Lecture 3 - Geometric and Exponential Growth

Transcript

1. Geometric and Exponential Growth
2. Today’s learning objectives The equations for geometric and exponential growth
3. Today’s learning objectives The equations for geometric and exponential growth The relationship between geometric growth and the BIDE model
4. Today’s learning objectives The equations for geometric and exponential growth The relationship between geometric growth and the BIDE model The difference between continuous and discrete time models population growth
5. Today’s learning objectives The equations for geometric and exponential growth The relationship between geometric growth and the BIDE model The difference between continuous and discrete time models population growth The definition of density independent population growth
6. What is Population Dynamics? The study of spatial and temporal variation in population size and structure Background and Review Geometric Growth Connection to BIDE Exponential Growth 3 / 20
7. Fundamental Question How does abundance go from Nt to Nt+1 ? Background and Review Geometric Growth Connection to BIDE Exponential Growth 4 / 20
8. Fundamental Question How does abundance go from Nt to Nt+1 ? Answer: The BIDE Model Nt+1 = Nt + Bt + It − Dt − Et B=Births, I=Immigrations, D=Deaths, E=Emigrations Background and Review Geometric Growth Connection to BIDE Exponential Growth 4 / 20
9. Fundamental Question How does abundance go from Nt to Nt+1 ? Answer: The BIDE Model Nt+1 = Nt + Bt + It − Dt − Et B=Births, I=Immigrations, D=Deaths, E=Emigrations Geometric growth is a simplification of BIDE. Background and Review Geometric Growth Connection to BIDE Exponential Growth 4 / 20
10. Fundamental Question How does abundance go from Nt to Nt+1 ? Answer: The BIDE Model Nt+1 = Nt + Bt + It − Dt − Et B=Births, I=Immigrations, D=Deaths, E=Emigrations Geometric growth is a simplification of BIDE. Exponential growth is a continuous time version of geometric growth. Background and Review Geometric Growth Connection to BIDE Exponential Growth 4 / 20
11. Reverend Thomas Malthus, 1766–1834 Background and Review Geometric Growth Connection to BIDE Exponential Growth 5 / 20
12. Relevance To Wildlife Biology and Management Charles Darwin (Origin of Species) “There is no exception to the rule that every organic being increases at so high a rate, that if not destroyed, the earth would soon be covered by the progeny of a single pair.” Background and Review Geometric Growth Connection to BIDE Exponential Growth 6 / 20
13. Relevance To Wildlife Biology and Management Charles Darwin (Origin of Species) “There is no exception to the rule that every organic being increases at so high a rate, that if not destroyed, the earth would soon be covered by the progeny of a single pair.” “Hence, as more individuals are produced than can pos- sibly survive, there must in every case be a struggle for existence. . . ” Background and Review Geometric Growth Connection to BIDE Exponential Growth 6 / 20
14. Aldo Leopold, Game Management 1946 “Every wild species has certain fixed habits which govern the reproductive process, and determine its maximum rate. [. . . ] Thus one pair of quail, if entirely unmolested in an “ideal” environment, would increase at this rate:” At End of Young Adults Total 1st year 14 2 16 2nd year (16/2)14=112 16 128 3rd year (128/2)14=896 128 1024 Background and Review Geometric Growth Connection to BIDE Exponential Growth 7 / 20
15. Aldo Leopold, Game Management 1946 “Every wild species has certain fixed habits which govern the reproductive process, and determine its maximum rate. [. . . ] Thus one pair of quail, if entirely unmolested in an “ideal” environment, would increase at this rate:” At End of Young Adults Total 1st year 14 2 16 2nd year (16/2)14=112 16 128 3rd year (128/2)14=896 128 1024 “The maximum rate of increase is of course never attained in nature. Part of it never takes place, part of it is absorbed by natural enemies, and part of it [. . . ] is absorbed by hunters.” Background and Review Geometric Growth Connection to BIDE Exponential Growth 7 / 20
16. So What Is Geometric Growth? Discrete time, t = 1, 2, . . . Nt = N0 (1 + r)t r = discrete-time version of intrinsic rate of increase Background and Review Geometric Growth Connection to BIDE Exponential Growth 8 / 20
17. So What Is Geometric Growth? Discrete time, t = 1, 2, . . . Nt = N0 (1 + r)t Or, for one time step: Nt+1 = Nt + Nt r r = discrete-time version of intrinsic rate of increase Background and Review Geometric Growth Connection to BIDE Exponential Growth 8 / 20
18. Example, Nt+1 = Nt + Nt r N0 = 3, r = 1 Time Population size (t) (Nt ) 0 3 q 0 2 4 6 8 10 0 500 1000 1500 2000 2500 3000 Time (t) Population size (N) Background and Review Geometric Growth Connection to BIDE Exponential Growth 9 / 20
19. Example, Nt+1 = Nt + Nt r N0 = 3, r = 1 Time Population size (t) (Nt ) 0 3 1 6 q q 0 2 4 6 8 10 0 500 1000 1500 2000 2500 3000 Time (t) Population size (N) Background and Review Geometric Growth Connection to BIDE Exponential Growth 9 / 20
20. Example, Nt+1 = Nt + Nt r N0 = 3, r = 1 Time Population size (t) (Nt ) 0 3 1 6 2 12 q q q 0 2 4 6 8 10 0 500 1000 1500 2000 2500 3000 Time (t) Population size (N) Background and Review Geometric Growth Connection to BIDE Exponential Growth 9 / 20
21. Example, Nt+1 = Nt + Nt r N0 = 3, r = 1 Time Population size (t) (Nt ) 0 3 1 6 2 12 3 24 q q q q 0 2 4 6 8 10 0 500 1000 1500 2000 2500 3000 Time (t) Population size (N) Background and Review Geometric Growth Connection to BIDE Exponential Growth 9 / 20
22. Example, Nt+1 = Nt + Nt r N0 = 3, r = 1 Time Population size (t) (Nt ) 0 3 1 6 2 12 3 24 4 48 q q q q q 0 2 4 6 8 10 0 500 1000 1500 2000 2500 3000 Time (t) Population size (N) Background and Review Geometric Growth Connection to BIDE Exponential Growth 9 / 20
23. Example, Nt+1 = Nt + Nt r N0 = 3, r = 1 Time Population size (t) (Nt ) 0 3 1 6 2 12 3 24 4 48 5 96 q q q q q q 0 2 4 6 8 10 0 500 1000 1500 2000 2500 3000 Time (t) Population size (N) Background and Review Geometric Growth Connection to BIDE Exponential Growth 9 / 20
24. Example, Nt+1 = Nt + Nt r N0 = 3, r = 1 Time Population size (t) (Nt ) 0 3 1 6 2 12 3 24 4 48 5 96 6 192 q q q q q q q 0 2 4 6 8 10 0 500 1000 1500 2000 2500 3000 Time (t) Population size (N) Background and Review Geometric Growth Connection to BIDE Exponential Growth 9 / 20
25. Example, Nt+1 = Nt + Nt r N0 = 3, r = 1 Time Population size (t) (Nt ) 0 3 1 6 2 12 3 24 4 48 5 96 6 192 7 384 q q q q q q q q 0 2 4 6 8 10 0 500 1000 1500 2000 2500 3000 Time (t) Population size (N) Background and Review Geometric Growth Connection to BIDE Exponential Growth 9 / 20
26. Example, Nt+1 = Nt + Nt r N0 = 3, r = 1 Time Population size (t) (Nt ) 0 3 1 6 2 12 3 24 4 48 5 96 6 192 7 384 8 768 q q q q q q q q q 0 2 4 6 8 10 0 500 1000 1500 2000 2500 3000 Time (t) Population size (N) Background and Review Geometric Growth Connection to BIDE Exponential Growth 9 / 20
27. Example, Nt+1 = Nt + Nt r N0 = 3, r = 1 Time Population size (t) (Nt ) 0 3 1 6 2 12 3 24 4 48 5 96 6 192 7 384 8 768 9 1536 q q q q q q q q q q 0 2 4 6 8 10 0 500 1000 1500 2000 2500 3000 Time (t) Population size (N) Background and Review Geometric Growth Connection to BIDE Exponential Growth 9 / 20
28. Example, Nt+1 = Nt + Nt r N0 = 3, r = 1 Time Population size (t) (Nt ) 0 3 1 6 2 12 3 24 4 48 5 96 6 192 7 384 8 768 9 1536 10 3072 q q q q q q q q q q q 0 2 4 6 8 10 0 500 1000 1500 2000 2500 3000 Time (t) Population size (N) Background and Review Geometric Growth Connection to BIDE Exponential Growth 9 / 20
29. Three Possible Outcomes, Nt+1 = Nt + Nt r If −1 ≤ r < 0 population goes extinct q q q q q q q q q q q 0 2 4 6 8 10 0 200 400 600 800 1000 Time (t) Population size (N) If r = 0 population is stable q q q q q q q q q q q 0 2 4 6 8 10 600 800 1000 1200 1400 Time (t) Population size (N) If r > 0 population grows to ∞ q q q q q q q q q q q 0 2 4 6 8 10 0 10000 20000 30000 40000 50000 Time (t) Population size (N) Background and Review Geometric Growth Connection to BIDE Exponential Growth 10 / 20
30. r and λ, Nt+1 = Nt + Nt r r is the discrete growth rate λ is the finite growth rate λ = Nt+1 Nt λ = 1 + r Background and Review Geometric Growth Connection to BIDE Exponential Growth 11 / 20
31. From BIDE To Geometric Growth Fundamental equation of population ecology Nt+1 = Nt + Bt + It − Dt − Et Nt = Abundance at year t B = Births I = Immigrations D = Deaths E = Emigrations Background and Review Geometric Growth Connection to BIDE Exponential Growth 12 / 20
32. From BIDE To Geometric Growth Ignore immigration and emigration Nt+1 = Nt + Bt − Dt Nt = Abundance in year t B = Births D = Deaths Background and Review Geometric Growth Connection to BIDE Exponential Growth 13 / 20
33. From BIDE To Geometric Growth Step 1: Divide both sides by Nt Nt+1 Nt = 1 + Bt Nt − Dt Nt Background and Review Geometric Growth Connection to BIDE Exponential Growth 14 / 20
34. From BIDE To Geometric Growth Step 1: Divide both sides by Nt Nt+1 Nt = 1 + Bt Nt − Dt Nt Step 2: Write in terms of per capita birth and death rates Nt+1 Nt = 1 + b − d = 1 + r = λ Background and Review Geometric Growth Connection to BIDE Exponential Growth 14 / 20
35. From BIDE To Geometric Growth Step 1: Divide both sides by Nt Nt+1 Nt = 1 + Bt Nt − Dt Nt Step 2: Write in terms of per capita birth and death rates Nt+1 Nt = 1 + b − d = 1 + r = λ Step 3: Geometric growth Nt+1 = Nt + Ntr Background and Review Geometric Growth Connection to BIDE Exponential Growth 14 / 20
36. So What Is Exponential Growth? Continuous time version of geometric growth Nt = N0 ert N0 = initial abundance r = intrinsic rate of increase t = time (any positive number) Background and Review Geometric Growth Connection to BIDE Exponential Growth 15 / 20
37. So What Is Exponential Growth? Continuous time version of geometric growth Nt = N0 ert Or, in terms of instantaneous rate of change: dN dt = rN N0 = initial abundance r = intrinsic rate of increase t = time (any positive number) Background and Review Geometric Growth Connection to BIDE Exponential Growth 15 / 20
38. So What Is Exponential Growth? Continuous time version of geometric growth Nt = N0 ert Or, in terms of instantaneous rate of change: dN dt = rN N0 = initial abundance r = intrinsic rate of increase t = time (any positive number) The exponential growth model is often considered more appropriate than the geometric growth model for birth flow populations in which reproduction occurs throughout the year. Background and Review Geometric Growth Connection to BIDE Exponential Growth 15 / 20
39. So What Is Exponential Growth? Continuous time version of geometric growth Nt = N0 ert Or, in terms of instantaneous rate of change: dN dt = rN N0 = initial abundance r = intrinsic rate of increase t = time (any positive number) The exponential growth model is often considered more appropriate than the geometric growth model for birth flow populations in which reproduction occurs throughout the year. However, geometric growth models can provide a good approximation of birth flow or birth pulse populations. Background and Review Geometric Growth Connection to BIDE Exponential Growth 15 / 20
40. Density Independent Growth Geometric and exponential growth are examples of density independent growth Background and Review Geometric Growth Connection to BIDE Exponential Growth 16 / 20
41. Density Independent Growth Geometric and exponential growth are examples of density independent growth Definition: Population growth rate (r) is not affected by population size (N). Background and Review Geometric Growth Connection to BIDE Exponential Growth 16 / 20
42. Density Independent Growth Geometric and exponential growth are examples of density independent growth Definition: Population growth rate (r) is not affected by population size (N). Implications: Resources are unlimited and there is no carrying capacity! Background and Review Geometric Growth Connection to BIDE Exponential Growth 16 / 20
43. Model Assumptions (1) Population is geographically closed No immigration No emigration Background and Review Geometric Growth Connection to BIDE Exponential Growth 17 / 20
44. Model Assumptions (1) Population is geographically closed No immigration No emigration (2) Reproduction occurs seasonally (for geometric growth) Background and Review Geometric Growth Connection to BIDE Exponential Growth 17 / 20
45. Model Assumptions (1) Population is geographically closed No immigration No emigration (2) Reproduction occurs seasonally (for geometric growth) (3) Constant birth rate (b) and death rate (d) No genetic variation among individuals No age- or stage-structure No time lags Background and Review Geometric Growth Connection to BIDE Exponential Growth 17 / 20
46. Model Assumptions (1) Population is geographically closed No immigration No emigration (2) Reproduction occurs seasonally (for geometric growth) (3) Constant birth rate (b) and death rate (d) No genetic variation among individuals No age- or stage-structure No time lags (4) No stochasticity No random variation in birth or death No random variation in environmental conditions Background and Review Geometric Growth Connection to BIDE Exponential Growth 17 / 20
47. Can We Apply The Model To Real Data? All models are wrong, but some are useful. (George Box) Background and Review Geometric Growth Connection to BIDE Exponential Growth 18 / 20
48. Can We Apply The Model To Real Data? All models are wrong, but some are useful. (George Box) Is exponential growth a useful model? Background and Review Geometric Growth Connection to BIDE Exponential Growth 18 / 20
49. Can We Apply The Model To Real Data? All models are wrong, but some are useful. (George Box) Is exponential growth a useful model? • Possibly for describing some populations during short time periods, e.g. invasive species or prey following removal of predators • Also useful as foundation for more realistic models Background and Review Geometric Growth Connection to BIDE Exponential Growth 18 / 20
50. Looking Ahead Density dependence and logistic growth Background and Review Geometric Growth Connection to BIDE Exponential Growth 19 / 20
51. Assignment Read pages 15–19 in Conroy and Carroll Be prepared for a quiz Background and Review Geometric Growth Connection to BIDE Exponential Growth 20 / 20