Lecture 04 - Logistic growth

Transcript

1. Logistic Population Growth
2. Topics The equation for logistic growth in discrete time
3. Topics The equation for logistic growth in discrete time The definition of density-dependent growth
4. Topics The equation for logistic growth in discrete time The definition of density-dependent growth Basic properties of the model
5. Topics The equation for logistic growth in discrete time The definition of density-dependent growth Basic properties of the model Strange behavior of the (discrete time) model, such as damped oscillations and chaos
6. From Geometric to Logistic Growth Geometric growth Nt+1 = Nt + Nt r Background Logistic growth Summary 3 / 12
7. From Geometric to Logistic Growth Geometric growth Nt+1 = Nt + Nt r Logistic growth Nt+1 = Nt + Nt rmax 1 − Nt K where • rmax is the growth rate when Nt is close to 0. • K is the carrying capacity Background Logistic growth Summary 3 / 12
8. Density-dependent growth Logistic growth is an example of density-dependent growth Background Logistic growth Summary 4 / 12
9. Density-dependent growth Logistic growth is an example of density-dependent growth Definition: Population growth rate is affected by population size (N). Background Logistic growth Summary 4 / 12
10. Density-dependent growth Logistic growth is an example of density-dependent growth Definition: Population growth rate is affected by population size (N). Implications: Resources are limited and there is a carrying capacity. Background Logistic growth Summary 4 / 12
11. Graphical depiction 0 20 40 60 80 100 0 20 40 60 80 100 Time (t) Population size (N) Geometric growth Background Logistic growth Summary 5 / 12
12. Graphical depiction 0 20 40 60 80 100 0 20 40 60 80 100 Time (t) Population size (N) Geometric growth Logistic growth Background Logistic growth Summary 5 / 12
13. Growth rate (λt = Nt+1 /Nt) 0 20 40 60 80 100 0 10 20 30 40 50 60 Time (t) Population size (N) Inflection point (K/2) 0 20 40 60 80 100 1.00 1.02 1.04 1.06 1.08 Time (t) Growth rate: (λ=Nt+1 Nt ) Background Logistic growth Summary 6 / 12
14. Growth (∆t = Nt+1 − Nt) 0 20 40 60 80 100 0 10 20 30 40 50 60 Time (t) Population size (N) Inflection point (K/2) 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Time (t) Growth: (∆N =Nt+1 − Nt ) Background Logistic growth Summary 7 / 12
15. Growth (∆t = Nt+1 − Nt) as a function of N 10 20 30 40 50 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Population size (N) Growth: (∆N =Nt+1 − Nt ) Inflection point (K/2) Background Logistic growth Summary 8 / 12
16. What happens when we change rmax? 0 20 40 60 80 100 0 20 40 60 80 100 Time (t) Population size (N) rmax=0.1 Background Logistic growth Summary 9 / 12
17. What happens when we change rmax? 0 20 40 60 80 100 0 20 40 60 80 100 Time (t) Population size (N) rmax=0.1 rmax=0.5 Background Logistic growth Summary 9 / 12
18. What happens when we change rmax? 0 20 40 60 80 100 0 20 40 60 80 100 Damped oscillation Time (t) Population size (N) rmax=0.1 rmax=0.5 rmax=2.0 Background Logistic growth Summary 9 / 12
19. What happens when we change rmax? 0 20 40 60 80 100 0 20 40 60 80 100 Chaos Time (t) Population size (N) rmax=0.1 rmax=0.5 rmax=2.0 rmax=3.0 Background Logistic growth Summary 9 / 12
20. Definitions Overcompensation Density-dependent response in which populations over- or under-shoot carrying capacity rather than approach it gradually Background Logistic growth Summary 10 / 12
21. Definitions Overcompensation Density-dependent response in which populations over- or under-shoot carrying capacity rather than approach it gradually Chaos Highly variable dynamics that are extremely sensitive to small changes in parameters Background Logistic growth Summary 10 / 12
22. Assumptions of basic model • K and rmax are constant • No sex or age effects or other sources of individual heterogeneity • No time lags • No stochasticity Background Logistic growth Summary 11 / 12
23. Summary and assignment Summary • Logistic growth is a form of density-dependent growth Background Logistic growth Summary 12 / 12
24. Summary and assignment Summary • Logistic growth is a form of density-dependent growth • Growth rate (λt = Nt+1 /Nt ) declines as N approaches K Background Logistic growth Summary 12 / 12
25. Summary and assignment Summary • Logistic growth is a form of density-dependent growth • Growth rate (λt = Nt+1 /Nt ) declines as N approaches K • Growth (∆t = Nt+1 − Nt ) peaks at K/2 (the inflection point) Background Logistic growth Summary 12 / 12
26. Summary and assignment Summary • Logistic growth is a form of density-dependent growth • Growth rate (λt = Nt+1 /Nt ) declines as N approaches K • Growth (∆t = Nt+1 − Nt ) peaks at K/2 (the inflection point) • The model isn’t mechanistic in the sense that it doesn’t include birth, mortality, and movement processes. Background Logistic growth Summary 12 / 12
27. Summary and assignment Summary • Logistic growth is a form of density-dependent growth • Growth rate (λt = Nt+1 /Nt ) declines as N approaches K • Growth (∆t = Nt+1 − Nt ) peaks at K/2 (the inflection point) • The model isn’t mechanistic in the sense that it doesn’t include birth, mortality, and movement processes. • But it does allow for complex dynamics that resemble patterns seen in nature. Background Logistic growth Summary 12 / 12
28. Summary and assignment Summary • Logistic growth is a form of density-dependent growth • Growth rate (λt = Nt+1 /Nt ) declines as N approaches K • Growth (∆t = Nt+1 − Nt ) peaks at K/2 (the inflection point) • The model isn’t mechanistic in the sense that it doesn’t include birth, mortality, and movement processes. • But it does allow for complex dynamics that resemble patterns seen in nature. Assignment Read pages 32–36 in Conroy and Carroll Background Logistic growth Summary 12 / 12