Population Math

What is a population A group of organisms of the same species occupying a particular space at a particular time

Population size N = number of individuals in a population N = the change in population size B = # of births D = # of deaths I = # of immigrants E = # of emigrants N = (B-D) + (I-E) in a closed population N = B-D

N = current population size How many eagles are there? N = 5 N may change with time It may increase It may decrease it may stay the same

r = growth rate r = negative Why?

2 things can cause the value of r to change. What are they? r = growth rate r = positive Why? Neglecting migration 2 things can cause the value of r to change. What are they? r = (births-deaths)/N

practice A population of 5 eagles has 3 births & 2 deaths. What is the growth rate? r = (B-D)/N = (3-2)/5 = ⅕ = 0.2

On a clean sheet of paper. Solve for r. Show equation & work In 1 year, 10 eagles die and 5 are born in a population that had 25 eagles at the beginning of the season. In one breeding season, 2 dandelions reproduced making 20 more, but you sprayed and killed 2 300 seniors started the school year, 3 had babies and 1 didn’t survive the year.

Dividing by “N” gives a per capita rate Dividing by “N” gives a per capita rate. In other words, the average change per individual in the population. If the value of r remains constant, you will have exponential growth. Describe the graph.

Back to our eagle population r = (3-2)/5 = 0.2 Nr = increase in population size 5 x 0.2 = 1 Add this to the previous population size 5+1 = 6 #4. Complete the table TIME (yr) N 5 1 6 2 3 4

Why exponential The table shows the “lag phase”, but if the growth rate stays the same (0.2) and the population size of eagles had increased to 100, what would be the population one year later? Two years later? Three years?

note if the growth rate is constant The change in population size increases every year. The larger the starting population, the bigger the increase. You could use a spreadsheet such as Excel to calculate out many generations quickly.

6. Complete the data if N=100 and r= -0.3 TIME N 100 1 2

#7. Recalculate our eagle population if r=0.5 TIME (yr) N 5 1 2 3 4

REMEMBER If the value of r is greater than zero, the growth is exponential.

What slows this growth? Limiting Factors How do limiting factors change a population growth graph? What type of growth is this?

Population density Population size divided by area (if terrestrial) Population size divided by volume (if aquatic) Perhaps because of its great size, the bald eagle requires a lot of space. Bald eagles defend nesting territories ranging from 275 to 3,200 acres with an average of about 500 acres. The Wichita Mountains Wildlife Refuge is about 60,000 acres. About than 10% of this is suitable habitat for eagles. 8. How many acres is suitable for bald eagles? 9. Use the average space needed to determine the number of eagles at the Refuge at carrying capacity.

Population density 10. Using the data in questions 7-10, in what year will the bald eagles exceed the carrying capacity of the Wildlife Refuge? 11. Using data in questions 7-10, calculate the population density for years 0, 1 and 2. (N/A or Number of eagles/acres of habitat)

DARWIN SAID MORE INDIVIDUALS IN A SPECIES WILL BE BORN THAN CAN SURVIVE. THOSE BEST SUITED TO THEIR ENVIRONMENT WILL BE BETTER ABLE TO SURVIVE AND REPRODUCE. THIS IS TRUE DUE TO LIMITING FACTORS WHICH CREATE COMPETITION.

K = carrying capacity - the maximum population size an ecosystem can support. If our initial 5 eagles eat trout, there’s still plenty of trout since there is more energy at the lower trophic level. But by the 4th year, 10 eagles are eating the trout

r-strategist have a high value for r Coral reproducing K-strategists reach a fairly stable carrying capacity beluga whales

Crude Birth Rate – births as a ratio to the total population per 1000 individuals in the population in a given amount of time (usually a year) Crude Death Rate – deaths as a ratio to the total population per 1000 individuals in the population in a given amount of time (year) %Annual Growth Rate (AGR) = CBR-CDR/10 Doubling Time – time for population size to double at current growth rate

Calculate CBR CBR = B/N * 1000 (divided by time if needed) Practice 10 births when N = 5000 10/5000 = 0.0002 * 1000 =2%

Calculate CDR CDR = D/N * 1000 (divided by time if needed) Practice 15 deaths when N = 45,000 15/45000 = 0.00033 * 1000 =0.33%

Calculate CBR CBR = B/N * 1000 (divided by time if needed) 1. 20 births when N = 4000 2. 40 births when N = 4000 Age Specific CBR = Live births to females in specific age range/number of females ages 12- 19 * 1000 3. 50 births, but only 10 to females over 35 in a population with 600 females over 35 years old.

Calculate CDR CDR = D/N * 1000 in a specified time (year) 4. 25 deaths in 2 years with a population of 500 5. 25 deaths in 6 months with a starting population of 500

Annual Growth Rate (AGR) Stated as a percentage (CBR-CDR)/10 6. Find AGR if CBR = 5 and CDR = 3 7. Find AGR if in a population of 100,000 there are 100 births and 25 deaths.

Doubling Time The time it takes a population size to double under a constant growth rate Rule of 70: Doubling Time = 70/AGR(%)

8. How many years will it take for a population of 5000 to double if the annual growth rate is 2%? 5%? 10% 9. In 2013 the population of deer was 500, in what year will the population be 1000 assuming unlimited resources and a constant growth rate of 7%? 10. A population of rats went from 1000 to 2000 over a period of 5 years. What is the AGR?

Exponent refresher When multiplying, add exponents When dividing, subtract exponents Place one non-zero digit to the left of the decimal Example: 2.0 x 109/2.0 x 106 = 1.0 x 103 Example: 1.2 x 105 * 2.0 x 104 = 2.4 x 109 Example: 9.0 x 103 * 2.0 x 105 = 18 x 108 = 1.8 x 109

Currently the world population is 7.1 x 10 9. The AGR is 1.14%. 11. During what year will the population double? 12. What will the world population size be when it doubles?

13. The U. S. population is 3. 2 x 10 8. The current AGR is 0. 7% 13. The U.S. population is 3.2 x 10 8. The current AGR is 0.7%. How many years until the population doubles in size? 14. What will be the U.S. population size in one year? 15. The area of the U.S. is 9.8 x 106 km2. What is the population density? 16. India’s population 1.3 x 109 is and the area is 3.3 x 106 km2. What is the population density of India?