MATH: Population Ecology

A population is a group of organisms of the same species that live together in a particular area. Population size is dependent on four things – births, deaths, immigration into the population, and emigration out of the population.

Factors that Influence Population Size

Why do populations of some species fluctuate greatly while others do not?

There are three fundamental characteristics of individuals in any population:

Density: The number of individuals per unit area. It is not a static quantity and often fluctuates as individual organisms move into and out of an area.
Dispersion: The pattern of spacing that occurs among individuals within a given area. Dispersion patterns can include clumped, uniform, and random. Factors that can influence density and dispersion patterns include ecological needs, the structure of the environment, and interactions among the individuals in a population.
Demographics: Shows how the populations change over time due to birth and death rates.

The image shows three patterns of dispersion in a population: uniform, random, and clumped

The data in a life table are used to prepare survivorship curves. Some typical survivorship curves are shown in the graph (notice the logarithmic scale on the y-axis). Different groups of species show distinctly different trends in survivorship. Ecologists classify these survivorship trends as Type I, Type II, and Type III. Remember that many organisms do not fit just one pattern of survivorship and that these curves are generalizations.

Click the types to reveal more information in the Survivorship learning object.

Population Growth

Exponential growth describes a hypothetical model for population growth in which space and resources are available in unlimited supply. As a result, the population growth rate increases with each new generation. Under this model, a population rapidly grows quite large and continues to grow indefinitely.

The image shows an exponential curve plotted on a set of axes. It is in the shape of a J.

In reality, there is simply not enough space or resources for natural populations to continue to grow unchecked. Limiting factors within every ecosystem, such as the availability of food or the effects of predation and disease, prevent a population from becoming too large. These limiting factors determine an ecosystem’s carrying capacity, or maximum population size the environment can support given all available resources.

Logistic growth describes a model for population growth that takes into account carrying capacity and is therefore a more realistic model for population growth. According to the logistic growth model, a population first grows exponentially because there are few individuals and plentiful resources. As the population gets larger and approaches the environment’s carrying capacity, resources become more scarce and the growth rate slows. This leads to the logistic growth model’s characteristic S-shaped curve.

The image shows a logistic curve plotted on a set of axes. It is in the shape of an S.

Can carrying capacities change?

Carrying capacities can change. An ecosystem’s carrying capacity may fluctuate based on seasonal changes, or it may change as a result of human activity or a natural disaster.

Limiting factors are environmental factors that can limit population size and lead to the population hovering around carrying capacity. There are generally two categories.

Density-dependent limiting factors are limiting factors that depend on population size. These factors come into play when a population is large, crowded, and dense. Density-dependent limiting factors include competition, predation, parasitism, and disease. These factors have the most effect when a population has reached or exceeded its carrying capacity. Parasites harm or kill their host and produce a similar effect to that of predators.

Density-independent limiting factors are limiting factors that do not depend on population size. These factors come into play and limit populations regardless of whether the population size is large or small. These factors include unusual weather, natural disasters, seasonal cycles, and certain human activities such as damming rivers or clear-cutting forests.

Sort the following examples into either density-dependent or density-independent limiting factors in the Density Sorting learning activity below.

Population Growth Equations

Population ecologists use a variety of mathematical methods to model population dynamics (how populations change in size and composition over time). Some of these models represent growth without environmental constraints, while others include "ceilings" determined by limited resources.

Check out the image below of the AP Biology equation sheet. Notice that these are written as differential equations. Please don’t worry if you haven’t taken calculus – you’ll still be able to do the math for these problems after working through this section. Note that the first equation, Rate = dY/dt is just there to remind you that any differential equation is modeling a rate of change. We do not need this equation for our purposes in the population growth lesson.

The image has the differential exponential growth equations from the AP biology equation sheet.

Watch the Pop Video 1 below to learn more about how we can model the population growth rate (dN/dt) using births and deaths.

How do we model the exponential growth of a population? As we mentioned briefly above, we get exponential growth when r_max for our population is positive and constant. While any positive r_max can lead to exponential growth, you will often see exponential growth represented by r_max as the maximum per capita rate of increase for a particular species under ideal conditions, and it varies from species to species. For instance, bacteria can reproduce much faster than humans and would have a higher maximum per capita rate of increase.

Watch the Pop Video 2 below to learn more about modeling exponential growth.

We can mathematically model logistic growth by modifying our equation for exponential growth, using an r_max (per capita growth rate) that depends on population size (N) and how close it is to carrying capacity (K). Note that the term (K-N)/K is added to the logistic growth equation. This term goes to 0 as N approaches K, which makes sense because the slope is 0 at carrying capacity.

Watch the Pop Video 3 below to learn more about modeling logistic growth.

Now, practice using the growth equations to calculate the answers below. Remember that if you know the carrying capacity OR the question states logistic growth, you will use the logistic growth equation. If not, use the exponential growth equation. Make sure you have your formula sheet and calculator handy.

Try the Growth Equations below to check your knowledge.

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