2. X X Population size or Density Births
Demography: the study of these processes
Population size or Density
Immigration
Emigration
General model of population growth:
Nt+1 = Nt + Bt – Dt + It – Et
X X
Deaths

3. Begin: Simple Model Nt = N0 * Rt All individuals identical
Define replacement rate as R (sometimes λ)
Example:
If R = 3, what is the population size at t=4?
N4 = N0* R* R* R* R
N4 = N0* R4
N4 = N0 34
Nt = N0 * Rt

4. Exponential growth R is the replacement rate (or lambda)
Discrete Time model: Populations reproduce only at limited times—
Nt = N0Rt
How do we describe the RATE of change with time?
Nt+1 = NtR
R is the replacement rate (or lambda)
λ = Nt+1 /Nt
Density-independent
(R does not change with pop size)
Resources not limiting
Or generations are non-overlapping
often accurate for insect invasions (pest spread at early stages)

5. Increasing the R (or lambda)

6. Begin: Simple Model Nt = N0 * Rt (discrete exponential) 3 = e1.099
Define replacement rate as R (sometimes λ)
Example:
If R = 3, what is the population size at t=4?
N4 = N0* R* R* R* R
N4 = N0* R4
N4 = N0 34
Nt = N0 * Rt (discrete exponential)
3 = e1.099
...thus R = er where r is intrinsic growth rate
Nt = N0ert (continuous exponential)

7. Exponential growth r is the intrinsic growth rate (per captia change)
Continuous time model: Population size changes continuously
Nt = N0ert
r is the intrinsic growth rate (per captia change)
The population growth rate is:
dN/dt = rN
Ln both sides (ln Nt = ln N0 + rt), then differentiate with respect to time to prove to yourself that 1/Nt * dNt/dt = r (change in N per unit time, per N)
Density-independent (r does not change with pop size)
Resources not limiting

8. General models: Exponential Growth
Describe how idealized populations would grow in infinite environments…
Is the population growing “un-checked” over the short term?
If yes, then density-independent model may be reasonable approx.
Two forms:
Geometric
Exponential
Discrete
Continuous
Two options: 1) increase to infinity or 2) decrease to zero….just a matter of time (rate)

9. As populations grow what happens to demographic rates?
Okay, but we know most populations don’t grow unchecked!
As populations grow what happens to demographic rates?
Intuition? N
Birth Rate
Death Rate

10. Limits on Population Growth
Density Dependent Limits?
Food/prey
Water
Shelter, nest sites, territories
Disease
Mates
Density Independent Limits?
Weather
Includes stochastic events: hurricanes, fires
Climate
But sometimes climate effects become density dependent….example: El nino in the Galapagos Is.

11. Logistic Population Growth
Exponential population growth with a linear decrease in r as a function of N
growth rate diminishes as limit is approached
Carrying capacity (K) = max # individuals that can be supported in the environment
dN/dt = r0 N(1- N/K)
r0 is the equivalent of the intrinsic growth rate
rrealized = r0 (1-N/K)
rate of growth slows to zero as K is reached
Make r a function of Nt. Linear decrease in r with N. dN/dt = r((K-N/K)N. ((K-N)/K) is often re-written as 1- (N/K)
Another common form Nt = K + 1/ be^rt (where b= K/2, or the half-saturation point similar to other sigmoid equations) 1- N
1000

12. Adding non-linear feedback
Non-linear effects of density (N) on r
Theta logistic model
dN/dt = r (1- (N/K)θ)
Where θvaries from 0 to infinity (shape parameter)
When θ = 1, linear function (same as exponential)

15. How to recognize density dependence
Manipulate density of an organism
Record individual performance across a range of densities (growth, survival, reproduction)
Pearl (1927) as a classic example
Or, observe the success of individuals as a function of the number of adults.
Examples-reproduction:
Fisheries stock-recruit relationships

16. One of the first laboratory ‘tests’
Pearl (1927)
Maintained Drosophila colonies in bottles with fixed amount of yeast

17. 2. Population rate of change # New added versus pop’n size (N)
Ways to look at simple dynamics of populations (logistic density dep. & density indep.)
1. Time series
Number of individuals (N) at each time t
2. Population rate of change
dN/dt = Nt+1-Nt
# New added versus pop’n size (N)
3. Per capita rate of change
dN/dt/N = (Nt+1-Nt)/Nt
Does pop’n growth rate change with N? N t N dN dt N dN dt

18. Exponential increase Time N dN/dt dN/dt/N 20 1 23 2 27 3 31 4 36 5 4220 1 23 2 27 3 31 4 36 5 42 6 49 7 57 8 66 9 77

19. Exponential increase Time N dN/dt dN/dt/N 20 1 23 2 27 3 31 4 36 5 4220 1 23 2 27 3 31 4 36 5 42 6 49 7 57 8 66 9 77

20. Exponential increase Time N dN/dt dN/dt/N 20 23-20 = 3 1 23 27-23 = 420
23-20 = 3 1 23
27-23 = 4 2 27 3 31 4 36 5 42 6 49 7 57 8 66 9 77

21. Exponential increase Time N dN/dt dN/dt/N 20 23-20 = 3 1 23 27-23 = 420
23-20 = 3 1 23
27-23 = 4 2 27 4 3 31 5 36 6 42 7 49 8 57 9 66 11 77
New recruits each timestep
Number

22. Exponential increase Time N dN/dt dN/dt/N 20 3 3/20 =0.15 1 23 420 3
3/20 =0.15 1 23 4
4/23=0.17 2 27
4/27=0.15 31 5 36 6 42 7 49 8 57 9 66 11 77

23. Exponential increase Time N dN/dt dN/dt/N 20 3 3/20 =0.15 1 23 420 3
3/20 =0.15 1 23 4
4/23=0.17 2 27
4/27=0.15 31 5
0.16 36 6
0.17 42 7 49 8 57 9 66 11 77

24. Exponential increase Time N dN/dt dN/dt/N 20 3 3/20 =0.15 1 23 420 3
3/20 =0.15 1 23 4
4/23=0.17 2 27
4/27=0.15 31 5
0.16 36 6
0.17 42 7 49 8 57 9 66 11 77
Rate of pop’n growth per individual (CONSTANT)
Number

25. Population abundance (N)
Time
dN/dt
Density (N)
dN/dt/N
Density (N)

26. Exponential Population Growth Population abundance (N)
Time
Accelerating population increase
dN/dt
Density (N)
Constant per capita increase
dN/dt/N
Density (N)

27. Logistic Time N dN/dt dN/dt/N 5 1 8 2 12 3 18 4 27 38 6 50 7 62 73 95 1 8 2 12 3 18 4 27 38 6 50 7 62 73 9 82

28. Logistic Time N dN/dt dN/dt/N 5 8-5=3 1 8 12-8=4 2 12 3 18 4 27 38 65
8-5=3 1 8
12-8=4 2 12 3 18 4 27 38 6 50 7 62 73 9 82

29. Logistic Time N dN/dt dN/dt/N 5 8-5=3 1 8 12-8=4 2 12 6 3 18 9 4 27 115
8-5=3 1 8
12-8=4 2 12 6 3 18 9 4 27 11 38 50 7 62 73 82
New recruits each timestep
Number

30. Logistic Time N dN/dt dN/dt/N 5 3 3/5=0.6 1 8 4 4/8=0.5 2 12 65 3
3/5=0.6 1 8 4
4/8=0.5 2 12 6
6/12=0.5 18 9 27 11 38 50 7 62 73 82

31. Logistic Time N dN/dt dN/dt/N 5 3 3/5=0.6 1 8 4 4/8=0.5 2 12 65 3
3/5=0.6 1 8 4
4/8=0.5 2 12 6
6/12=0.5 18 9
0.5 27 11
0.4 38
0.32 50
0.24 7 62
0.18 73
0.12 82
0.07
Rate of pop’n growth per individual (NOT CONSTANT)
Number

32. Logistic Population Growth
Population abundance (N)
Time
Highest population increase at intermediate densities
dN/dt
Density (N)
Declining per capita contribution
dN/dt/N
Density (N)