[LS2-1] Carrying Capacity Mathematics
This standard focuses on applying mathematical concepts in order to better understand the phenomena of carrying capacity.
Resources for this Standard:
Here’s the Actual Standard:
Use mathematical and/or computational representations to support explanations of factors that affect carrying capacity of ecosystems at different scales.
This standard is one of several NGSS high school standards that is highly involved with math. These standards can make for great collaborative projects between biology and math teachers, including projects that span multiple classes. Here, we will focus more on the biology of carrying capacity and the biotic and abiotic factors that influence carrying capacity.
Carrying capacity is the number of individuals a given environment can support, based on a number of interrelated factors. First, let’s look at abiotic factors like sunlight, water, and substrate. Since the globe gets uneven amounts of sunlight, both the northern and southern poles receive the least light and considerable seasonal differences. This limits the amount of plant life present, which in turn lowers the carrying capacity of other organisms. Precipitation levels and the amount of nutrients in the ground substrate further determine how many plants can grow, influencing higher levels of the food web.
Biotic factors include other organisms within the environment, as well as the organic material they release when they die. If a population had no predators and an abundant food supply, it would experience exponential growth. However, at a certain point, the abiotic factors of the environment are stretched thin and plant life is restricted. This restricts the herbivores, which then restrict the population of carnivores. As such, all environments have this hard limit – or carrying capacity – that defines how many organisms can live in a given place.
A little clarification:
The standard contains this 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, and population changes gathered from simulations or historical data sets.
Let’s look at this clarification a little closer:
Relationships among Independent Factors
The key here is to not only determine how independent factors will affect a single population but to also see how multiple independent factors can play into a much more complex situation. For example, if you only look at precipitations, some clear patterns emerge. In general, places with high precipitation are also areas that have a lot of plant life. However, if we also look at the factor of temperature, we see that very cold regions have much lower carrying capacities because the cold temperatures inhibit plant growth.
We can also look at phenomena such as predator-prey dynamics, which shows how populations of predators and prey oscillate in similar patterns as each population can limit the other. Depending on what population of animals is in focus, there are many biological interactions that directly affect carrying capacity including competition, predation, parasitism, mutualism, and other population-level relationships between organisms.
There are a wide variety of ways to visualize these relationships mathematically. You can graph different populations to estimate carrying capacity. You can create histograms of different traits of a population to show how populations are adapting to carrying capacity limits. You can either create this data to show the theory, or you can have students work on historical datasets that show a real situation.
What to Avoid
This NGSS standard also contains the following Assessment Boundary:
Assessment does not include deriving mathematical equations to make comparisons.
Here’s a little more specificity on what that means:
Deriving Mathematical Equations:
When focusing on graphical representations of carrying capacity, important mathematical concepts include slope, identifying units, and estimating carrying capacity based on population dynamics. However, this does not mean that students need to create or express complex mathematics. At this point, visual estimation and arguments based on the change of slope should be sufficient.
Advanced students can be introduced to calculus and trigonometry theories that can shed even more light on estimating carrying capacity. This material should not be assessed for students to demonstrate an understanding of this standard.