1. FW364 Ecological Problem Solving Lab 5: Density Dependence and
Allee Effects

2. Log onto computers please Open up Excel, download today’s files from

3. Outline for Today
Example of Paramecium density dependent population growth
Learn how to recognize types of density dependence from data
Estimate parameters needed for models with density dependence
Lab 5: Investigate how density dependence affects growth dynamics
Investigate deterministic fluctuations caused by density dependence types with two different population growth rates
Investigate how density dependence types combine with stochastic fluctuations to affect risk of population extinction / explosion

4. Scramble – Population Forecast Scramble Population Dynamics
Population moves toward equilibrium point (carrying capacity) in smaller and smaller jumps each time step… back-and-forth approach
Carrying capacity = intersection of the function with steady-state line
on replacement curve
Population overshoots carrying capacity then falls below carrying capacity
Repeated until carrying capacity is finally hit
Let’s look at a Contest Example

5. Contest – Population Forecast Contest Replacement Curve
Time Nt 1 2 3 …
100
275
425
480
500
K = 500
N0 = 100
Let’s look at population trajectory

6. Contest – Population Forecast Contest Population Trajectory
Time Nt 1 2 3 …
100
275
425
480
500
K = 500

7. Contest – Population Forecast Contest Population Dynamics
Population moves toward equilibrium point (carrying capacity) in smaller and smaller jumps each time step… steady approach
Carrying capacity = intersection of the function with steady-state line
on replacement curve
Population never overshoots carrying capacity
Examples starting above K

8. Scramble – Population Forecast Scramble Replacement Curve
Time Nt 1 2 3 4 …
1000
200
525
480
510
500
K = 500
Large fluctuations
N0 = 1000

9. Contest – Population Forecast Contest Replacement Curve
Time Nt 1 2 …
1000
550
510
500
K = 500
Slowing decline
N0 = 1000

10. Paramecium Example
Goal: Create a model of density dependent growth from Paramecium data
Given these data
(and our knowledge of
density dependent growth):
What type of density dependence does Paramecium display?
Scramble or Contest?
What are the parameters we will need for this model?

11. Paramecium Example To Excel
Goal: Create a model of density dependent growth from Paramecium data
Given these data
(and our knowledge of
density dependent growth):
What type of density dependence does Paramecium display?
Scramble or Contest?
What are the parameters we will need for this model?
 N0, K, λmax
 We will do an example of how to estimate these parameters from real data
To Excel

12. Density Dependence Functions
Note: This is not the density dependence function for the Paramecium data
0.0
1.0
2.0
3.0
4.0
5.0
6.0
200
400
600
800
1000
1200 N
y-intercept:
λmax
scramble λ
contest
steady state
Carrying capacity (K)

13. Lab 5 - Density Dependence
Goal:
Investigate how density dependence affects population growth dynamics
General Process:
Build models with scramble and contest density dependence
Compare models with two different growth rates (λmax)
Compare models with and without stochasticity

14. Lab 5 - Density Dependence Scramble (extreme effect)
Part A: Investigating deterministic fluctuations caused by density dependence types with two different population growth rates (λmax)
Scramble and contest cause different deterministic fluctuations
Fluctuations are due to model equations, NOT stochasticity
Scramble (extreme effect)
Contest
(1 – Nt/K)
Nt+1 = Nt ( λmax )
λmax K
Nt+1 = Nt
Nt λmax – Nt + K

15. Lab 5 - Density Dependence
General Comments:
Most of the in-class work (and work for lab reports) involves
the interpretation of results
I am, as always, happy to chat about interpretation
For Reports:
Exercise instructions give a lot of guidance regarding:
Figures and tables to include in report
Tips for interpreting results
Lab is due on Oct. 29, 2013