HS-LS2-1: Using Math to Explain Carrying Capacity in Ecosystems
In plain English: HS-LS2-1 is the NGSS high school performance expectation that asks students to use graphs, charts, histograms and simulations to explain what limits the carrying capacity of an ecosystem. Students compare how resources, boundaries, climate and competition change the largest population a habitat can support, from a lab culture to an island. It is usually taught in Biology or Environmental Science.
Use mathematical and/or computational representations to support explanations of factors that affect carrying capacity of ecosystems at different scales.
Clarification Statement: Emphasis is on quantitative analysis and comparison of the relationships among interdependent factors including boundaries, resources, climate and competition. Examples of mathematical comparisons could include graphs, charts, histograms, or population changes gathered from simulations or historical data sets.
Assessment Boundary: Assessment does not include deriving mathematical equations to make comparisons.
Next Generation Science Standards · Disciplinary Core Idea: Ecosystems: Interactions, Energy, and Dynamics · Official standard
Every population eventually runs into limits. In this lesson students use numbers to explain what sets the carrying capacity of a habitat, the largest population it can support over time. They read carrying capacity from population graphs, estimate it from the amount of food or space available, build and read a histogram of yearly counts, run a hand simulation, and compare habitats of very different size, from a flask of yeast to an island. Each time, the math is the evidence for an explanation about a factor: resources, boundaries, climate or competition.
The NGSS Clarification Statement says: "Emphasis is on quantitative analysis and comparison of the relationships among interdependent factors including boundaries, resources, climate and competition. Examples of mathematical comparisons could include graphs, charts, histograms, or population changes gathered from simulations or historical data sets." The Assessment Boundary says: "Assessment does not include deriving mathematical equations to make comparisons." So students use representations they are given or build from data (graphs, tables, histograms, simple ratios such as food supply divided by food needed per animal), but they never derive a growth equation. Every data set on this page is invented for teaching and labeled as invented.
Learning Objectives
By the end of this lesson, students will be able to:
Estimate the carrying capacity of a habitat from a population graph or table by averaging the years after the population levels off
Estimate carrying capacity from resource data by dividing the supply of a limiting resource by the amount one individual needs
Use a histogram of yearly counts to describe how a population varies around its carrying capacity
Compare carrying capacity across habitats of different sizes using totals and densities, and explain which factor (resources, boundaries, climate or competition) accounts for the difference
Run a simple simulation, record the population changes and use the results to support an explanation
Prior Knowledge Required
Students should already be comfortable with:
How resource availability affects individuals and populations MS-LS2-1
Ratios, rates and percent change 6.RP.A.3
Finding the mean of a data set and reading a histogram 6.SP.B.5
Reading a line graph and describing where it rises steeply, slowly or not at all
Show a photo of a pond covered edge to edge with a floating green plant. Ask students to think about how it got that way, starting from a few plants in spring.
Warm-Up Prompt
"In April a pond had a few floating plants. By August the surface is covered. Sketch a graph of the number of plants from April to October. Does the curve keep rising? If not, what stops it, and what would you measure to find out?"
Collect sketches. Many students draw a curve that rises forever or one that rises and suddenly stops. Keep a list of the limits they name (space on the surface, nutrients, light, cold weather, other plants): the lesson sorts them into resources, boundaries, climate and competition.
Direct Instruction15-20 minutes
Carrying capacity from data. Carrying capacity is the largest population of a species that a habitat can support over time with the resources, space and conditions it has. A population that starts small often grows slowly, then quickly, then more slowly as each individual gets a smaller share of food or space, and finally levels off. Because real counts go up and down from year to year, the best estimate of carrying capacity is the average of the counts after the curve levels off, not the highest single count. A second estimate comes from resources: divide the supply of the limiting resource by the amount one individual needs. When two independent estimates agree, they support the explanation that this resource is the limit. Use Diagram 1 with the first example.
Reading carrying capacity from a graph (island scale)
20 deer are released on an island with no predators. Counts every 2 years from year 0 to year 22 (invented): 20, 35, 60, 100, 155, 215, 262, 290, 296, 305, 298, 301.
Result: The counts level off after year 14. The average of years 16-22 is (296 + 305 + 298 + 301) ÷ 4 = 300 deer. The largest 2-year increase is +60 deer (years 8-10), when the herd was near half of 300.
Estimating carrying capacity from a resource
Winter browse (twigs and buds) on the same island averages 81,000 kg. An adult deer eats about 1.8 kg of browse per day during a 150-day winter.
Result: Need per deer = 1.8 × 150 = 270 kg. Carrying capacity ≈ 81,000 ÷ 270 = 300 deer. This matches the graph, which supports the explanation that winter food limits the herd.
Climate changes carrying capacity
A drought summer cuts the next winter's browse by 40%.
Result: Browse = 81,000 × 0.60 = 48,600 kg, so carrying capacity ≈ 48,600 ÷ 270 = 180 deer. A herd of 300 is now 120 deer above what the island can feed, so the model predicts a decline toward 180.
Boundaries and competition (rock scale)
Invented data: barnacles need about 4 cm² of bare rock each. A rock face has 2.0 m² (20,000 cm²) of surface. Mussels, which compete for the same space, cover 30% of it.
Result: Without mussels: 20,000 ÷ 4 = 5,000 barnacles. With mussels: 20,000 × 0.70 = 14,000 cm², so 14,000 ÷ 4 = 3,500 barnacles.
Comparing habitats at different scales
Invented data: a 2 ha pond supports about 180 sunfish year after year. A 50 ha lake supports about 3,000.
Result: Pond: 180 ÷ 2 = 90 fish per ha. Lake: 3,000 ÷ 50 = 60 fish per ha. The lake holds more fish in total, but the shallow pond supports more per hectare, so size alone does not set carrying capacity.
After each example, ask: "Which number is the evidence, and which factor does it point to?" Stress two ideas. First, carrying capacity belongs to a habitat and a species together: it changes when resources, boundaries, climate or competitors change. Second, a single high count is not the carrying capacity: a population can briefly overshoot and then fall back.
Guided Practice15 minutes
Pairs analyze invented counts of breeding pairs of a burrow-nesting seabird on a small rocky island, recorded for 20 years after the colony leveled off. The island has 450 burrow sites. Pairs build a histogram, then compare it with Diagram 2.
Guided practice data: breeding pairs per year (invented)
Years 1-5
Years 6-10
Years 11-15
Years 16-20
428
432
412
446
441
425
435
421
436
438
440
437
419
444
427
431
446
430
433
439
Sort the 20 counts into the bins 410-419, 420-429, 430-439 and 440-449 and draw a histogram.
Find the bin with the most years and the mean of the 20 counts (433 pairs). Where does the colony spend most of its time?
The mean is 17 pairs below the 450 burrow sites. Give two factors other than burrows that could hold the colony below the number of sites.
Write one sentence that uses the histogram as evidence for an explanation of what limits this colony.
Independent Practice10-15 minutes
Each student analyzes invented data from a lab experiment: yeast grown in two flasks of the same broth, one with 2% sugar and one with 1% sugar. The table gives millions of cells per mL.
Independent practice data: yeast cells (millions per mL, invented)
Time (h)
0
6
12
18
24
30
36
42
2% sugar
1
4
14
30
38
39
40
41
1% sugar
1
4
11
17
19
20
21
19
Students estimate each carrying capacity as the average of the last three counts (40 and 20 million cells per mL), compare the ratio of the two estimates with the ratio of the sugar amounts, and write a claim, evidence and reasoning paragraph about the limiting factor in the flasks. Ask them why the two flasks grew at the same rate for the first 6 hours.
Closure5 minutes
Exit ticket: "A meadow produces 4,500 kg of seeds a year that mice can eat, and each mouse eats about 1.5 kg a year. Estimate the carrying capacity. A parking lot then paves half of the meadow. What happens to the estimate, and which factor is this?" (3,000 mice, then 1,500: a boundary that removes habitat.) Return to the warm-up sketches and ask students to mark the carrying capacity on their pond graph.
Differentiation Strategies
For Struggling Students
Give population tables with an extra column for the change from one count to the next, so students can see growth speed up and then slow down
Provide a frame for resource estimates: "Supply ___ ÷ need per individual ___ = about ___ individuals"
Start with the island deer graph before the histogram, and let students use a calculator for every average
For Advanced Students
Have students set up the bean simulation in a spreadsheet, change one factor at a time and compare the level-off values across runs
Ask students to compare two habitats using both total carrying capacity and carrying capacity per hectare, and explain when each comparison is the fairer one
Ask students to find a real long-term population data set from a park or wildlife agency and judge whether it shows a carrying capacity
Assessment Guidance
What to Look For
Check that students estimate carrying capacity from the level part of a graph and use an average, not the peak or the last single count. Look for explanations that name one factor and point to a specific number as evidence. Listen for the idea that carrying capacity is a fixed property of a species: it is not, it changes with the habitat. When students compare habitats of different sizes, check that they compare densities as well as totals.
02
Classroom Activities
3 Activities
1
Bean Simulation: Deer on a Winter Island
25 minGroups of 4-6
Groups run a hand simulation in which dried beans stand for winter food. Each round is one year. Each deer needs 3 beans to survive the winter, and a deer that gathers 6 or more beans also produces one offspring. Groups record the herd size for 15 rounds, then change one factor and see how the level-off value changes.
Procedure
Spread 36 beans on a tray. Each deer is a paper cup; start with 4. In each round, students take turns: for each cup, one student has 5 seconds to pick up beans one at a time and drop them into that cup. Change the order of the cups every round.
Count each cup: fewer than 3 beans, the deer dies; 3-5 beans, it survives; 6 or more, it survives and adds one new deer. Record the new herd size and return all beans to the tray.
Run 10 rounds with 36 beans, then 5 "drought" rounds with 24 beans.
Safety: keep beans on the tray and pick up any that fall so no one slips. Do not eat the beans: dried beans are not safe to eat raw. Paper squares work as well and suit students with food allergies.
Show that with 36 beans the herd can never be larger than 12 deer. (Each survivor uses 3 beans and each offspring needs 3 more.) Rounds 5-10 average 11.5 deer: why is the average a little below 12?
After the drought starts, the herd never goes above 8. Which factor did you change, and how does 24 ÷ 3 explain the new level?
How would you change the simulation to model a competitor that also eats the beans, or a fence that closes off part of the tray?
Modification for Distance Learning
Students run the same rules in a spreadsheet, using a random number for the beans each deer gathers, and post their 15-round graph with a two-sentence explanation.
2
Duckweed Lab: Space and Nutrients
25 min setup, then 5 min counts twice a week for 4 weeksGroups of 3
Groups grow duckweed, a tiny floating plant, in containers that differ in surface area or in nutrient strength. They count fronds (the small leaf-like bodies) twice a week for 4 weeks, graph the counts and compare the level-off values.
Procedure
Container A: a clear cup with about 50 cm² of water surface and full-strength nutrient solution. Container B: a container with about 100 cm² of surface and the same solution at the same depth. Container C: like A but with half-strength solution.
The teacher prepares the solutions from aquarium plant fertilizer diluted as the label directs, wearing goggles and gloves. Students add 10 fronds to each container and place all three the same distance under an LED lamp.
Count fronds with a magnifying lens on days 0, 4, 7, 11, 14, 18, 21, 25 and 28. Top up with distilled water to the same mark.
Safety: use duckweed from a biological supply company or aquarium store, not from a local pond. Wear gloves or wash hands after handling plants and water. Never pour duckweed or its water into a sink, storm drain, pond or stream, because it spreads quickly: at the end, seal it in a bag and put it in the trash. Keep lamp cords away from the water and wipe up spills.
On day 28, B has 428 fronds and A has 216. Doubling the surface area about doubled the count, but B also holds twice the volume of solution at the same depth, so it has twice the nutrients too. Which two factors could explain the doubling, and how could you change the setup to separate them?
C, with half-strength nutrients, ended at 120 fronds, a little more than half of A. What does this tell you about nutrients in container A?
B grew by 36 fronds in its last 3 days while A grew by only 4. Has B reached its carrying capacity yet? What would you do to find out?
Modification for Distance Learning
Students graph the sample data above, estimate each level-off value and write an explanation. A student with access to duckweed can run one container at home and share photos of the counts.
3
Data Stations: Carrying Capacity at Four Scales
25 minGroups of 3-4
Four stations each hold an invented data set from a habitat of a different size. At each station, groups estimate a carrying capacity or a change in it, name the factor, and write one claim-evidence-reasoning card. Groups rotate every 6 minutes.
Station Data
Station 1, a 250 mL culture flask (competition): Protist A alone levels off at 120 cells per mL, and protist B alone at 90. Grown together, A levels off at 70 and B at 45.
Station 2, a mountain stream (climate): Trout per 100 m of stream in four cool summers: 46, 50, 47, 49. In four hot summers: 20, 25, 21, 22.
Station 3, a 12,000 ha elk winter range (boundary): The herd held at about 1,800 elk. A housing development then removed 3,000 ha of winter range, and the herd leveled off at about 1,300.
Station 4, a historical-style data set (overshoot): Sheep on an island, counted every 5 years from year 0 to year 40: 100, 450, 1,600, 2,400, 1,100, 1,500, 1,350, 1,450, 1,400.
Discussion Questions
At Station 1, how much lower is each protist's level-off value when the two share a flask? What resource might they compete for?
At Station 2, the average trout count in cool summers is 48 and in hot summers 22. What else would you want to know before blaming temperature?
At Station 3, 1,800 elk on 12,000 ha is 0.15 elk per ha. At that density the remaining 9,000 ha would hold 1,350 elk. The herd leveled off a little lower. Suggest a reason.
At Station 4, why is the year-15 peak of 2,400 a poor estimate of carrying capacity? What is a better estimate?
Challenge Variation
Groups rank the four stations by how strong the evidence is for the named factor, and describe one extra data set that would make the weakest station convincing.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Island Deer Population Leveling Off at Its Carrying Capacity
Invented counts of deer on an island, every 2 years, drawn to scale. Growth is fastest in the middle (years 8-10) and slows as each deer gets less winter browse. After year 14 the counts stay close to 300, the same value the browse supply predicts (81,000 kg ÷ 270 kg per deer).
Diagram 2: Histogram of Yearly Seabird Counts
The 20 guided-practice counts of breeding pairs, sorted into bins of 10 and drawn to scale. Most years fall in the 430-439 bin, and the mean is 433 pairs, below the 450 burrow sites: something besides burrows also holds the colony down.
04
Homework Assignment
~30 min
HS-LS2-1 Homework: Carrying Capacity from Data
Directions: All data sets are invented for practice. Show each calculation with units, and end each problem with a one- or two-sentence explanation that names the factor and the number you used as evidence.
Part 1: Reading and Estimating Carrying Capacity (Problems 1-2)
Sea otters return to a kelp bay. Counts every 2 years from year 0 to year 18 are: 12, 22, 40, 68, 104, 136, 152, 158, 154, 160. (a) Estimate the carrying capacity of the bay using the average of the last four counts. (b) Between which two counts did the population grow the most? (c) Explain why growth slowed near the end.
A 400 ha grassland pasture grows about 1,460 kg of edible grass per hectare each year, but only half of it can be eaten without harming the plants. A sheep eats about 1.6 kg of grass per day. (a) How much grass can be eaten each year? (b) Estimate the carrying capacity for sheep. (c) In a wet year the grass grows 25% more. What is the new estimate, and which factor changed?
Part 2: Comparing Factors (Problems 3-4)
A histogram shows the number of hawk pairs nesting in a valley. Years 1-15, before a highway was built: 20-24 pairs in 1 year, 25-29 pairs in 4 years, 30-34 pairs in 8 years, 35-39 pairs in 2 years. Years 16-30, after the highway: 15-19 pairs in 3 years, 20-24 pairs in 9 years, 25-29 pairs in 3 years. (a) Which bin holds the most years in each period? (b) Using the midpoints of those bins, estimate the percent change in carrying capacity. (c) Which factor does the highway represent, and what other data would strengthen your explanation?
Native crayfish in a 500 m² stretch of stream averaged 7.6, 8.4, 8.1 and 7.9 per m² in four years before an invasive crayfish arrived, and 3.1, 2.8, 3.3 and 2.8 per m² in four years after. (a) Find both averages. (b) Find the percent change. (c) About how many fewer native crayfish does the stretch hold? (d) Which factor is at work, and what data would show it is not just a run of bad weather?
Part 3: Simulations and Arguments (Problems 5-6)
A computer simulation of a fish population in a lake is run with three food supplies. The last five years of each run give: 100% food: 812, 790, 805, 798, 795 fish; 75% food: 598, 606, 590, 604, 602 fish; 50% food: 402, 396, 405, 399, 398 fish. (a) Estimate the carrying capacity in each run. (b) Describe the pattern. (c) Use the pattern to predict the carrying capacity with 60% food.
A student claims: "A bigger forest always has a bigger carrying capacity." An 80 ha forest with about 12 dead standing trees per ha holds 24 pairs of woodpeckers. A 200 ha forest of young trees with about 2 dead standing trees per ha holds 20 pairs. Woodpeckers nest in holes they dig in dead trees. Compute the pairs per 100 ha in each forest and write a claim, evidence and reasoning response to the student.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
Using the Representation
Reads the graph, table or histogram correctly and uses an average or a resource ratio with units
Correct method with one reading or calculation error
Naming the Factor
Names resources, boundaries, climate or competition and ties it to a specific number
Names a factor but does not connect it to the data
Explanation and Evidence
Explanation cites specific numbers and says what further data would strengthen it
Explanation is stated but the evidence is general
05
Quiz: 20 Questions
Interactive, with answers
Instructions
All data sets are invented. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
A biologist counts the same population in a habitat every year for 30 years. Which pattern in the counts would show that the population has reached the carrying capacity of the habitat?
Answer: B
Near carrying capacity births and deaths roughly balance, so the counts stop climbing and vary around one level: B. Choices A and D describe a population still growing, far below its limit. Choice C describes a population that crashed and died out, which shows the habitat could not support it at all, not a steady limit.
Question 2 of 20 · Multiple Choice
Lizards on a small island were counted every year (invented data): 50, 90, 150, 210, 240, 252, 248, 250, 251. What is the best estimate of the carrying capacity?
Answer: C
The last four counts, 252, 248, 250 and 251, average 250.25, so about 250 lizards. Choice A is half of that, where growth was fastest, not where it stopped. Choice B adds all the counts together. Choice D misreads the data: from the sixth count on, the counts stay between 248 and 252.
Question 3 of 20 · Multiple Choice
A marsh produces about 36,000 kg of seeds each winter that geese can eat. Each goose eats about 45 kg of seeds over the winter. If seeds are the limiting factor, what is the carrying capacity for geese?
Answer: A
36,000 kg ÷ 45 kg per goose = 800 geese. Choice B multiplies the supply by the need instead of dividing. Choice C gives the need per goose, not a number of geese. Choice D misplaces the decimal point, dividing by 4.5 kg instead of 45 kg.
Question 4 of 20 · Multiple Choice
A pond supports about 2,000 snails, and algae is their limiting food. A cool, cloudy summer cuts algae growth by 30%. What is the best estimate of the new carrying capacity?
Answer: D
If algae limits the snails, 30% less algae supports 30% fewer snails: 2,000 × 0.70 = 1,400. Choice B is the size of the drop, not the new level. Choice C adds 30% instead of taking it away. Choice A ignores that climate changed the food supply.
Question 5 of 20 · Multiple Choice
Salamanders live evenly across a 60 ha forest, which supports about 1,500 of them. A new road and houses remove 20 ha of the forest. What is the best estimate of the new carrying capacity?
Answer: B
The density is 1,500 ÷ 60 = 25 per ha, and 40 ha remain, so 25 × 40 = 1,000. Choice A is the number lost, not the number left. Choice C subtracts 20 salamanders instead of 20 ha. Choice D divides by the remaining fraction instead of multiplying by it.
Question 6 of 20 · Multiple Choice
A moose population held steady near 900 for ten years (invented data). After a winter of very deep snow it fell to about 590, then climbed back to about 900 within four years, with no change in food or predators. Which explanation do the data best support?
Answer: A
The population returned to the same level once the snow year passed, so the long-run limit did not change. Choice B is contradicted by the recovery to about 900. Choice C ignores ten steady years. Choice D names a factor that the data give no evidence for: food and predators did not change, and no competitor is mentioned.
Question 7 of 20 · Multiple Choice
A histogram of 25 yearly counts of a marmot colony (invented data) shows: 60-69 marmots in 2 years, 70-79 in 6 years, 80-89 in 13 years and 90-99 in 4 years. Which range is the best estimate of the carrying capacity?
Answer: C
More than half the years (13 of 25) fall in the 80-89 bin, so the colony spends most of its time there. Choice B picks the highest bin, which describes good years, not the usual level. Choice A picks the rarest low years. Choice D ignores that the counts cluster in one bin.
Question 8 of 20 · Multiple Choice
A 3 ha pond supports about 240 frogs and a 20 ha lake supports about 1,000 frogs of the same species. Which statement do the data support?
Answer: D
Pond: 240 ÷ 3 = 80 frogs per ha. Lake: 1,000 ÷ 20 = 50 frogs per ha. So the lake has the larger total but the pond has the higher density. Choice A compares totals and calls it density. Choice C assumes that smaller means overcrowded, which the steady counts do not show.
Question 9 of 20 · Multiple Choice
A native minnow averaged 42 fish per pool before a competing fish arrived and 28 per pool afterward (invented data). By what percent did the carrying capacity for the minnow fall?
Answer: B
The drop is 42 - 28 = 14 fish, and 14 ÷ 42 ≈ 0.33, so about 33%. Choice A gives the drop in fish as a percent. Choice C divides the drop by the new value (14 ÷ 28). Choice D is the fraction that remains, 28 ÷ 42.
Question 10 of 20 · Multiple Choice
A computer simulation of rabbits on a pasture is run twice with the same food supply. One run starts with 50 rabbits and the other with 900. Both level off near 600. What does this result show?
Answer: A
Both runs end at the same value, so the limit comes from the food supply in the model. Choice B is contradicted: the larger start did not end higher. Choice C expects the start to matter, which is the idea the result disproves. Choice D is contradicted by the run that grew from 50 to 600.
Question 11 of 20 · Multiple Choice
Three ponds of the same size produce 2, 4 and 6 kg of algae per week and support about 300, 600 and 900 snails. Use the pattern in these data to predict the carrying capacity of a similar pond that produces 5 kg of algae per week.
Answer: C
Each pond supports 150 snails per kg of weekly algae (300 ÷ 2 = 600 ÷ 4 = 900 ÷ 6 = 150), so 5 kg supports about 150 × 5 = 750. Choice A uses the 4 kg pond without adjusting. Choice B multiplies 300 by 5. Choice D adds 300 snails for the one extra kilogram, but in the data each extra 2 kg adds 300 snails.
Question 12 of 20 · Multiple Choice
A student wants to compare the carrying capacity of the same bird species in five different habitats. Which representation would make the comparison clearest?
Answer: D
A bar chart puts one level-off value per habitat side by side, which is exactly the comparison needed. Choice A shows only one habitat. Choice B describes the birds, not the populations. Choice C shows parts of a whole, and the habitats are not parts of one total.
Question 13 of 20 · Multiple Choice
A deer herd near its carrying capacity was counted in four years (invented data): 510, 470, 530 and 490. A student uses 530 as the carrying capacity. What is a better estimate?
Answer: A
The mean is (510 + 470 + 530 + 490) ÷ 4 = 500. Near carrying capacity, counts rise and fall around the limit, so the average describes it better than the best year (choice B) or the worst year (choice C). Choice D adds the counts instead of averaging them.
Question 14 of 20 · Multiple Choice
Students count periwinkle snails in five 1 m² quadrats on a salt marsh: 22, 18, 25, 20 and 15 snails. The whole marsh zone where they live is 2,000 m². About how many snails does the zone hold?
Answer: C
The mean is (22 + 18 + 25 + 20 + 15) ÷ 5 = 20 snails per m², and 20 × 2,000 = 40,000. Choice A is the total in the five quadrats. Choice B multiplies that total by the area, which counts each square meter five times. Choice D is the density, not the number in the zone.
Question 15 of 20 · Short Answer
A rabbit population on a pasture grew from 100 to 160 in one year when it was small, and from 400 to 420 in one year when it was large (invented data). Find the percent growth in each year and use the numbers to explain what is happening.
Small population: (160 - 100) ÷ 100 = 60%. Large population: (420 - 400) ÷ 400 = 5%. The growth rate per rabbit fell sharply as the population grew. Each rabbit gets a smaller share of grass and space, so fewer young survive: the population is approaching the carrying capacity of the pasture.
Question 16 of 20 · Short Answer
A pair of barn owls needs about 2,500 small rodents a year. The fields in a valley produce about 30,000 rodents a year that owls can catch. (a) Estimate the carrying capacity for owl pairs. (b) Farmers start using rodent poison, which cuts the rodents available by 40%. Estimate the new carrying capacity and name the factor.
(a) 30,000 ÷ 2,500 = 12 pairs. (b) 30,000 × 0.60 = 18,000 rodents, and 18,000 ÷ 2,500 = 7.2, so about 7 pairs: a fraction of a pair is not possible, and 8 pairs would need 20,000 rodents. The factor is resources (food). The poison could also harm owls that eat poisoned rodents, which would lower the population further.
Question 17 of 20 · Short Answer
Three reserves of the same size have average rainfall of 300, 450 and 600 mm per year. Each supports a steady population of gazelles: 150, 225 and 300 (invented data). Describe the pattern, predict the carrying capacity of a similar reserve with 400 mm of rain, and name the factor.
Each reserve supports 0.5 gazelle per mm of yearly rain (150 ÷ 300 = 225 ÷ 450 = 300 ÷ 600 = 0.5), so a 400 mm reserve would support about 200 gazelles. The factor is climate: more rain grows more grass, which is the food that limits the gazelles. The prediction assumes the reserves differ only in rainfall.
Question 18 of 20 · Short Answer
A songbird nests only in tree holes. A biologist thinks the number of holes, not food, limits the population in a set of woodlots. Plan an investigation that could test this idea. Say what you would change, what you would measure, and what result would support the idea.
Choose several similar woodlots and add nest boxes to about half of them, leaving the others as controls. Count the breeding pairs in every woodlot for several years before and after adding the boxes, and measure food (for example, insect counts) in both groups. If the pairs increase in the woodlots with boxes and stay level in the controls while food stays the same, the data support the idea that nest holes limit the population. If both groups stay the same, something else, such as food, is the limit.
Question 19 of 20 · Short Answer
In a computer simulation, each rabbit needs 50 kg of grass a year and the pasture grows 9,000 kg a year. Five runs level off at 172, 168, 175, 170 and 165 rabbits. Compare the simulation results with the carrying capacity the food supply predicts, and suggest one reason for the difference.
The food supply predicts 9,000 ÷ 50 = 180 rabbits. The five runs average (172 + 168 + 175 + 170 + 165) ÷ 5 = 170 rabbits, about 10 fewer. The runs level off below 180 because the model includes chance: in some years rabbits do not find all the grass, or some die of other causes, so the population stays a little below the food limit.
Question 20 of 20 · Short Answer
Water fleas grown in a 1 L jar of pond water level off at about 60 per liter. A student predicts that a 1,000,000 L lake would hold 60,000,000 water fleas. Explain why this scale-up could be wrong.
The student multiplies 60 per liter by 1,000,000 L to get 60,000,000, which assumes every liter of the lake is like the jar. A lake is not: deep water gets little light, so less algae grows there for the water fleas to eat, and the lake has fish and insects that eat water fleas, which the jar did not. Temperature and oxygen also change with depth. Carrying capacity depends on the conditions of the habitat, not only its volume, so the lake could hold far fewer per liter.
0 of 20 answered · 0 correct
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Frequently Asked Questions
10 Questions
What does HS-LS2-1 mean?
HS-LS2-1 asks students to use math, such as graphs, charts, histograms and simulations, to explain what limits the carrying capacity of an ecosystem. The focus is on using numbers as evidence for an explanation, not on memorizing a definition. The code means High School, Life Science, core idea 2 (Ecosystems: Interactions, Energy, and Dynamics), performance expectation 1.
Is HS-LS2-1 taught in biology or environmental science?
It is usually taught in the ecology unit of high school Biology, and it also fits Environmental Science or an integrated science course. The math needed is averages, ratios, percent change and reading graphs, so students do not need anything beyond middle school math.
What is carrying capacity in simple terms?
It is the largest population of a species that a habitat can keep supporting over time. Below it, a population usually grows; near it, births and deaths roughly balance. It belongs to a particular species in a particular habitat, and it changes when the habitat changes.
What is the assessment boundary for HS-LS2-1?
The official Assessment Boundary says that assessment does not include deriving mathematical equations to make comparisons. Students use graphs, tables, histograms and simple calculations such as averages and ratios, and they may use a simulation that someone else built, but they are not asked to derive a population growth equation. Deriving one is beyond this standard.
What factors affect carrying capacity?
The Clarification Statement names boundaries, resources, climate and competition. Boundaries include fences, roads, shorelines and the edge of a habitat. Resources include food, water, nutrients, light and nesting sites. Climate includes temperature, rainfall and severe weather. Competition comes from other individuals or other species that use the same resources. Predators and disease also matter in real ecosystems.
What does "ecosystems at different scales" mean?
It means comparing habitats of very different sizes, from a flask or a jar in the lab to a pond, a forest, an island or a whole region. The same reasoning works at every scale, but a fair comparison often needs densities (individuals per liter or per hectare) as well as totals, because a big habitat is not always a better one.
Can a population go above its carrying capacity?
Yes, for a while. A population that grows fast can overshoot, especially when food regrows slowly, and then it falls back, sometimes below the carrying capacity if the habitat was damaged. That is why students should estimate carrying capacity from several years of counts rather than from the highest one.
How do you find carrying capacity on an HS-LS2-1 graph?
Find the part of the population graph where the curve stops rising and the counts stay near one level, then average those counts. Check the estimate against a second source when you can, such as the food supply divided by the food one individual needs. Two estimates that agree make the explanation much stronger.
What simulations can students use for carrying capacity?
A hand simulation with beans or paper squares works well and costs almost nothing. A spreadsheet is a good next step: each row is a year and simple rules decide births and deaths. Many free online population simulations also let students change food, space or competitors. Whatever the tool, students should record the output and use it as evidence, and remember that a simulation is only as good as its rules.
How does HS-LS2-1 connect to other standards?
It builds on MS-LS2-1, where students analyze data on how resources affect populations. HS-LS2-2 extends the same math to biodiversity, and HS-LS2-6 asks students to evaluate claims about how ecosystems stay stable or change. On the math side, it uses the same skills as the statistics standards on representing and comparing data.
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Related Standards
5 standards
These standards connect to HS-LS2-1: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
MS-LS2-1Prerequisite
Analyze data on how resource availability affects organisms and populations
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