Start out slow:
Shallow slope

13. Let’s consider a driving example:
Calculus Refresher
Let’s consider a driving example:
Average speed for trip:
Slope of line
connecting end points
Slope =
20 mi
30 min
20 mi
0.5 hr =
= 40 mi/hr
Finite rates are average rates of change over some definite interval
e.g., between two locations; between two years

14. Calculus Refresher
We can determine our speed over shorter intervals by
calculating the slope of a tangent to the curve at any point

15. Calculus Refresher
To obtain our exact speed at any given moment in time
(i.e., the speed on our speedometer at any moment),
we need a method to determine the slope of the curve at each moment
If we define a moment as an infinitesimally small interval of time,
we have a definition of our instantaneous rate of speed
 slope of function over infinitely small interval
Trick to finding the slope: calculus!
We can find the instantaneous rate
(which in our example is not a single number b/c the slope constantly changes)
by differentiating the distance vs. time function (i.e., taking the derivative)
and plugging in a value for time

16. Calculus Refresher
To obtain our exact speed at any given moment in time
(i.e., the speed on our speedometer at any moment),
we need a method to determine the slope of the curve at each moment
If we define a moment as an infinitesimally small interval of time,
we have a definition of our instantaneous rate of speed
 slope of function over infinitely small interval
Trick to finding the slope: calculus!
We can find the instantaneous rate
(which in our example is not a single number b/c the slope constantly changes)
by differentiating the distance vs. time function (i.e., taking the derivative)
and plugging in a value for time
We’ll use the tangent method to approximate (for exercises),
and software (Stella) to do calculus for labs
We are not doing the differentiation in this class

17. Calculus Refresher
“Distance vs. time is great for a physics major, but what does this have to do with ecology and management?”

18. Calculus Refresher
“Distance vs. time is great for a physics major, but what does this have to do with ecology and management?”
Curve could describe dingo population growth after introduction to a new area
Abundance, N
(years)

19. Calculus Refresher
“Distance vs. time is great for a physics major, but what does this have to do with ecology and management?”
To determine the instantaneous net growth rate
for the population, we would need to define an equation:
N = f(time)
 Taking the derivative of that function,
dN/dt (remember: dN/dt is like ΔN/Δt),
would give us the instantaneous net growth rate of population
Abundance, N
(years)

20. Population Growth Example
Let’s do a population growth exercise with Daphnia

21. Population Growth Example
Q1: What is the average population growth rate for the 30 day period?

22. Population Growth Example
Q1: What is the average population growth rate for the 30 day period?
900 – 600 Daphnia / m3
Average =
= 10 Daphnia / m3 / day
30 days

23. Population Growth Example
Q2: What is the growth rate at points A and B (using tangent method)? B A

24. Population Growth Example
Q2: What is the growth rate at points A and B (using tangent method)? B A
1000 – 800 Daphnia / m3
Slope A =
= 20 Daphnia / m3 / day
day

25. Population Growth Example
Q2: What is the growth rate at points A and B (using tangent method)? B A
920 – 1020 Daphnia / m3
Slope B =
= -10 Daphnia / m3 / day
days

26. Population Growth Example
Q3: For what days was the (instantaneous) growth rate:
a) negative, b) positive, and c) zero?

27. Population Growth Example
Q3: For what days was the (instantaneous) growth rate:
a) negative, b) positive, and c) zero?
dN/dt Positive
dN/dt Negative
dN/dt Zero

28. Instantaneous Rates & Predator-Prey Models
We’ll need to use instantaneous rates to analyze predator-prey dynamics!
Why?
We are interested in how predators and prey interact
 Interactions will happen in continuous time
i.e., want to investigate linked changes in
prey abundance over time (ΔV / Δt or dV / dt)
and changes in predator abundance over time (ΔP / Δt or dP / dt)
Let’s continue to build our predator prey models 

29. Instantaneous Rates & Predator-Prey Models
Victim, V:
Vt+1 = Vt + Bv - Dv
Predator, P:
Pt+1 = Pt + Bp - Dp
Start with the simple models we built last class
Well do prey first, then predators

30. Instantaneous Rates & Predator-Prey Models
Victim, V:
Vt+1 = Vt + Bv - Dv
Predator, P:
Pt+1 = Pt + Bp - Dp
Rearrange to:
Vt+1 - Vt = Bv - Dv
And substitute ΔV for Vt+1 - Vt
ΔV = Bv - Dv
Still a model of discrete change in V
To make continuous, change time to “infinitely small time intervals”
(~instantaneous time step)
dV/dt = Bv - Dv

31. Instantaneous Rates & Predator-Prey Models
Victim, V:
Vt+1 = Vt + Bv - Dv
Predator, P:
Pt+1 = Pt + Bp - Dp
Rearrange to:
Rearrange to:
Vt+1 - Vt = Bv - Dv
Pt+1 - Pt = Bp - Dp
And substitute ΔV for Vt+1 - Vt
And substitute ΔP for Pt+1 - Pt
ΔV = Bv - Dv
ΔP = Bp - Dp
Still a model of discrete change in V
Still a model of discrete change in P
To make continuous, change time to “infinitely small time intervals”
(~instantaneous time step)
To make continuous, change time to “infinitely small time intervals”
(~instantaneous time step)
dV/dt = Bv - Dv
dP/dt = Bp - Dp

32. Instantaneous Rates & Predator-Prey Models
Victim, V:
dV/dt = Bv - Dv
Predator, P:
dP/dt = Bp - Dp
A few comments on these equations:
When changing from discrete to continuous time, we added time units:
dV/dt, dP/dt, Bv, Bp, Dv, and Dp units are:
Numbers/time
dV/dt and dP/dt can be positive or negative,
depending on whether population is increasing or decreasing
Next step:
Express B and D as products of per capita birth (b) and death (d) rates
and population sizes (P and V)

33. Instantaneous Rates & Predator-Prey Models
Victim, V:
dV/dt = Bv - Dv
Predator, P:
dP/dt = Bp - Dp
Birth rate:
Bv = bvV
where bv is the per capita birth rate
Units:
bv =
# prey (babies)
1 prey (adult) * time = 1
time
bvV =
prey
time
i.e., # prey born in
pop per unit time

34. Instantaneous Rates & Predator-Prey Models
Victim, V:
dV/dt = Bv - Dv
Predator, P:
dP/dt = Bp - Dp
Birth rate:
Bv = bvV
where bv is the per capita birth rate
Birth rate:
Bp = bpP
where bp is the per capita birth rate
Units:
bv =
# prey (babies)
1 prey (adult) * time = 1
time
Units:
bp =
# predator (babies)
1 pred. (adult) * time = 1
time
bvV =
prey
time
bpP =
predators
time
i.e., # prey born in
pop per unit time
i.e., # preds born in
pop per unit time

35. Instantaneous Rates & Predator-Prey Models
Victim, V:
dV/dt = bvV - Dv
Predator, P:
dP/dt = bpP - Dp

36. Instantaneous Rates & Predator-Prey Models
Victim, V:
dV/dt = bvV - Dv
Predator, P:
dP/dt = bpP - Dp
Death rate:
Dv = dvV
where dv is the per capita death rate
Units:
dv =
[Prob. individual dies]
time = 1
dvV =
prey
time
i.e., # prey that die in
pop per unit time

37. Instantaneous Rates & Predator-Prey Models
Victim, V:
dV/dt = bvV - Dv
Predator, P:
dP/dt = bpP - Dp
Death rate:
Dv = dvV
where dv is the per capita death rate
Death rate:
Dp = dpP
where dp is the per capita death rate
Units:
dv =
[Prob. individual dies]
time = 1
Units:
dp =
[Prob. individual dies]
time = 1
dvV =
prey
time
i.e., # prey that die in
pop per unit time
dpP =
predators
time
i.e., # preds that die in
pop per unit time

38. Instantaneous Rates & Predator-Prey Models
Victim, V:
dV/dt = bvV - dvV
Predator, P:
dP/dt = bpP - dpP
Look familiar?
These equations are the same as what we did earlier when we were studying individual populations with continuous growth
i.e., dN/dt = bN - dN
A few notes:
Remember: b and d do not have the primes ( ’ ) because continuous time
Equations describe rates of change in predator and prey populations
Next step: Link equations

39. Link predator-prey equations Handing back exams in Lab Today
Looking Ahead
No class tomomrrow!
Next Class (Monday):
Link predator-prey equations
Discuss assumptions
Homework
Isle Royale Wolf Paper
Handing back exams in Lab Today