Name: Population Growth: Exponential and Logistic Growth and Predator-Prey Model - Concept
Uploaded: 2019-01-29T11:04:46.000Z
Duration: 4 min 45 s
Channel: JoVE

K12 Standards

Louisiana Student Standards for Science

High School

Environmental Science

Resources And Resource Management

HS-EVS1-2

Population Growth

lab manual sub articles

Lab Manual Sub Articles

<p>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. </p>
<p>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. </p>
<p>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. </p>
<p>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&#39; 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. </p>
<p>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. </p>

population growth

<h4>Population Models</h4>
<p>An ecological population is a group of individuals of a single species living in an area at the same time. To persist, a population must either grow or maintain its size. Population ecology is the study of how population size and age distribution change over time through interactions with other species and the environment as well as with individuals of their own species. A particularly important feature that is studied by ecologists is the rate at which a population grows or declines. A population grows when its birth rate is greater than its death rate. That is, more offspring are being produced than individuals are dying. If the opposite is true and the death rate is greater than the birth rate, the population is in decline and if it continues to decrease it will eventually go extinct.</p>
<p>Population ecologists build models to predict how the size of populations will change over time. Two simple models of population growth are exponential (geometric) growth models and logistic growth models. Exponential population models are density-independent, meaning that per capita birth rates and death rates will remain constant regardless of how many individuals are in the population, therefore these populations increase to infinity. However, logistic growth models are density-dependent, therefore the growth rate of the population is dependent on how many individuals are currently in the population. Geometric or exponential growth can only occur when resources are readily available and in unlimited supply. For example, under optimal growing conditions, bacteria follow an exponential growth pattern, with each organism producing two additional reproducing organisms in the next generation<sup>1</sup>.</p>
<h4>Exponential Growth</h4>
<p>For a population that follows an exponential growth pattern, population ecologists can estimate the number of individuals in the next generation based on the size of the initial population and the growth rate, which represents the proportion of new offspring produced per individual for each generation. With a growth rate of 0.05, 5% of individuals produce one offspring in the next generation. At this rate it would only take 23 generations to produce over a million offspring from 2 initial offspring under exponential growth. Some bacteria can grow and divide very rapidly, and thus would be able to reach large population numbers relatively quickly. For instance, <em>Escherichia coli</em> can double its population in approximately 20 minutes, whereas the rate for <em>Syntrophobacter fumaroxidans</em> is 140 hours<sup>2</sup>.</p>
<p>Geometric or exponential growth cannot go on forever, as eventually resources like food or space will be exhausted, and organisms begin to die more quickly or slow reproduction. The maximum number of individuals in a population that can be supported based on the resources in the environment is known as the carrying capacity (<em>K</em>). When more individuals exist in an area than the area can support, the carrying capacity is said to be exceeded. Natural populations may exceed carrying capacity only briefly, before the lack of resources leads to increased death rates and/or decreased reproduction.</p>
<h4>Logistic Growth</h4>
<p>Populations of many species follow a logistic growth pattern, where the growth rate of the population is dependent on its current size. Therefore, the growth rate of the logistic growth is variable as opposed to the constant growth rate of the exponential model. As the population nears its carrying capacity, its growth begins to decrease, until the number of individuals reaches or briefly exceeds carrying capacity and subsequently maintains a size near carrying capacity. By incorporating a carrying capacity that represents the limiting resources of a population, logistic models consider a more realistic relationship between an organism and its environment.</p>
<h4>Population Dynamics</h4>
<p>Sophisticated models of population dynamics include variables representing real-world interactions between distinct populations. One such example which represents two interacting populations is the predator-prey model that was independently developed by mathematicians Alfred J. Lotka and Vito Volterra in the 1920s. In fact, predator-prey models have been adapted for a variety of species interactions and continue to be used in adapted forms almost a century after they were developed. One population is a prey species, like hares, and the other is a predator, like wolves, that feed on the prey. The species are closely interacting, because the wolf requires the hare as a food source, thus their birth rate is dependent on the amount of prey they consume and how effectively they convert this food into producing new offspring. The prey populations are dependent on the predator population, because their death rate is dependent on the number of prey that are eaten by predators in a generation. The close interaction of the species can result in oscillating patterns shown between their populations.</p>
<p>Models of population dynamics have a variety of applications in conservation biology. Altering models to predict the extinction risk of species is particularly important for helping to protect endangered species, since the loss of one species can result in losses to other species in the food web. The extinction of predator species worldwide is of great concern and can result in instability and extinctions of prey species. Sophisticated modeling and conservation planning will help to protect species in a changing world.</p>
<p>The predator-prey models have also been adapted to study competitive interactions, in which multiple species are competing with each other for resources. These models can be used to predict which species will &#8220;win&#8221; competition interactions and have been applied to the spread of non-native species to predict if a species will be able to establish in a new area, and whether they are likely to become invasive. In most natural applications, models must consider many resources that are available to species, and a complex web of interactions between organisms. For example, lionfish which are native to the Indo-Pacific are invasive in the Atlantic and the Caribbean. Lionfish can reproduce all year long, have no natural predators, and can consume numerous native fish species at unsustainable rates which permanently alters the Atlantic and Caribbean ecosystems. Using population growth models has enabled scientists to determine that approximately a quarter of the entire lionfish population has to be removed from the ecosystem every month just to stunt population growth<sup>3</sup>.</p>
<p>Population growth models have been adapted to fit human populations using demography, which is the study of birth rates, death rates, and statistics related to population sizes based on age groupings. The growth rate of countries and other political districts can be forecast using demographic data, and this is important for planning infrastructure in localities. Thus, application of population growth models is important to planning for the growth of the human population. Further, Lotka-Volterra model has also been applied to study and predict the competition of businesses, further demonstrating the versatility of the population growth models to fields outside of biology<sup>4</sup>.</p>
<h4>References</h4>
<ol class="references">
	<li><strong>Ghislain Y. Gangwe Nana, Camille Ripoll, Armelle Cabin-Flaman, David Gibouin, Anthony Delaune, Laurent Janniere, Gerard Grancher, Gaelle Chagny, Corinne Loutelier-Bourhis, Esther Lentzen, Patrick Grysan, Jean-Nicolas Audinot, Vic Norris.</strong> Division-Based, Growth Rate Diversity in Bacteria. <em>Front Microbiol. </em>2018, Vol. 9, 849 (doi: 10.3389/fmicb.2018.00849).</li>
	<li><strong>Beth Gibson, Daniel J. Wilson, Edward Feil, Adam Eyre-Walker.</strong> The distribution of bacterial doubling times in the wild. <em>Proc Biol Sci. . </em>2018, Vol. 285, 1880 (20180789).</li>
	<li><strong>Rice, James A. MorrisJr.Email authorKyle W. ShertzerJames A.</strong> A stage-based matrix population model of invasive lionfish with implications for control. <em>Biological Invasions. </em>2011, Vol. 13, 1 (7-12).</li>
	<li><strong>Ofer Malcai, Ofer Biham, Peter Richmond, and Sorin Solomon.</strong> Theoretical analysis and simulations of the generalized Lotka-Volterra model. <em>Phys. Rev. E. </em>2002, Vol. 66, 3.</li>
</ol>

Population Growth: Exponential and Logistic Growth and Predator-Prey Model - Concept

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<ul><li><a href="#" data-time="0.0">Exponential Population Growth Model
</a></li><li><a href="#" data-time="267">Logistic Population Growth Model
</a></li><li><a href="#" data-time="519">Predator-Prey Model
</a></li><li><a href="#" data-time="800">Results
</a></li></ul>

<ol class="note-items"><li data-note-item="1" data-parent-collapse="">Exponential Population Growth Model
<button class="expand">Expand</button><ul class="lab-items"><li data-sub-note-item="1-1"><div class="lab-step">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.
</div></li><li data-sub-note-item="1-2"><div class="lab-step">To build an exponential population growth model, first open a new workbook in Excel.
</div></li><li data-sub-note-item="1-3"><div class="lab-step">Save the file to a secure location and name it &#8220;exponential growth&#8221;.
</div></li><li data-sub-note-item="1-4"><div class="lab-step">Label cell A1 &#8220;generation&#8221;, and cell B1 &#8220;population size&#8221;.
</div></li><li data-sub-note-item="1-5"><div class="lab-step">Type the number zero into cell A2. This row will represent the initial generation.
</div></li><li data-sub-note-item="1-6"><div class="lab-step">In cell A3 type this formula: = 1 + A2
</div></li><li data-sub-note-item="1-7"><div class="lab-step">Now hit enter.
</div></li><li data-sub-note-item="1-8"><div class="lab-step">Click on cell A3 and drag the small box in the bottom right corner to cell A12 for 10 generations.
</div></li><li data-sub-note-item="1-9"><div class="lab-step">Type the number two in cell B2. This will represent the initial population size in the model.
</div></li><li data-sub-note-item="1-10"><div class="lab-step">Label cell E1 &#8220;growth rate&#8221;, 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.
</div></li><li data-sub-note-item="1-11"><div class="lab-step">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.
</div></li><li data-sub-note-item="1-12"><div class="lab-step">Enter the exponential growth equation in cell B3 as the formula shown here: = B2 + (1 + E2) * B2
</div></li><li data-sub-note-item="1-13"><div class="lab-step">Now, hit enter.
</div></li><li data-sub-note-item="1-14"><div class="lab-step">Click cell B3 and drag the small square in the bottom right corner down to cell B12 to simulate 10 generations of exponential growth.
</div></li><li data-sub-note-item="1-15"><div class="lab-step">To observe the results, select all data, along with the headers, in columns A and B.
</div></li><li data-sub-note-item="1-16"><div class="lab-step">Click &#8220;insert&#8221; and select &#8220;scatter plot with smooth lines and markers&#8221; 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.
</div></li><li data-sub-note-item="1-17"><div class="lab-step">Label the Y axis &#8220;population size&#8221; and the X axis &#8220;generation&#8221;. Title the chart &#8220;exponential growth of a population&#8221;.
</div></li><li data-sub-note-item="1-18"><div class="lab-step">Observe how the population increases in a classic exponential growth pattern.
</div></li><li data-sub-note-item="1-19"><div class="lab-step">Next, create Table 1 labeling the &#8220;growth rate R&#8221; as the header of column A, and &#8220;population size after 10 generations&#8221; as the second column.
</div></li><li data-sub-note-item="1-20"><div class="lab-step">Record the population size at generation 10 for the growth rate of 0.05, the rate we just simulated.
</div></li><li data-sub-note-item="1-21"><div class="lab-step">Now, set the growth rate in cell E2 to 0.1, and repeat the steps to generate a new scatter plot.
</div></li><li data-sub-note-item="1-22"><div class="lab-step">Record the population size at generation 10.
</div></li><li data-sub-note-item="1-23"><div class="lab-step">Continue to alter the growth rate using the values in table one, noticing how the Y axis changes to accommodate the increasing values.
</div></li><li data-sub-note-item="1-24"><div class="lab-step">After completing Table 1, set the growth rate back to 0.05 and change the initial population size in cell B2 to 100 individuals.
</div></li><li data-sub-note-item="1-25"><div class="lab-step">Record the population size at generation 10 in a new table, Table 2.
</div></li><li data-sub-note-item="1-26"><div class="lab-step">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.
</div></li></ul></li><li data-note-item="2" data-parent-collapse="">Logistic Population Growth Model
<button class="expand">Expand</button><ul class="lab-items"><li data-sub-note-item="2-1"><div class="lab-step">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.
</div></li><li data-sub-note-item="2-2"><div class="lab-step">To build a logistic population growth model, open a new spreadsheet in Excel.
</div></li><li data-sub-note-item="2-3"><div class="lab-step">Save the file as &#8220;logistic growth&#8221; in the same folder as before.
</div></li><li data-sub-note-item="2-4"><div class="lab-step">Then label cell A1 &#8220;generation&#8221; and B1 &#8220;population size&#8221;.
</div></li><li data-sub-note-item="2-5"><div class="lab-step">Type the number zero in cell A2. This row will represent the initial generation.
</div></li><li data-sub-note-item="2-6"><div class="lab-step">In cell A3 type the formula: = 1 + A2
</div></li><li data-sub-note-item="2-7"><div class="lab-step">Now, hit enter.
</div></li><li data-sub-note-item="2-8"><div class="lab-step">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.
</div></li><li data-sub-note-item="2-9"><div class="lab-step">Type the number two in cell B2 to represent the initial population size.
</div></li><li data-sub-note-item="2-10"><div class="lab-step">Label cell E1 as &#8220;max growth rate&#8221; and cell F1 as &#8220;carrying capacity&#8221;.
</div></li><li data-sub-note-item="2-11"><div class="lab-step">Finally, set the max growth rate in cell E2 to 0.05. This value will represent the maximum growth rate the population may achieve &#8211; &#8220;R max&#8221; 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.
</div></li><li data-sub-note-item="2-12"><div class="lab-step">Now set the carrying capacity in cell F2 to 50.
</div></li><li data-sub-note-item="2-13"><div class="lab-step">Enter the logistic growth equation in cell B3 as the formula below: = B2 + ((1 + E$2) * B2) * ((F$2 - B2) /F$2)
</div></li><li data-sub-note-item="2-14"><div class="lab-step">Now hit enter.
</div></li><li data-sub-note-item="2-15"><div class="lab-step">Drag the formula down to cell B12 to add it to all 10 generations.
</div></li><li data-sub-note-item="2-16"><div class="lab-step">To observe the results of this exercise, highlight all data and headers in columns A and B, then click &#8220;insert&#8221; and select &#8220;scatter plot with smooth lines and markers&#8221;.
</div></li><li data-sub-note-item="2-17"><div class="lab-step">Label the Y axis population size and the X axis generation.
</div></li><li data-sub-note-item="2-18"><div class="lab-step">Title the chart as logistic growth of a population.
</div></li><li data-sub-note-item="2-19"><div class="lab-step">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.
</div></li><li data-sub-note-item="2-20"><div class="lab-step">Now set the max growth rate in cell E2 to 1.
</div></li><li data-sub-note-item="2-21"><div class="lab-step">Notice how the population responds when it rises over carrying capacity.
</div></li><li data-sub-note-item="2-22"><div class="lab-step">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.
</div></li><li data-sub-note-item="2-23"><div class="lab-step">Now set the max growth rate back to 0.05 and set the carrying capacity to 1,000.
</div></li><li data-sub-note-item="2-24"><div class="lab-step">Notice how the graph changes as the growth approaches the carrying capacity.
</div></li><li data-sub-note-item="2-25"><div class="lab-step">Drag the formulas in A12 and B12 down to A14 and B14.
</div></li><li data-sub-note-item="2-26"><div class="lab-step">Adjust the plot to include these new data points to show when the population reaches carrying capacity.
</div></li><li data-sub-note-item="2-27"><div class="lab-step">Save the Excel sheet and close it.
</div></li></ul></li><li data-note-item="3" data-parent-collapse="">Predator-Prey Model
<button class="expand">Expand</button><ul class="lab-items"><li data-sub-note-item="3-1"><div class="lab-step">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&#39; population size.
</div></li><li data-sub-note-item="3-2"><div class="lab-step">To build a predator-prey model, open a new spreadsheet in Excel.
</div></li><li data-sub-note-item="3-3"><div class="lab-step">Save the file in the same folder as the other two models and name it &#8220;predator-prey model&#8221;.
</div></li><li data-sub-note-item="3-4"><div class="lab-step">Label cell A1 &#8220;time&#8221;, cell B1 &#8220;prey&#8221; and cell C1 &#8220;predator&#8221;.
</div></li><li data-sub-note-item="3-5"><div class="lab-step">In cell A2 type zero and hit enter.
</div></li><li data-sub-note-item="3-6"><div class="lab-step">In cell A3 type the formula below: = 1 + A2
</div></li><li data-sub-note-item="3-7"><div class="lab-step">Now hit enter.
</div></li><li data-sub-note-item="3-8"><div class="lab-step">Drag the formula down to cell A802 as the model will require many time points to visualize changes of both populations.
</div></li><li data-sub-note-item="3-9"><div class="lab-step">Now type 100 into cell B2 to represent the initial prey population and type 25 into cell C2 as the initial predator population.
</div></li><li data-sub-note-item="3-10"><div class="lab-step">Label cell E1 &#8220;prey growth rate&#8221;, cell F1 &#8220;consumption of prey by predator&#8221;, cell G1 &#8220;conversion efficiency&#8221; and cell H1 &#8220;starvation of predator&#8221;.
</div></li><li data-sub-note-item="3-11"><div class="lab-step">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.
</div></li><li data-sub-note-item="3-12"><div class="lab-step">Type 0.0005 in cell F2 for the attack rate in both the prey population and the predator population model.
</div></li><li data-sub-note-item="3-13"><div class="lab-step">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.
</div></li><li data-sub-note-item="3-14"><div class="lab-step">Now type the prey growth equation in cell B3 as the formula below: =B2+(E$2*B2)-(F$2*C2*B2)
</div></li><li data-sub-note-item="3-15"><div class="lab-step">Hit enter.
</div></li><li data-sub-note-item="3-16"><div class="lab-step">Then type the prey growth equation in cell C3 as the displayed formula: =C2 + (F$2*G$2*B2*C2)-(H$2*C2)
</div></li><li data-sub-note-item="3-17"><div class="lab-step">Hit enter.
</div></li><li data-sub-note-item="3-18"><div class="lab-step">Hold down the shift keys and click to select both cells B3 and C3.
</div></li><li data-sub-note-item="3-19"><div class="lab-step">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.
</div></li><li data-sub-note-item="3-20"><div class="lab-step">To visualize the data, select all of the data and headers in columns A, B and C.
</div></li><li data-sub-note-item="3-21"><div class="lab-step">Click &#8220;insert&#8221; and select &#8220;scatter plot with smoothed lines&#8221;.
</div></li><li data-sub-note-item="3-22"><div class="lab-step">A chart showing the population change of the predator and the prey populations should appear.
</div></li><li data-sub-note-item="3-23"><div class="lab-step">Label the Y axis of the plot population size and the X axis time.
</div></li><li data-sub-note-item="3-24"><div class="lab-step">Title the plot &#8220;predator-prey interactions&#8221;.
</div></li><li data-sub-note-item="3-25"><div class="lab-step">Observe the interactions between the predator and prey populations.
</div></li><li data-sub-note-item="3-26"><div class="lab-step">Set the initial predator population to zero and observe what happens in the plot.
</div></li><li data-sub-note-item="3-27"><div class="lab-step">Now set the initial predator population to 120 and observe what happens to the predator population when there are more predators than prey.
</div></li><li data-sub-note-item="3-28"><div class="lab-step">Finally set the initial prey population to zero and observe what happens to the population of the predators.
</div></li><li data-sub-note-item="3-29"><div class="lab-step">Experiment with other values to see what other interactions can be created.
</div></li><li data-sub-note-item="3-30"><div class="lab-step">When finished, save the workbook and close it.
</div></li></ul></li><li data-note-item="4" data-parent-collapse="">Results
<button class="expand">Expand</button><ul class="lab-items"><li data-sub-note-item="4-1"><div class="lab-step">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.
</div></li><li data-sub-note-item="4-2"><div class="lab-step">Note how the growth rate R affected the population size after 10 generations in the exponential growth model.
</div></li><li data-sub-note-item="4-3"><div class="lab-step">Also consider how the initial population size affected the population size after 10 generations in the exponential growth model.
</div></li><li data-sub-note-item="4-4"><div class="lab-step">Now, consider the logistic growth model, and observe the shape of this graph, noting how it differs from the exponential growth model.
</div></li><li data-sub-note-item="4-5"><div class="lab-step">Then, note how the populations behave as they approach the carrying capacity in your logistic growth model.
</div></li><li data-sub-note-item="4-6"><div class="lab-step">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.
</div></li><li data-sub-note-item="4-7"><div class="lab-step">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.
</div></li><li data-sub-note-item="4-8"><div class="lab-step">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.
</div></li></ul></li></ol>

Population Growth: Exponential and Logistic Growth and Predator-Prey Model - Procedure

Exponential Population Growth Model
ExpandNOTE: In these experiments you will use computer software to simulate different types of growth models. ...

<p>In these experiments you will use computer software to simulate different types of growth models. First you will examine logistic growth. 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.</p><p>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.</p><p>Label cell A1 generation, and cell B1 population size. Type zero into cell A2.This row will represent the initial generation. In cell A3 type the formula equals one plus A2 and hit enter.</p><p>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.</p><p>This value represents the proportion of new offspring produced by each individual for each generation, or R in the exponential growth model. Here T is the generation. So NT is the number of organisms in the current generation and NT plus one is the number of individuals in the next generation.</p><p>Enter the exponential growth equation in cell B3 as the formula shown here and 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 with headers in columns A and B.Click insert and select scatter plot with smooth lines and markers from the toolbar.</p><p>A scatter plot should appear in the sheet. 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.</p><p>Title the chart exponential growth of a population. Observe how the population increases in a classic exponential growth pattern. Next create table one labeling the growth rate R as the header of column A, and population size after 10 generations as the second column.</p><p>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.</p><p>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 one, 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 two as shown.</p><p>Proceed to change the initial population size using the values in table two. Once table two is complete, save the Excel sheet and close it. 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.</p><p>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 spread sheet in Excel. Save the file as logistic growth in the same folder as before.</p><p>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 equals one plus A2 and hit enter.</p><p>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.</p><p>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. In this equation the term K is carrying capacity.</p><p>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 displayed formula and hit enter.</p><p>Then 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.</p><p>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 as shown. In this example, the population size exceeds its carrying capacity at the seventh generation at the maximum growth rate of 0.05.</p><p>Now set the max growth rate in cell E2 to one. Notice how the population responds when it rises over carrying capacity. Continue to set the max growth rate to the values in table three and record the generation at which the population rises two or above carrying capacity.</p><p>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.</p><p>Adjust the plot to includes these new data points to show when the population reaches carrying capacity. Finally save the Excel sheet and close it. In our third simulation of predatory-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.</p><p>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.</p><p>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 equals one plus A2 and hit enter.</p><p>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.</p><p>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. 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.</p><p>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 displayed formula and hit enter. Then type the prey growth equation in cell C3 as the displayed formula, again hitting enter.</p><p>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.</p><p>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.</p><p>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.</p><p>Experiment with other values to see what other interactions can be created. When finished save the workbook and close it. In the exponential growth model built in the first activity, what is the shape of the resulting population growth curve?</p><p>Do you think this type of growth could be sustained in nature? How did the growth rate R affect the population size after 10 generations in the exponential growth model? How did the initial population size affect the population size after 10 generations in the exponential growth model?</p><p>Now consider the logistic growth model. What is the shape of this curve? How did populations behave as they approach the carrying capacity in your logistic growth model?</p><p>How did the maximum per capita growth rate, R max, influence 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 predatory-prey model, how do prey populations respond to increasing predator populations?</p><p>How do predators respond to decreasing prey populations?</p>

<ul><li><a href="#" data-time="0.0">Preparation for the Population Growth Computer Simulation
</a></li></ul>

<ol class="note-items"><li data-note-item="1" data-parent-collapse="">Preparation for the Population Growth Computer Simulation
<button class="expand">Expand</button><ul class="lab-items"><li data-sub-note-item="1-1"><div class="lab-step">For this exercise, ensure that spreadsheet software is installed on student computers before starting the lab. NOTE: The instructions for this lab are written for Microsoft Excel, but Google Sheets is a free online alternative that can be used with minor adaptations to the instructions described here. 
</div></li><li data-sub-note-item="1-2"><div class="lab-step">Throughout this procedure, remind the students to frequently save their spreadsheet files to avoid loss of data.
</div></li></ul></li></ol>

<div class="section-question"><h4 class="question-sect">What is a population?</h4><p>A population is a group of individuals of a single species living in an area at the same time.
</p></div><div class="section-question"><h4 class="question-sect">What is population ecology?</h4><p>Population ecology is the study of how population size and age distribution change over time through interactions with the environment and individuals from their own species as well as others.
</p></div><div class="section-question"><h4 class="question-sect">What is a carrying capacity?</h4><p>The carrying capacity is the maximum population size allowed by the resources in a habitat.
</p></div><div class="section-question"><h4 class="question-sect">What are exponential and logistic growth models?</h4><p>Exponential growth models are independent of population density and follow a J-shaped curve. Logistic growth is limited by carrying capacity therefore depends on the population density. Since population growth slows down near carrying capacity, logistic growth curves are S-shaped.
</p></div><div class="section-question"><h4 class="question-sect">What is the importance of growth models?</h4><p>Mathematical models of population growth allow estimation of population sizes in various situations. For instance, we can calculate changes in predator and prey populations. Moreover, they are useful to estimate how populations will fare when outside influences (habitat loss, climate change, etc.) affect variables such as growth rate. In addition, scientists can calculate growth of invasive species as well as how native species will be affected.
</p></div>

Population Growth: Exponential and Logistic Growth and Predator-Prey Model - Prep

Preparation for the Population Growth Computer Simulation
ExpandFor this exercise, ensure that spreadsheet software is installed on student computers ...

<p>For this exercise, ensure that spreadsheet software is installed on student computers before starting the lab. The instructions for this lab are written for Microsoft Excel, but Google Sheets is a free online alternative that can be used with minor adaptations to the instructions described here. Throughout this procedure, remind the students to frequently save their spreadsheet files to avoid loss of data.</p><p>Please take time to review the materials list and the student protocol video to be fully prepped for the lab.</p>

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Population Growth

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Population Models

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