1. The Exponential Growth Model It’s the Malthusian Model in the words of Newton …

2. Population size N N(t+1) = N(t)

3. Population size N

4. The continuous-time exponential growth model: One  parameter: r = b - d  the instantaneous per capita population growth rate.

5. The continuous-time exponential growth model: To use this equation to predict population size some time in the future, one needs to integrate it over time.

6. Solving the exponential growth model by integration: 1. rearrange: 2. integrate both sides: 3. solve integral: Rules used:

7. 4. rearrange: 5. take the exponent: 6. rearrange: 3. solve integral:

8. Malthusian Growth Model (discrete-time model ) Exponential Growth Model (continuous-time model) One parameter  = 1 + b – d   population grows    population level    population declines One parameter  r = b - d r > 0  population grows r = 0  population level r < 0  population declines

9. What is the numerical relationship between the  and r of the two models? N(t) …………..N N(t+1)…………N t t0t0 The “1” in the Malthusian model needs to be specified:  = t - t 0

10. Re-arrange:

11. Given this relationship: if the doubling time of a population is 10 days, what is the value of r?

12. 1)How do we know if a population follows the exponential growth model? 2)Under conditions of “unchecked” growth, are there species-specific differences in how fast a population can grow?

13. Time Numbers of individuals Is this growth exponential?

14. The exponential growth model: y = b + s * x Plotting ln N v time, should result in a straight line where the slope is r!

15. The exponential growth model: time ln N t0t0 intercept = ln(N t 0 ) slope = r

16. Destruction Island pheasants: Doubling time :  = ln(2)/r = 0.69/1.11 = 0.62 years y = 1.93 + 1.11x r 2 = 0.985 Years after introduction Ln N

17. Cheatgrass in the Western USA: Doubling time :  = ln(2)/r = 0.69/0.132 = 5.25 years y = -3.72 + 0.132x r 2 = 0.985

18. Year AD ln(N) Estimated worldwide human population size: Humans grow at faster than exponential rates.

19. Adult bluefin tuna in the western North Atlantic

20. Nest box occupancy (%) Smallwood et al. 2009, Journal of Raptor Research 43(4):274-282. American Kestrel (Falco sparverius)

21. Loggerhead turtles on the coast of Turkey Ilgaz et al. 2007. Biodivers Conserv 16:1027–1037

22. Loggerhead Turtle: year No. turtlesln (N) 19931204.79 19941605.08 1995804.38 1996904.50 19971204.79 19981004.61 19991104.70 20001154.74 2001804.38 20021054.65 2003604.09 Year y = 90 - 0.043x r 2 = 0.2971 Halving time :  = ln(1/2)/r = -0.69/-0.043 = 16 years

23. When do we see exponential decline? 1.In general, when birth rates cannot keep up with death rates. 2.Exponential decline can occur : when critical resources disappear, and no substitute is found, when habitats disappear, and the species cannot find or move to adequate habitat elsewhere (e.g. islands), when a competitively superior species or a predator suddenly appears or becomes more abundant. when climate suddenly changes and the species cannot adapt, By chance, in small, isolated populations. 3.Exponential decline can lead to extinction.

24. Excel Worksheets: Exponential Growth Analysis

25. Unchecked population growth rates vary across 5-6 orders of magnitude among organisms

26. James Brown, New Scientists (2004)

28. For all the diversity among the spcies of earth, some regular patterns arise: 1.Larger organisms consume more energy and grow faster, but not exactly in proportion to their size. A 200 kg elk, 10,000x heavier than a 20 g mouse, eats only 1000x more food. Mouse metabolism is therefore 10x faster, but elk live 10x longer. 2.Smaller organisms have shorter time spans and reproduce faster than larger organisms. 3.Smaller organisms achieve higher population densities.

29. Summary: 1.Two equivalent models express unchecked population growth: the Malthusian model, and the exponential model. 2.Both models have one parameter that describes a population’s potential for growth, when resources are unlimited. 3.The population growth rates of earth’s organisms vary over many orders of magnitude. 4.Smaller organisms acquire resources faster, live shorter lives, and have shorter doubling times. 5.The Human population explosion over the past 2000 years can be linked to dramatic increases in the average energy consumption of individuals.