Logistic and Exponential Growth Chapter 40 Logistic and Exponential Growth

You Must Know The differences between exponential and logistic models of population growth. How to calculate and graph growth

Per Capita Rate of Increase − Change in population size Births Immigrants Deaths Emigrants   If immigration and emigration are ignored, a population’s growth rate (per capita increase) equals birth rate minus death rate. 3

Population Changes number of births over a period of time Per-capita birth rate (b) = ÷ population 1,500 ÷ 10,000 = 0.15 number of deaths over a period of time Per-capita death rate (d) = ÷ population 500 ÷ 10,000 = 0.05 Per-capita rate of increase (r) (b) - = (d) 0.15 - 0.05 = 0.10 or 10%

r  b − d r equals 0 (population stays the same size) r is positive (population increases) r is negative (population decreases) 5

N is the change in population size, t is the time interval.  rN N is the change in population size, t is the time interval. Example 0.04 individuals per year Population’s growth rate N = population size = 2,000 individuals X = an increase of 80 individuals per year 6

Exponential population growth is population increase under idealized conditions. Under these conditions, the rate of increase is at its maximum, denoted as rmax.

Exponential population growth results in a J-shaped curve. Exponential Growth dt dN  rmax N Population size (N) Number of generations Exponential population growth results in a J-shaped curve. 8

2,000 dt dN  1.0N 1,500 dt dN  0.5N Population size (N) 1,000 500 5 Figure 40.17 2,000 dt dN  1.0N 1,500 dt dN  0.5N Population size (N) 1,000 500 5 10 15 Number of generations 9

The J-shaped curve of exponential growth characterizes populations in new environments or rebounding populations. 8,000 6,000 Elephant population 4,000 Figure 40.18 Exponential growth in the African elephant population of Kruger National Park, South Africa The elephant population in Kruger National Park, South Africa, grew exponentially after hunting was banned. 2,000 1900 1910 1920 1930 1940 1950 1960 1970 Year 10

Bozeman Science video on Exponential Growth https://www.youtube.com/watch?v=c6pcRR5Uy6w

xkcd This is a joke…

The Logistic Growth Model Carrying capacity (K) is the maximum population size the environment can support. K  carrying capacity Population size (N) dt dN (K − N) K  rmax N Exponential growth cannot be sustained for long in any population. A more realistic population model limits growth by incorporating carrying capacity. Carrying capacity varies with the abundance of limiting resources. In the logistic population growth model, the per capita rate of increase declines as carrying capacity is reached. The logistic model starts with the exponential model and adds an expression that reduces per capita rate of increase as N approaches K. The logistic model of population growth produces a sigmoid (S-shaped) curve. Number of generations 13

Table 40.2 25 1.0 0.98 0.98 + 25 250 1.0 0.83 0.83 + 208 750 1.0 0.50 0.50 + 375 1,000 1.0 0.33 0.33 + 333 1,500 1.0 0.00 0.00 14

Exponential growth 2,000 dt dN  1.0N 1,500 K  1,500 Logistic growth Figure 40.19 Exponential growth 2,000 dt dN  1.0N 1,500 K  1,500 Logistic growth dt dN (1,500  N) 1,500 Population size (N)  1.0N 1,000 Population growth begins slowing here. Figure 40.19 Population growth predicted by the logistic model 500 5 10 15 Number of generations 15

Number of Paramecium/mL Figure 40.20 1,000 180 150 800 120 Number of Daphnia/50 mL Number of Paramecium/mL 600 90 400 60 200 30 5 10 15 20 40 60 80 100 120 140 160 Time (days) Time (days) The growth of many laboratory populations, including paramecia, fits an S-shaped curve when resources are limited . These organisms are grown in a constant environment lacking predators and competitors. Some populations overshoot K before settling down to a relatively stable density. Some populations fluctuate greatly and make it difficult to define K. (a) A Paramecium population in the lab (b) A Daphnia (water flea) population in the lab 16

Bozeman Science video on Logistic Growth https://www.youtube.com/watch?v=rXlyYFXyfIM

You will be given this information for the test and the AP Exam