What is the exponential growth model and how does it work?

The exponential growth model simulates population growth where each individual produces new offspring at a constant reproductive rate, R, each generation. The population size increases dramatically over time following the equation NT+1 = NT + (1 + R) × NT. This model demonstrates unrestricted growth that cannot be sustained indefinitely in nature due to resource limitations.

How does the logistic growth model differ from exponential growth?

The logistic growth model incorporates carrying capacity, K, which limits population growth as resources become scarce. Populations grow initially but slow as they approach carrying capacity, creating an S-shaped curve. Unlike exponential growth, logistic growth reflects realistic population dynamics where growth rate decreases as population size increases relative to available resources.

What happens to a population when it exceeds carrying capacity in the logistic model?

When a population exceeds carrying capacity, the population size decreases in the subsequent generation. The logistic equation accounts for this by reducing growth rate as population approaches or surpasses carrying capacity. This negative feedback mechanism prevents unlimited growth and stabilizes the population near the carrying capacity limit.

How do predator and prey populations interact in the predator-prey model?

In the predator-prey model, prey populations increase when predators are scarce, but as predators consume more prey, predator populations grow. Increased predators then reduce prey populations, causing predators to subsequently decline from starvation. This creates cyclical population fluctuations where each species' growth depends on the other's population size, demonstrating interdependent population dynamics.

What role does the growth rate R play in exponential population growth?

The growth rate R represents the proportion of new offspring produced by each individual per generation. Higher R values produce larger population sizes after a fixed number of generations. In exponential growth simulations, increasing R dramatically accelerates population expansion, demonstrating how reproductive rate directly determines population trajectory independent of carrying capacity constraints.

How does initial population size affect long-term population growth?

Initial population size influences the absolute population numbers achieved after a set number of generations in exponential growth models. Larger starting populations result in proportionally larger final populations when growth rate remains constant. However, the growth pattern shape remains similar regardless of initial size, as the exponential growth rate determines the trajectory.

What parameters control how quickly a population reaches carrying capacity?

The maximum growth rate, R max, determines how quickly populations approach carrying capacity in logistic growth models. Higher R max values cause populations to reach carrying capacity faster, while lower values result in slower approach. Carrying capacity itself sets the population limit, but R max controls the speed at which populations converge to that limit.

Population Growth

Concepts Instructor Prep Student Protocol

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Have you ever considered the environmental effects of farm-raised, or wild-caught fish? We can study the ecological impacts of these practices, by looking at population growth. A group of individuals of a single species, living in an area at the same time, is called a population.

In nature, fish populations grow at a high rate, when resources are plentiful. However, population growth slows down when its size nears the carrying capacity, k, which is the maximum population size allowed by the resources in a habitat. This type of population growth is known as logistic growth. If we know the population size of the current generation, or Nt, the maximum growth rate, r-max, which is the reproductive rate of one individual when no competition exists, and the carrying capacity, we can use the logistic growth formula to estimate the population size of the next generation.

At fish farms, resources like extra food and oxygen are artificially added to a relatively small living area, allowing population densities above natural carrying capacities. However, overcrowding releases excess biomaterials, and this can lead to algal blooms. If these resources remain high, cyanobacteria, or blue-green algae can continue to divide at the same growth rate. And this growth is not limited to a carrying capacity. Due to the continuous increase of resources. Therefore, it increases exponentially, and the size of the next generation in exponential growth can be calculated using this formula. Algal blooms like this deplete dissolved oxygen in the water and are detrimental to farmed fish and the ecosystem around the facility, necessitating cautious farming practices.

Fishing wild populations may impact ecosystems as well. If fishermen remove individuals faster than the population can replenish them with new fish, then the populations become smaller, and may struggle to rebound if predation pressure continues at the same rate. This may, in turn, disrupt the balance between predator and prey species. Scientists can estimate the sizes of future prey and predator populations, or Vt +1, and Ct +1, respectively, as long as they know current sizes of both populations, Vt and Ct, predator and prey growth rates, f, and r, and the predators' attack and starvation rates, a, and q. Typically, a large predator population reduces the prey population size. But then, the smaller available pool of prey will in turn reduce the predator population. Having less predators allows prey to thrive again, and the two populations will naturally oscillate back and forth, over time. The oscillation rule is a good way to visualize many natural predator-prey interactions, but, occasionally, the rule does not apply.

In invasive species, like the lionfish in the Atlantic and the Caribbean, far from their native waters of the Indo-Pacific oceans, they have no natural predators, and can reproduce all year long. But as a generalist predator, the consume numerous native species at unsustainable rates, permanently altering the Atlantic and Caribbean ecosystems. Applying the exponential growth model, that we saw in the algae example, scientists determined that approximately one quarter of the entire lionfish population would have to be removed every month, to keep their growth sustainable. Therefore, eating wild-caught Atlantic lionfish, and other edible invasive species, actually supports the health of the ecosystems.

Print Procedure Steps

Exponential Population Growth Model
- NOTE: In these experiments you will use computer software to simulate different types of growth models. HYPOTHESES: The experimental hypothesis might be that if we increase either R, the reproductive rate or the initial population size, NT, that this will increase the population size after 10 generations in an exponential growth model compared to unmanipulated control populations. The null hypothesis may be that populations of different R and initial population size will have the same population size as control groups after 10 generations.
- To build an exponential population growth model, first open a new workbook in Excel.
- Save the file to a secure location and name it “exponential growth”.
- Label cell A1 “generation”, and cell B1 “population size”.
- Type the number zero into cell A2. This row will represent the initial generation.
- In cell A3 type this formula: = 1 + A2
- Now hit enter.
- Click on cell A3 and drag the small box in the bottom right corner to cell A12 for 10 generations.
- Type the number two in cell B2. This will represent the initial population size in the model.
- Label cell E1 “growth rate”, and in cell E2 type the number 0.05. This value represents the proportion of new offspring produced by each individual for each generation, or R in the exponential growth model.
- NOTE: In the classic exponential growth equation, T is the generation, and NT is the number of organisms in the current generation. NT+1 will be is the number of individuals in the next generation.
- Enter the exponential growth equation in cell B3 as the formula shown here: = B2 + (1 + E2) * B2
- Now, hit enter.
- Click cell B3 and drag the small square in the bottom right corner down to cell B12 to simulate 10 generations of exponential growth.
- To observe the results, select all data, along with the headers, in columns A and B.
- Click “insert” and select “scatter plot with smooth lines and markers” from the toolbar. A scatter plot should appear in the sheet. TIP: If it does not, check to see if a new sheet was created in the workbook.
- Label the Y axis “population size” and the X axis “generation”. Title the chart “exponential growth of a population”.
- Observe how the population increases in a classic exponential growth pattern.
- Next, create Table 1 labeling the “growth rate R” as the header of column A, and “population size after 10 generations” as the second column.
- Record the population size at generation 10 for the growth rate of 0.05, the rate we just simulated.
- Now, set the growth rate in cell E2 to 0.1, and repeat the steps to generate a new scatter plot.
- Record the population size at generation 10.
- Continue to alter the growth rate using the values in table one, noticing how the Y axis changes to accommodate the increasing values.
- After completing Table 1, set the growth rate back to 0.05 and change the initial population size in cell B2 to 100 individuals.
- Record the population size at generation 10 in a new table, Table 2.
- Proceed to change the initial population size using the values in Table 2. Once the table is complete, save the Excel sheet and close it.
Logistic Population Growth Model
- HYPOTHESES: In a logistic growth model, the experimental hypothesis could be that populations will approach and meet the carrying capacity, and any populations with sizes greater than the carrying capacity will decrease in the subsequent generation. The null hypothesis might be that population growth will not follow a logistic S-shaped curve in the logistic models, and that generations with population sizes greater than the carrying capacity will continue to increase or stay the same in the next generation.
- To build a logistic population growth model, open a new spreadsheet in Excel.
- Save the file as “logistic growth” in the same folder as before.
- Then label cell A1 “generation” and B1 “population size”.
- Type the number zero in cell A2. This row will represent the initial generation.
- In cell A3 type the formula: = 1 + A2
- Now, hit enter.
- Now, click on cell A3 and drag the small box in the bottom right corner to cell A12 to number up to the 10th generation.
- Type the number two in cell B2 to represent the initial population size.
- Label cell E1 as “max growth rate” and cell F1 as “carrying capacity”.
- Finally, set the max growth rate in cell E2 to 0.05. This value will represent the maximum growth rate the population may achieve – “R max” in the discrete logistic equation. NOTE: In the classic logistic growth equation the term K represents carrying capacity. As the population size of the current generation or NT, approaches the carrying capacity, the growth of the population begins to slow.
- Now set the carrying capacity in cell F2 to 50.
- Enter the logistic growth equation in cell B3 as the formula below: = B2 + ((1 + E$2) * B2) * ((F$2 - B2) /F$2)
- Now hit enter.
- Drag the formula down to cell B12 to add it to all 10 generations.
- To observe the results of this exercise, highlight all data and headers in columns A and B, then click “insert” and select “scatter plot with smooth lines and markers”.
- Label the Y axis population size and the X axis generation.
- Title the chart as logistic growth of a population.
- Observe the growth of the population as it nears carrying capacity and record the generation that the population meets or exceeds carrying capacity in a new table.
- Now set the max growth rate in cell E2 to 1.
- Notice how the population responds when it rises over carrying capacity.
- Continue to set the max growth rate to the values in Table 3 and record the generation at which the population rises two or above carrying capacity.
- Now set the max growth rate back to 0.05 and set the carrying capacity to 1,000.
- Notice how the graph changes as the growth approaches the carrying capacity.
- Drag the formulas in A12 and B12 down to A14 and B14.
- Adjust the plot to include these new data points to show when the population reaches carrying capacity.
- Save the Excel sheet and close it.
Predator-Prey Model
- Hypotheses: In our third simulation of predator-prey logistics, the experimental hypothesis could be that predator populations will increase as prey populations increase and that the two populations will experience growth and decline dependent on the population of the other species. The null hypothesis may be that the two populations will show patterns of growth unrelated to the other species' population size.
- To build a predator-prey model, open a new spreadsheet in Excel.
- Save the file in the same folder as the other two models and name it “predator-prey model”.
- Label cell A1 “time”, cell B1 “prey” and cell C1 “predator”.
- In cell A2 type zero and hit enter.
- In cell A3 type the formula below: = 1 + A2
- Now hit enter.
- Drag the formula down to cell A802 as the model will require many time points to visualize changes of both populations.
- Now type 100 into cell B2 to represent the initial prey population and type 25 into cell C2 as the initial predator population.
- Label cell E1 “prey growth rate”, cell F1 “consumption of prey by predator”, cell G1 “conversion efficiency” and cell H1 “starvation of predator”.
- Type the number 0.02 in cell E2. This value represents the growth rate of the prey species or R in the prey population model. NOTE: Here the prey population is represented as VT and the predator population as CT. The value A is the per capita attack rate of the predator species on the prey. In the predator population model, variables are introduced for predators produced per prey consumed, F or conversion efficiency as well as for the starvation rate of the predators noted as Q.
- Type 0.0005 in cell F2 for the attack rate in both the prey population and the predator population model.
- Then type 0.8 in cell G2 to represent the conversion efficiency, and 0.05 in cell H2 for the starvation rate in the predator population equation.
- Now type the prey growth equation in cell B3 as the formula below: =B2+(E$2*B2)-(F$2*C2*B2)
- Hit enter.
- Then type the prey growth equation in cell C3 as the displayed formula: =C2 + (F$2*G$2*B2*C2)-(H$2*C2)
- Hit enter.
- Hold down the shift keys and click to select both cells B3 and C3.
- Click the small box at the bottom right corner of cell B3 and drag the two formulas down to row 802 to simulate the predator and prey populations over 800 generations.
- To visualize the data, select all of the data and headers in columns A, B and C.
- Click “insert” and select “scatter plot with smoothed lines”.
- A chart showing the population change of the predator and the prey populations should appear.
- Label the Y axis of the plot population size and the X axis time.
- Title the plot “predator-prey interactions”.
- Observe the interactions between the predator and prey populations.
- Set the initial predator population to zero and observe what happens in the plot.
- Now set the initial predator population to 120 and observe what happens to the predator population when there are more predators than prey.
- Finally set the initial prey population to zero and observe what happens to the population of the predators.
- Experiment with other values to see what other interactions can be created.
- When finished, save the workbook and close it.
Results
- In the exponential growth model built in the first activity, observe the shape of the resulting population growth curve, and consider whether you think this type of growth could be sustained in nature.
- Note how the growth rate R affected the population size after 10 generations in the exponential growth model.
- Also consider how the initial population size affected the population size after 10 generations in the exponential growth model.
- Now, consider the logistic growth model, and observe the shape of this graph, noting how it differs from the exponential growth model.
- Then, note how the populations behave as they approach the carrying capacity in your logistic growth model.
- Decide if you think that the maximum per capita growth rate, R max, influenced how quickly populations reached carrying capacity in the logistic growth model.
- Also, consider what happens to population size when populations are over carrying capacity in the logistic growth model, or what might happen if the carrying capacity is lowered.
- In the predator-prey model, note whether the prey population responded to increasing predator population. Conversely, consider how the predators then respond to decreasing prey populations.

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