1. FW364 Ecological Problem Solving Class 11: Population Regulation
October 13, 2013

2. Outline for Today
Continue to make population growth models more realistic
by adding in density dependence
Objectives from Last Monday:
Introduce density dependence (modeling style)
Discuss scramble and contest competition
Objectives for Today:
Continue discussion of scramble and contest competition
Introduce Allee effects (inverse density dependence)
Text (optional reading):
Chapter 3
Note: Midterm 2 is going to be moved to Nov. 6th, Revised syllabus available online.

3. Recap from Last Week How does exponential growth come to an end?
Two broadly defined limits to population growth:
Density independent factors
Do not regulate populations
Density dependent factors
 Regulate populations
Scramble
versus
Contest
Exploitative, free-for-all competition
Interference competition
Resources shared equally
Resources shared unequally
Density dependence affects
all individuals equally
Density dependence affects
some individuals lightly (territory owners)
other individuals strongly (non-owners)

4. r (continuous) lnλ (discrete) N Contest and Scramble Comparison
y-intercept:
2.00
ln λmax
Contest growth rate declines faster
1.50
1.00
0.50
steady state
0.00
r (continuous)
lnλ (discrete)
-0.50
Carrying capacity (K)
-1.00
contest
Scramble growth rate does not slow down
-1.50
-2.00
scramble
-2.50
200
400
600
800
1000
1200 N
Let’s look at λ versus population size…

5. Contest and Scramble Comparison
0.0
1.0
2.0
3.0
4.0
5.0
6.0
200
400
600
800
1000
1200 N
y-intercept:
Contest: Population growth rate declines faster at first, then slows more
λmax
scramble
λ (discrete)
contest
steady state
Carrying capacity (K)

6. Distinguishing Density Dependence Types
How can we tell which type of density dependence best
describes the population we are interested in?
Make detailed behavioral observations OR
Do a model fitting exercise
i.e., Collect data on population density over time for many generations
Fit the data to models of scramble and contest competition
Determine which model (i.e., scramble vs. contest) fits best N λ
contest
scramble

7. Distinguishing Density Dependence Types
How can we tell which type of density dependence best
describes the population we are interested in?
Make detailed behavioral observations OR
Do a model fitting exercise N λ
contest
scramble
Keep in mind:
Are working with ideal categories… fit will not be perfect…
…but categorizing is still helpful for modeling and making predictions

8. Population Dynamics
Why do the different density dependence types matter?
Scramble vs. contest has big difference on population stability
i.e., change in population size through time
Scramble (extreme effect)
Contest
Large fluctuations
Increased extinction risk
Difficult to predict
Stable population
Reduced extinction risk
Easier to predict

9. Population Dynamics
Why do the different density dependence types matter?
Scramble vs. contest has big difference on population stability
i.e., change in population size through time
Scramble (extreme effect)
Contest
Large fluctuations
Increased extinction risk
Difficult to predict
Stable population
Reduced extinction risk
Easier to predict
Let’s investigate population dynamics more closely…
… need to look at equations

10. General Approach to Equations
Goal: Want equations for how population growth rate (λ, r) changes with density N
lnλ (discrete)
r (continuous)
steady state
Carrying capacity (K)
ln λmax
contest
scramble
Want equations to describe these functions for contest and scramble

11. General Approach to Equations
Goal: Want equations for how population growth rate (λ, r) changes with density
Discrete growth:
Continuous growth:
Nt+1 = Nt λ
dN/dt = r N
Start with our exponential growth equations:
We want to “build” equations for λ and r…
Need to define new parameters:
λmax and K
rmax and K
λmax and rmax are maximum population growth rates
Occur when population size is very small
(as density approaches 0, population growth rates approach maximum)
K is carrying capacity
 Maximum sustainable population size

12. Scramble Equation Nt+1 = Nt λ dN/dt = r N λ = ( λmax ) (1 – N/K)
Goal: Want equations for how population growth rate (λ, r) changes with density
Scramble
Discrete growth:
Continuous growth:
Nt+1 = Nt λ
dN/dt = r N
(1 – Nt/K)
λ = ( λmax )
(1 – N/K)
r = rmax
Logistic growth equations
Identical to Ricker stock-recruitment equation

13. Scramble Equation Nt+1 = Nt λ dN/dt = r N λ = ( λmax ) (1 – N/K)
Goal: Want equations for how population growth rate (λ, r) changes with density
Scramble
Discrete growth:
Continuous growth:
Nt+1 = Nt λ
dN/dt = r N
(1 – Nt/K)
λ = ( λmax )
(1 – N/K)
r = rmax
Challenge: What happens to λ when:
Nt is very small?
Nt equals carrying capacity (K)?
Nt is larger than carrying capacity (K)?

14. Scramble Equation Nt+1 = Nt λ dN/dt = r N λ = ( λmax ) (1 – N/K)
Goal: Want equations for how population growth rate (λ, r) changes with density
Scramble
Discrete growth:
Continuous growth:
Nt+1 = Nt λ
dN/dt = r N
(1 – Nt/K)
λ = ( λmax )
(1 – N/K)
r = rmax
Challenge: What happens to λ when:
Nt is very small?  λ ≈ λmax (exponent ≈ 1)
Nt equals carrying capacity (K)?  λ = 1 (exponent = 0)
Nt is larger than carrying capacity (K)?  λ < 1 (exponent < 0)

15. Scramble Equation Nt+1 = Nt λ dN/dt = r N λ = ( λmax ) (1 – N/K)
Goal: Want equations for how population growth rate (λ, r) changes with density
Scramble
Discrete growth:
Continuous growth:
Nt+1 = Nt λ
dN/dt = r N
(1 – Nt/K)
λ = ( λmax )
(1 – N/K)
r = rmax
Challenge: What happens to r when:
N is very small?
N equals carrying capacity (K)?
N is larger than carrying capacity (K)?

16. Scramble Equation Nt+1 = Nt λ dN/dt = r N λ = ( λmax ) (1 – N/K)
Goal: Want equations for how population growth rate (λ, r) changes with density
Scramble
Discrete growth:
Continuous growth:
Nt+1 = Nt λ
dN/dt = r N
(1 – Nt/K)
λ = ( λmax )
(1 – N/K)
r = rmax
Challenge: What happens to r when:
N is very small?  r ≈ rmax
N equals carrying capacity (K)?  r = 0
N is larger than carrying capacity (K)?  r < 0

17. Beverton-Holt equation
Contest Equation
Goal: Want equations for how population growth rate (λ, r) changes with density
Contest
Discrete growth:
Continuous growth:
Nt+1 = Nt λ
dN/dt = r N
λmax K
λ =
Nt λmax – Nt + K
No continuous analog
Beverton-Holt equation

18. Contest Equation Nt+1 = Nt λ dN/dt = r N λmax K λ = Nt λmax – Nt + K
Goal: Want equations for how population growth rate (λ, r) changes with density
Contest
Discrete growth:
Continuous growth:
Nt+1 = Nt λ
dN/dt = r N
λmax K
λ =
Nt λmax – Nt + K
No continuous analog
Challenge: What happens to λ when:
Nt is very small?
Nt equals carrying capacity (K)?
Nt is larger than carrying capacity (K)?

19. Contest Equation Nt+1 = Nt λ dN/dt = r N λmax K λ = Nt λmax – Nt + K
Goal: Want equations for how population growth rate (λ, r) changes with density
Contest
Discrete growth:
Continuous growth:
Nt+1 = Nt λ
dN/dt = r N
λmax K
λ =
Nt λmax – Nt + K
No continuous analog
Challenge: What happens to λ when:
Nt is very small?  λ ≈ λmax (denominator ≈ K)
Nt equals carrying capacity (K)?  λ = 1 (denominator ≈ λmax K)
Nt is larger than carrying capacity (K)?  λ < 1 (denominator > numerator)

20. Predicting Population Size
We now have equations for population growth with density dependence!
Just considering discrete growth:
Scramble
Contest
λmax K
Nt+1 = Nt
Nt λmax – Nt + K
(1 – Nt/K)
Nt+1 = Nt ( λmax )
We can use these equations to calculate population size between any two consecutive years
 We’ll do this in a graphically interesting way using replacement curves in Lab tomorrow

21. Replacement Curves Nt+1 Nt Nt+1 > Nt λ > 1
Plot of next year’s population (Nt+1) size against current population size (Nt)
(for right now, think about points having the ability to fall anywhere in this space)
Helpful for seeing how different types of
density dependence influence actual population dynamics
1 : 1 line  Steady state line
Nt+1 = Nt λ = 1
Nt+1 > Nt
λ > 1
Nt+1
Time does not “move” from left to right in these figures!!!
Nt+1 < Nt
λ < 1
Example in Lab tomorrow Nt

22. Scramble vs. Contest Comparison
Comparison of scramble and contest density dependence
Similarity:
Both scramble and contest density dependence (competition) result in population regulation
Regulation slows population growth at high density…
… which draws the population toward an equilibrium (carrying capacity)
Differences:
Scramble: Additional individuals always reduce the population growth rate
linear relationship between lnλ (r) and N
Population will overshoot K
Contest: Effect of each additional individual on population growth decreases as more and more individuals are added
Population will not overshoot K

23. Scramble vs. Contest Comparison
Comparison of scramble and contest density dependence
Differences Con’t:
For two populations with same λmax and K
0.0
1.0
2.0
3.0
4.0
5.0
6.0
200
400
600
800
1000
1200 N
Population increases faster (λ) for scramble when N < K
Population decreases faster (λ) for scramble when N > K
Combination drives population fluctuations for scramble (Lab 5)
scramble λ
contest
steady state K

24. Another Type of Density Dependence
So far, density dependence has meant:
“Increased density leads to lower per capita population growth (lower λ)”
The reverse can also be true in some situations!
i.e., Increased density leads to higher per capita population growth (higher λ)
Inverse density dependence: Allee effects
Only occurs at low population sizes
(at high densities, near K, regular density dependence always applies)

25. How Alee Effects Happen
Per capita reproduction may decrease at low population sizes
 Mate limitation: density gets so low that mates cannot find each other
 Inbreeding: density so low that inbreeding occurs
 For plants, low density may mean fewer pollinators attracted
Mate limitation
Inbreeding
Pollinator attraction
Per capita mortality may decrease at higher population sizes (i.e., increase at low size)
 “Safety in numbers” situation, e.g. colony nesting
 For plants, greater density may mean greater soil stability (less erosion)
Colony nesting
Soil erosion

26. How Alee Effects Happen
Constant harvest amount also functions like an Allee effect
Lab 3: Nt+1 = Nt (1 + b’ – d’) - H
When a constant number is harvested,
the proportion of the population harvested changes as population size changes
For H = 100
When N = 1000, the proportion harvested is ( = 100 / 1000)
When N = 400, the proportion harvested is ( = 100 / 400)
When N = 200, the proportion harvested is ( = 100 / 200)
Proportional harvest mortality increases
as population decreases
(like a death rate increasing at low population sizes)
… inverse density dependence
 Constant harvests are a risky practice
Blue whale harvest

27. Alee Effects Dynamics
Allee effects involve positive feedback at low population sizes
i.e., population decline leads to further decline
Result: Dip in the population growth rate versus density curve
λ > 1
scramble without Allee effect
scramble with Allee effect
Allee effect gets stronger from top to bottom curve
λ < 1
Based on Fig. 3.13

28. Alee Effects Dynamics
Allee effects involve positive feedback at low population sizes
i.e., population decline leads to further decline
Result: Dip in the population growth rate versus density curve
λ > 1
scramble without Allee effect
has one equilibrium = K
scramble with Allee effect
has two equilibria
λ < 1
Based on Fig. 3.13

29. Allee Effects – Population Forecast Allee Replacement Curve
Stable
equilibrium
Unstable
equilibrium
N0 below first equilibrium
 Population crash
N0 above first equilibrium
 Carrying capacity

30. Allee Effects – Population Forecast Allee Replacement Curve
Stable
equilibrium
Unstable
equilibrium
Fate of population depends on starting conditions (starting size)

31. Allee Effects – Population Forecast
Allee Replacement Curve
λ > 1
Increase
to K
Decrease
to K
Decline to extinction
λ < 1
Population fate is extinction if it falls below unstable equilibrium

32. Allee Effects – Wrap Up
Allee effects have major implications for management of endangered species and fisheries
 It is crucial that Allee effects be identified and quantified
 Must manage to keep population far above “the well”
(unstable equilibrium)
Possible that Allee effects are the cause of
collapses of some overharvested fisheries
Bluefin tuna
Even if population does not decline at low densities, a slowed rate of growth will keep the population low for longer periods (slow to recover)
 More vulnerable to random extinction

33. Lab Tomorrow
We will be working with some data in excel and doing some practice problems similar to what you will be seeing on Midterm 2 (now scheduled for Nov. 6th)
 We will look at scramble and contest competition
 We will look at Allee effects
You won’t need to bring the RAMAS software, as we will be working with excel and mainly analyzing data.