The document discusses exponential and logistic population growth models, highlighting the mathematical relationships between birth and death rates and factors influencing population dynamics such as density dependence. It includes comparisons of continuous and discrete growth equations, explorations of population behaviors like oscillations and chaos, and notes the implications of life history traits on population growth patterns. Assumptions of constant environments and resource availability are emphasized in models of growth and projections.

1. Population Growth
Exponential:
Continuous addition of births and
deaths at constant rates (b & d)
Such that r = b - d
Problem: no explicit prediction is made
Solution: isolate N terms on left, and integrate

2. Result of the integration:
r=0.05

3. Exponential growth relationships
Slope of this curve
Increases with density
Slope of
Curve on
left
Density
Slope of line = r

4. Exponential growth, log scale
Linear increase of log
values with time is a
sign of exponential growth

5. Geometric Growth
Time is measured in discrete (contant) chunks
Simplest approach: Generations are the time unit
R0: Average number of offspring produced per individual,
per lifetime-- Factor that a population will be multiplied by
for each generation. Often called the Net Rate of Increase.
Time is measured in generations in this equation.

6. Relationship between R0 and r
A population growing for one generation should show the
same result using either of the following equations:
Continuous, where t=t
(t = “generation time”)
Discrete, where T=1
generation
If these give the same result, then

7. R0 and r
So! Information about R and t can lead us to r

8. Ways of finding R0 and t

9. Cohort study

10. Survivorship calculations

11. Fecundity calculations

12. Age-specific reproduction

13. Generation Time, t

14. Approximate r

15. Assumptions of exponential or
geometric growth projections
Constant lx and mx schedules
This implies that reproduction and survival
will not change with density
This also implies that any changes in physical
or chemical environment have no influence on
survival or reproduction
No important interactions with other species
if age-specific data are used, assume stable age
distribution.

16. Suppose we let lx, mx and t vary
with density
Bottom line: let r (per capita growth rate) vary with N
dN/Ndt
N
r
K
0
0

17. Density-dependent growth
dN/Ndt
N
r
K
-r/K
Y = A + BX
0
0

18. Logistic equation
Predictive form:

19. Human rates of change vs N

20. Projection based on Logistic model:

21. Earlier US projection, similar
approach:

22. Logistic Examples
Full-loop (2x the bacteria)
Half-loop (half that on right)
Paramecium, 2 species, growing for 8 days at high <r> and low <l> resource
levels. Scale has been stretched on right to be equivalent to that on the left

23. More logistic examples
Growth of a zooplankton crust-
acean, Moina, at different
temperatures
Growth of flour beetles in flower,
In containers holding different amts
of flour

24. Drosophila studies

25. Evolution of K in Drosophila
Post-radiation
Control
Hybrid
Inbred
Results suggest that K responds to an increase in genetic variation,
And that it changes gradually through time in response to selection.

26. Assumptions of Logistic Growth
Constant environment (r and K are constants)
Linear response of per capita growth rate to density
Equal impact of all individuals on resources
Instantaneous adjustment of population growth to change in N
No interactions with species other than those that are food
Constantly renewed supply of food in a constant quantity

27. Discrete Model for Limited Growth
Same assumptions, except population grows in bursts with each
Generation-- built-in time lag
Models of this sort show the potential influence that a time lag can
have on population change.

28. Simple model, complex behavior
R = 0.1, K = 1000

29. Simple model, complex behavior
R = 1.9, K = 1000
Damped oscillation

30. Simple model, complex behavior
r= 2.2, K = 1000
Limit cycle

31. Simple model, complex behavior
r= 2.5, K = 1000
4-point cycle

32. Simple model, complex behavior
r= 2.58, K = 1000
8-point cycle

33. Simple model, complex behavior
r= 2.7, K = 1000
Erratic

34. Chaos
r= 3, K = 1000

35. Overshoot, Crash, Extinction
r= 3.000072, K = 1000

36. Concerns about Chaos
Biological populations don’t appear to have the growth
capacity to generate chaos, but this shows the potential
importance of time lags.
More complicated models can be even more sensitive
Some systems might be completely unpredictable

37. Evolution of Life Histories
Life history features:
Rates of birth, death, population growth
Patterns of reproduction and mortality
Behavior associated with reproduction
Efficiency of resource use, and carrying capacity
Anything that affects population growth

38. Patterns

39. More patterns

40. Tradeoff