1. Population Dynamics Focus on births (B) & deaths (D)
B = bNt , where b = per capita rate (births per individual per time)
D = dNt
N = bNt – dNt = (b-d)Nt

2. Discrete birth intervals (Birth Pulse)
Exponential Growth
Density-independent growth models
Discrete birth intervals (Birth Pulse)
vs.
Continuous breeding (Birth Flow)

3.  > 1
 < 1
 = 1
Nt = N0 t

5. Geometric Growth
When generations do not overlap, growth can be modeled geometrically.
Nt = Noλt
Nt = Number of individuals at time t.
No = Initial number of individuals.
λ = Geometric rate of increase.
t = Number of time intervals or generations.

6. Exponential Growth Birth Pulse Population (Geometric Growth)
e.g., woodchucks
(10 individuals to 20 indivuals)
N0 = 10, N1 = 20
N1 =  N0 ,
where  = growth multiplier = finite rate of increase
 > 1 increase
 < 1 decrease
 = 1 stable population

7. Exponential Growth Birth Pulse Population N2 = 40 = N1 
Nt = N0 t
Nt+1 = Nt 

8. Continuous breeding (Birth Flow)
Exponential Growth
Density-independent growth models
Discrete birth intervals (Birth Pulse)
vs.
Continuous breeding (Birth Flow)

9. Exponential Growth
Continuous population growth in an unlimited environment can be modeled exponentially.
dN / dt = rN
Appropriate for populations with overlapping generations.
As population size (N) increases, rate of population increase (dN/dt) gets larger.

10. Exponential Growth
For an exponentially growing population, size at any time can be calculated as:
Nt = Noert
Nt = Number individuals at time t.
N0 = Initial number of individuals.
e = Base of natural logarithms =
r = Per capita rate of increase.
t = Number of time intervals.

11. Exponential Population Growth

12. Exponential Population Growth

13. Nt = N0ert
Difference Eqn
Note: λ = er

14. Exponential growth and change over time
N = N0ert
dN/dt = rN
Number (N)
Slope (dN/dt)
Time (t)
Number (N)
Slope = (change in N) / (change in time)
= dN / dt

15. ON THE MEANING OF r rm - intrinsic rate of increase – unlimited
resourses
rmax – absolute maximal rm
- also called rc = observed
r > 0
r < 0
r = 0

17. Intrinsic Rates of Increase
On average, small organisms have higher rates of per capita increase and more variable populations than large organisms.

18. Growth of a Whale Population
Pacific gray whale (Eschrichtius robustus) divided into Western and Eastern Pacific subpopulations.
Rice and Wolman estimated average annual mortality rate of and calculated annual birth rate of 0.13.
= 0.041
Gray Whale population growing at 4.1% per yr.

19. Growth of a Whale Population
Reilly et.al. used annual migration counts from to obtain 2.5% growth rate.
Thus from , pattern of growth in California gray whale population fit the exponential model:
Nt = Noe0.025t

20. What values of λ allow What values of r allow Population Growth
Stable Population Size
Population Decline
What values of r allow
Population Growth
Stable Population Size
Population Decline
λ > 1.0
r > 0
λ = 1.0
r = 0
λ < 1.0
r < 0

21. Logistic Population Growth
As resources are depleted, population growth rate slows and eventually stops
Sigmoid (S-shaped) population growth curve
Carrying capacity (K): number of individuals of a population the environment can support
Finite amount of resources can only support a finite number of individuals

22. Logistic Population Growth

23. Logistic Population Growth
dN / dt = rN
dN/dt = rN(1-N/K)
r = per capita rate of increase
When N nears K, the right side of the equation nears zero
As population size increases, logistic growth rate becomes a small fraction of growth rate
Highest when N=K/2
N/K = Environmental resistance

24. Exponential & Logistic Growth (J & S Curve)

25. Logistic Growth

26. Actual Growth

27. Populations Fluctuate

28. Limits to Population Growth
Environment limits population growth by altering birth and death rates
Density-dependent factors
Disease, Resource competition
Density-independent factors
Natural disasters

29. Galapagos Finch Population Growth

31. Logistic Population Model
Nt = 2, R = 0.15, K = 450
A. Discrete equation
- Built in time lag = 1
- Nt+1 depends on Nt

32. I. Logistic Population Model
B. Density Dependence

33. Logistic Population Model C. Assumptions
No immigration or emigration
No age or stage structure to influence births and deaths
No genetic structure to influence births and deaths
No time lags in continuous model

34. Logistic Population Model C. Assumptions
Linear relationship of per capita growth rate and population size (linear DD) K

35. Logistic Population Model C. Assumptions
Linear relationship of per capita growth rate and population size (linear DD)
Constant carrying capacity – availability of resources is constant in time and space
Reality?

36. I. Logistic Population Model
Discrete equation
Nt = 2, r = 1.9,
K = 450
Damped Oscillations
r <2.0

37. I. Logistic Population Model
Discrete equation
Nt = 2, r = 2.5,
K = 450
Stable Limit Cycles
2.0 < r < 2.57
* K = midpoint

38. I. Logistic Population Model
Discrete equation
Nt = 2, r = 2.9,
K = 450
Chaos
r > 2.57
Not random
change
Due to DD
feedback and time
lag in model

40. Underpopulation or Allee Effect
Opposite type of DD
population size down and population growth down
b=d
b=d d
Vital rate b
b<d
r<0 N* K N

41. Review of Logistic Population. Model D. Deterministic vs. Stochastic
Review of Logistic Population Model D. Deterministic vs. Stochastic Models
Nt = 1, r = 2,
K = 100
* Parameters set
deterministic
behavior same

42. Review of Logistic Population. Model D. Deterministic vs. Stochastic
Review of Logistic Population Model D. Deterministic vs. Stochastic Models
Nt = 1, r = 0.15, SD = 0.1; K = 100, SD = 20
* Stochastic
model, r and K
change at
random each
time step

43. Review of Logistic Population. Model D. Deterministic vs. Stochastic
Review of Logistic Population Model D. Deterministic vs. Stochastic Models
Nt = 1, r = 0.15, SD = 0.1; K = 100, SD = 20
* Stochastic
model

44. Review of Logistic Population. Model D. Deterministic vs. Stochastic
Review of Logistic Population Model D. Deterministic vs. Stochastic Models
Nt = 1, r = 0.15, SD = 0.1; K = 100, SD = 20
* Stochastic
model

45. Environmental Stochasticity A. Defined
Unpredictable change in environment occurring in time & space
Random “good” or “bad” years in terms of changes in r and/or K
Random variation in environmental conditions in separate populations
Catastrophes = extreme form of environmental variation such as floods, fires, droughts
High variability can lead to dramatic fluctuations in populations, perhaps leading to extinction

46. Environmental Stochasticity A. Defined
Unpredictable change in environment occurring in time & space
Random “good” or “bad” years in terms of changes in r and/or K
Random variation in environmental conditions in separate populations
Catastrophes = extreme form of environmental variation such as floods, fires, droughts
High variability can lead to dramatic fluctuations in populations, perhaps leading to extinction

47. Environmental Stochasticity A. Defined
Unpredictable change in environment occurring in time & space
Random “good” or “bad” years in terms of changes in r and/or K
Random variation in environmental conditions in separate populations
Catastrophes = extreme form of environmental variation such as floods, fires, droughts
High variability can lead to dramatic fluctuations in populations, perhaps leading to extinction

48. Environmental Stochasiticity B. Examples – variable fecundity
Relation Dec-Apr rainfall and number of juvenile California quail per adult (Botsford et al in Akcakaya et al. 1999)

49. Environmental Stochasiticity B. Examples - variable survivorship
Relation total rainfall pre-nesting and proportion of Scrub Jay nests to fledge (Woolfenden and Fitzpatrik 1984 in Akcakaya et al. 1999)

50. Environmental Stochasiticity B. Examples – variable rate of increase
Muskox population on Nunivak Island, (Akcakaya et al. 1999)

51. Environmental Stochasiticity - Example of random K
Serengeti wildebeest data set – recovering from Rinderpest outbreak
Fluctuations around K possibly related to rainfall

52. Exponential vs. Logistic
No DD DD
All populations same
All populations same
No Spatial component

53. Space Is the Final Frontier in Ecology
History of ecology = largely nonspatial
e.g., *competitors mixed perfectly with prey
*homogeneous ecosystems with uniform distributions of resources
But ecology = fundamentally spatial
ecology = interaction of organisms with their [spatial] environment

54. Incorporating Space
Metapopulation: a population of subpopulations linked by dispersal of organisms
Two processes = extinction & recolonization
subpopulations separated by unsuitable habitat (“oceanic island-like”)
subpopulations can differ in population size & distance between

55. Metapopulation Model (Look familiar?)
p = habitat patch (subpopulation)
c = colonization
e = extinction

56. Metapopulation Model (Look familiar?)

59. Rescue Effect

60. Another Population Model
Source-sink Dynamics: grouping of multiple subpopulations, some are sinks & some are sources
Source Population = births > deaths = net exporter
Sink Population = births < deaths

61. <1
>1

62. Metapopulations Classic Metapopulation Definition of Population?
Groups of populations within which there is a significant amount of movement of individuals via dispersal
Classic
Metapopulation

63. Metapopulation Con’t
This kind of population structure applies when there are “groups” of populations occupying habitat that occurs in discrete patches (patchy).
These patches are separated by areas of inhospitable habitat, but connected by routes for dispersal.
Populations fluctuate independently of each other

64. The probability of dispersal from one patch to another depends on:
Distance between patches
Nature of habitat corridors linking the patches
Ability of the species to disperse (vagility or mobility) – dependent on body size

65. Who Cares? Why bother discussing these models?
Metapopulations & Source-sink Populatons highlight the importance of:
habitat & landscape fragmentation
connectivity between isolated populations
genetic diversity