1. Population Ecology I.Attributes of Populations II.Distributions III. Population Growth – change in size through time A. Calculating Growth Rates 1. Discrete Growth With discrete growth, N (t+1) = N (t) λ Or, N t = N o λ t

2. Population Ecology I.Attributes of Populations II.Distributions III. Population Growth – change in size through time A. Calculating Growth Rates 2. Exponential Growth – continuous reproduction With discrete growth: N (t+1) = N (t) λ or N t = N o λ t Continuous growth: N t = N o e rt Where r = intrinsic rate of growth (per capita and instantaneous) and e = base of natural logs (2.72) So, λ = e r

3. Population Ecology I.Attributes of Populations II.Distributions III. Population Growth – change in size through time A. Calculating Growth Rates 3. Equivalency If λ is between zero and 1, the r < 0 and the population will decline. If λ = 1, then r = 0 and the population size will not change. If λ >1, then r > 0 and the population will increase.

4. Population Ecology I.Attributes of Populations II.Distributions III. Population Growth – change in size through time A. Calculating Growth Rates 3. Equivalency The rate of population growth is measured as: The derivative of the growth equation: N t = N o e rt dN/dt = rN o

5. III. Population Growth – change in size through time A. Calculating Growth Rates B. The Effects of Age Structure 1. Life Table - static: look at one point in time and survival for one time period

6. III. Population Growth – change in size through time A. Calculating Growth Rates B. The Effects of Age Structure 1. Life Table

7. III. Population Growth – change in size through time A. Calculating Growth Rates B. The Effects of Age Structure 1. Life Table Why λ ? (discrete breeding season and discrete time intervals)

8. III. Population Growth – change in size through time A. Calculating Growth Rates B. The Effects of Age Structure 1. Life Table - dynamic (or “cohort”) – follow a group of individuals through their life xnxlxdxqxLmex 01151.00900.7870.01.01 1250.2260.2422.01.86 2190.1770.3715.51.29 3120.10100.837.00.75 420.0210.501.51.00 510.0111.000.50.50 600---- Song Sparrows Mandarte Isl., B.C. (1988)

9. Age classes (x): x = 0, x = 1, etc. Initial size of the population: n x, at x = 0. xnxlxdxqxLmex 0115 1 2 3 4 5 6

10. Age classes (x): x = 0, x = 1, etc. Initial size of the population: n x, at x = 0. Number reaching each birthday are subsequent values of nx xnxlxdxqxLmex 0115 125 219 312 42 51 60

11. Age classes (x): x = 0, x = 1, etc. Initial size of the population: n x, at x = 0. Survivorship (l x ): proportion of population surviving to age x. xnxlxdxqxLmex 01151.00 1250.22 2190.17 3120.10 420.02 510.01 600

12. Age classes (x): x = 0, x = 1, etc. Initial size of the population: n x, at x = 0. Survivorship (l x ): proportion of population surviving to age x. Mortality: dx = # dying during interval x to x+1. Mortality rate: mx = proportion of individuals age x that die during interval x to x+1. xnxlxdxmxLmex 01151.00900.78 1250.2260.24 2190.1770.37 3120.10100.83 420.0210.50 510.0111.00 600--

13. Survivorship Curves: Describe age-specific probabilities of survival, as a consequence of age-specific mortality risks.

14. Age classes (x): x = 0, x = 1, etc. Initial size of the population: n x, at x = 0. Survivorship (l x ): proportion of population surviving to age x. Number alive DURING age class x: Lm = (n x + (n x+1 ))/2 xnxlxdxqxLmex 01151.00900.7870.0 1250.2260.2422.0 2190.1770.3715.5 3120.10100.837.0 420.0210.501.5 510.0111.000.5 600---

15. Age classes (x): x = 0, x = 1, etc. Initial size of the population: n x, at x = 0. Survivorship (l x ): proportion of population surviving to age x. Number alive DURING age class x: Lm = (n x + (n x+1 ))/2 Expected lifespan at age x = e x - T = Sum of Lm's for age classes =, > than age (for 3, T = 9) - e x = T/n x (number of individuals in the age class) ( = 9/12 = 0.75) - e x = the number of additional age classes an individual can expect to live. xnxlxdxqxLmex 01151.00900.7870.01.01 1250.2260.2422.01.86 2190.1770.3715.51.29 3120.10100.837.00.75 420.0210.501.51.00 510.0111.000.50.50 600----

16. III. Population Growth – change in size through time A. Calculating Growth Rates B. The Effects of Age Structure 1. Life Tables 2. Age Class Distributions

17. III. Population Growth – change in size through time A. Calculating Growth Rates B. The Effects of Age Structure 1. Life Tables 2. Age Class Distributions When these rates equilibrate, all age classes are growing at the same single rate – the intrinsic rate of increase of the population (r m )

18. III. Population Growth – change in size through time A. Calculating Growth Rates B. The Effects of Age Structure C. Growth Potential Net Reproductive Rate = Σ(l x b x ) = 2.1 Number of daughters/female during lifetime. If it is >1 (“replacement”), then the population has the potential to increase multiplicatively (exponentially).

19. III. Population Growth – change in size through time A. Calculating Growth Rates B. The Effects of Age Structure C. Growth Potential Generation Time – T = Σ(xl x b x )/ Σ(l x b x ) = 1.95

20. III. Population Growth – change in size through time A. Calculating Growth Rates B. The Effects of Age Structure C. Growth Potential Intrinsic rate of increase: r m (estimated) = ln(R o )/T = 0.38 Pop growth dependent on reproductive rate (R o ) and age of reproduction (T).

21. III. Population Growth – change in size through time A. Calculating Growth Rates B. The Effects of Age Structure C. Growth Potential Intrinsic rate of increase: r m (estimated) = ln(R o )/T = 0.38 Pop growth dependent on: reproductive rate (R o ) and age of reproduction (T). Doubling time = t2 = ln(2)/r = 0.69/0.38 = 1.82 yrs

22. III. Population Growth – change in size through time A. Calculating Growth Rates B. The Effects of Age Structure C. Growth Potential Northern Elephant Seals: <100 in 1900  150,000 in 2000. r = 0.073, λ = 1.067 Malthus: All populations have the capacity to expand exponentially

23. xlxbxlxbxxlxbx 01.0000 10.52010 20--- R = 10 T = 1 r = 2.303 III. Population Growth – change in size through time A. Calculating Growth Rates B. The Effects of Age Structure C. Growth Potential D. Life History Redux Net Reproductive Rate = Σ(l x b x ) = 10 Generation Time – T = Σ(xl x b x )/ Σ(l x b x ) = 1.0 r m (estimated) = ln(R o )/T = 2.303

24. xlxbxlxbxxlxbx 01.0000 10.52010 20--- R = 10 T = 1 r = 2.303 xlxbxlxbxxlxbx 01.0000 10.52211 20--- R = 11 T = 1 r = 2.398 III. Population Growth – change in size through time A. Calculating Growth Rates B. The Effects of Age Structure C. Growth Potential D. Life History Redux - increase fecundity, increase growth rate (obvious)

25. xlxbxlxbxxlxbx 01.0220 10.52010 20--- R = 12 T = 0.833 r = 2.983 xlxbxlxbxxlxbx 01.0000 10.52211 20--- R =11 T = 1 r = 2.398 III. Population Growth – change in size through time A. Calculating Growth Rates B. The Effects of Age Structure C. Growth Potential D. Life History Redux - increase fecundity, increase growth rate (obvious) - decrease generation time (reproduce earlier) – increase growth rate

26. R = 20 T = 1.5 r = 2.00 xlxbxlxbxxlxbx 01.0000 10.52211 20--- R = 11 T = 1 r = 2.398 xlxbxlxbxxlxbx 01.0000 10.52010 20.5201020 30--- III. Population Growth – change in size through time A. Calculating Growth Rates B. The Effects of Age Structure C. Growth Potential D. Life History Redux - increase fecundity, increase growth rate (obvious) - decrease generation time (reproduce earlier) – increase growth rate - increasing survivorship – DECREASE GROWTH RATE (lengthen T)

27. R = 230 T = 2.30 r = 2.36 Original r = 2.303 xlxbxlxbxxlxbx 01.0 000 10.9 000 20.8200160320 30.7100 70210 40--- III. Population Growth – change in size through time A. Calculating Growth Rates B. The Effects of Age Structure C. Growth Potential D. Life History Redux - increase fecundity, increase growth rate (obvious) - decrease generation time (reproduce earlier) – increase growth rate - increasing survivorship – DECREASE GROWTH RATE (lengthen T) - survivorship adaptive IF: - necessary to reproduce at all - by storing E, reproduce disproportionately in the future

28. III. Population Growth – change in size through time A. Calculating Growth Rates B. The Effects of Age Structure C. Growth Potential D. Life History Redux E. Limits on Growth: Density Dependence Robert Malthus 1766-1834

29. Premise: - as population density increases, resources become limiting and cause an increase in mortality rate, a decrease in birth rate, or both... DENSITY RATE BIRTH DEATH r > 0 r < 0

30. Premise: - as population density increases, resources become limiting and cause an increase in mortality rate, a decrease in birth rate, or both... DENSITY RATE BIRTH DEATH r > 0 r < 0

31. Premise: - as population density increases, resources become limiting and cause an increase in mortality rate, a decrease in birth rate, or both... DENSITY RATE BIRTH DEATH r > 0r < 0

32. As density increases, successful reproduction declines And juvenile suvivorship declines (mortality increases)

33. Lots of little plants begin to grow and compete. This kills off most of the plants, and only a few large plants survive.

34. Premise: Result: There is a density at which r = 0 and DN/dt = 0. THIS IS AN EQUILIBRIUM.... DENSITY RATE BIRTH DEATH r = 0 K

36. 1.Premise 2. Result 3. The Logistic Growth Equation: Exponential: dN/dt = rN N t

37. 1.Premise 2. Result 3. The Logistic Growth Equation: Exponential:Logistic: dN/dt = rNdN/dt = rN [(K-N)/K] N t N t K

38. N t K When a population is very small (N~0), the logistic term ((K-N)/K) approaches K/K (=1) and growth rate approaches the exponential maximum (dN/dt = rN). 1.Premise 2. Result 3. The Logistic Growth Equation (Pearl-Verhulst Equation, 1844-45): Logistic: dN/dt = rN [(K-N)/K]

39. N t K As N approaches K, K-N approaches 0; so that the term ((K-N)/K) approaches 0 and dN/dt approaches 0 (no growth). 1.Premise 2. Result 3. The Logistic Growth Equation: Logistic: dN/dt = rN [(K-N)/K]

40. N t K Should N increase beyond K, K-N becomes negative, as does dN/dt (the population will decline in size). 1.Premise 2. Result 3. The Logistic Growth Equation: Logistic: dN/dt = rN [(K-N)/K]

41. N t K Minimum viable population size add (N-m)/N m 1.Premise 2. Result 3. The Logistic Growth Equation: Logistic: dN/dt = rN [(K-N)/K] [(N-m)/K]

42. III. Population Growth – change in size through time A. Calculating Growth Rates B. The Effects of Age Structure C. Growth Potential D. Life History Redux E. Limits on Growth: Density Dependence F. Temporal Dynamics

43. ROUGHLY, growth per generation is: log(N t+1 ) - log(N t ) = R[log(K) - log(N t )] N t K SO: when R < 1, the population will grow only a fraction of the difference between K and Nt. Asymptotic approach to K. NtNt F. Temporal Dynamics - DISCRETE GROWTH

44. N t K SO: If R = 1, then the population reaches K exactly. NtNt F. Temporal Dynamics - DISCRETE GROWTH ROUGHLY, growth per generation is: log(N t+1 ) - log(N t ) = R[log(K) - log(N t )]

45. N t K SO: If 1.0 < R < 2.0, then the population overshoots by progressively smaller amounts... convergent oscillation. NtNt ROUGHLY, growth per generation is: log(N t+1 ) - log(N t ) = R[log(K) - log(N t )] F. Temporal Dynamics - DISCRETE GROWTH

46. N t K SO: If 1.0 < R < 2.0, then the population overshoots by progressively smaller amounts... convergent oscillation. NtNt

47. N t K SO: If 1.0 < R < 2.0, then the population overshoots by progressively smaller amounts... convergent oscillation. NtNt

48. N t K SO: when 2 < R < 2.5, oscillations increase each time interval; divergent oscillation initially. But at low N, the linearity breaks down and this equation is not descriptive. End up with a stable limit cycle. Over 2.5? Chaotic. NtNt ROUGHLY, growth per generation is: log(N t+1 ) - log(N t ) = R[log(K) - log(N t )] F. Temporal Dynamics - DISCRETE GROWTH

49. F. Temporal Dynamics - DISCRETE GROWTH

50. N t K Lags occur because of developmental time between acquisition of resources FOR breeding and the event of breeding itself. NtNt Breed here, and may be out of balance with resources F. Temporal Dynamics - CONTINUOUS GROWTH

51. F. Temporal Dynamics - CONTINUOUS GROWTH Population cycles are a function of R and the lag time (t). Either high R or long lags increase the amplitude 1.6

52. F. Temporal Dynamics - CONTINUOUS GROWTH Growth curves are not an intrinsic characteristic of a species – they can change with environmental conditions. Daphnia

53. III. Population Growth – change in size through time A. Calculating Growth Rates B. The Effects of Age Structure C. Growth Potential D. Life History Redux E. Limits on Growth: Density Dependence F. Temporal Dynamics G. Spatial Dynamics and Metapopulations

54. III. Population Growth – change in size through time A. Calculating Growth Rates B. The Effects of Age Structure C. Growth Potential D. Life History Redux E. Limits on Growth: Density Dependence F. Temporal Dynamics G. Spatial Dynamics and Metapopulations Dealing with small populations with increase chance of stochastic extinction (e). The survival of the metapopulation is dependent on a rate of migration that can ‘rescue’ extinct at a compensatory rate.

55. III. Population Growth – change in size through time A. Calculating Growth Rates B. The Effects of Age Structure C. Growth Potential D. Life History Redux E. Limits on Growth: Density Dependence F. Temporal Dynamics G. Spatial Dynamics and Metapopulations Dealing with small populations with increase chance of stochastic extinction (e). The survival of the metapopulation is dependent on a rate of migration that can ‘rescue’ extinct at a compensatory rate. If e > c, pop goes extinct. p = proportion of suitable habitats occupied e = extinction rate ep = prop of habitats being vacated Rate of colonization depends on fraction of patches that are empty (1-p), and number of occupied patches dispersing colonists. c = migration rate, and so rate of colonization = cp(1-p) Change in patch occupancy: dp/dt = cp(1-p) - ep Peq = 1 – e/c

56. G. Spatial Dynamics and Metapopulations What influences e and c? Extinction probability is strongly influenced by population size.

57. G. Spatial Dynamics and Metapopulations What influences e and c? Extinction probability is strongly influenced by population size. Degree of patch isolation affects colonization probability

58. Greater Patch area means more resources, larger populations, and lower e Closer patches mean higher c Low e/c High e/c ??

59. Why is this important?