Modeling Population Growth in Excel

The Malthus and Condorcet Equations, simple formulas that model relatively complex trends in population growth, are now accessible with an Excel calculator that allows the user full control over every component of the equations. Students can use the Excel file to model human population growth under the assumption that a human carrying capacity exists.

The Malthus Equation expresses the growth rate of a population as a function of the current population size and current carrying capacity. Specifically, the growth rate of a population is equal to a Malthusian parameter multiplied by the current population size multiplied by the difference between the current carrying capacity and the current population size. This relationship creates a high growth rate once a population is large enough to reproduce at its full potential, but remains a low growth rate when the population is very small or when a population is nearing its carrying capacity and feeling the effect of constrained resources. The Malthusian parameter is almost invariably between zero and one because a negative Malthusian parameter would lead to a population’s gradual extinction while a Malthusian parameter greater than one would lead to explosive population growth that would greatly exceed the carrying capacity. In the latter situation, unrealistically rapid and extreme periods of growth and contraction would ensue.

The Condorcet Equation expresses the growth rate of the carrying capacity of a population as equal to the growth rate of the population multiplied by a constant termed the Condorcet parameter. The logic behind this mathematical relationship is that the carrying capacity of a population increases or decreases proportionally with the growth rate of a population because an additional person in a population can have a positive or negative effect on the carrying capacity. This implies that a Condorcet parameter greater than one results from a society where an additional individual somehow increases the number of people that can be supported even when taking into account the resources that additional individual consumes; this could result from a situation where there are increasing returns to labor. If doctors cure diseases better when more of them work together, this is reflected by a Condorcet parameter greater than one. A Condorcet parameter between zero and one is most realistic for human populations because the contribution of another person will probably grow the carrying capacity but not by more than one. A negative parameter implies that an additional person would actually lower the carrying capacity; perhaps every additional person would consume natural resources at a rate greater than the previous individual’s rate.

As Cohen (1995 Science 269: 341-346) points out, the equations are not necessarily realistic models of human population growth. There is no consensus about whether or not a human carrying capacity exists. In theory, we as a species might be able to continually develop technology at such a rate that we are unable to approach a carrying capacity. A slowdown in overall human population growth is more likely due to a global increase in income per capita that leads to altered reproductive strategies.

With r=0.1 and c=0.1 as parameters, the population experiences a positive but steadily decreasing growth rate because the carrying capacity increases at 1/10th the rate of population growth, and since population growth slows as the population size approaches the carrying capacity, we observe almost asymptotic behavior. This is a realistic pattern for human population growth if a carrying capacity exists.

Figure 1: with r=0.1 and c=0.1 as parameters, the population experiences a positive but steadily decreasing growth rate because the carrying capacity increases at 1/10th the rate of population growth, and since population growth slows as the population size approaches the carrying capacity, we observe almost asymptotic behavior. This is a realistic pattern for human population growth if a carrying capacity exists.

The calculator defines the Malthus Equation as dP(t)/dt=rP(t)[K(t)-P(t)] and the Condorcet Equation as dK(t)/dt=c dP(t)/dt (See Cohen 1995: 343). The user may enter values for the initial states of r (the “Malthusian parameter”), P(t), (population size), K(t) (carrying capacity), c (“Condorcet parameter”), t_0 (the starting time for the model) and dt (the length of one interval in time) that determine all of the future changes in population size. The rates of change of population and carrying capacity at time t, dP(t)/dt and dK(t)/dt respectively, are determined by the equations. The Malthusian and Condorcet parameters are constant in a growth model provided that there are no exogenous shocks that affect the nature of population or carrying capacity growth. Because of this, they do not vary as a function of t.

To explore the Malthus-Condorcet calculator, please follow this link to an automatic download of the Excel spreadsheet containing the calculator.

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Visualizing Mass Communications and State Institutions in Wartime China (1937-45)

In China, the study of history has always gone hand-in-hand with the study of geography. When studying China’s modern history, however, focus has shifted toward large-scale processes, such as revolution, and large-scale sociological transformations, such as changing class relations. More recently, however, some historians are starting to bring geography back in. Pathbreaking endeavors such as the China Historical GIS project and Harvard University WorldMap platform-based ChinaMap allow researchers to visualize the transformation of China across space and time. The result has been a new understanding of China and Chinese history highlighting the spatial distribution of ethnic and linguistic diversity, economic development, elite networks, and state institutions. One exciting result of this new understanding is that it allows students and researchers alike to visualize large-scale processes across time periods, which can in turn lead to new questions about how different places might have experienced the same era or event. Through the use of spatial approaches, we are challenged to rethink the applicability of national historical narratives to local human landscapes.

As a teacher and researcher of East Asian history, much of what I do focuses on how media, institutions, and person-to-person networks have connected the modern Chinese state to populations both inside and outside of China. Working in tandem with DASIL, I have begun to build and visualize datasets which describe how the “connective tissue” of state-building looked during the period of China’s War of Resistance to Japan (1937-1945)—a period of intense destruction and dislocation which some historians have also described as key period of modernization. This data is drawn from two editions of The China Handbook: a publication of the Chinese Ministry of Information released in 1943 and again in 1946. I discovered this publication quite by happenstance while searching the Grinnell College Library collections for local gazetteer data related to the period of China’s Republican Era (1911-1949). The value of The China Handbook is that it provides comprehensive provincial and urban data for a number of indicators of state development; here we (myself and DASIL’s outstanding post-bac fellow, Bonnie Brooks ’15) have focused on data concerning communications, education, and health care. To be fair, and as admitted by The China Handbook’s original editor, Hollington K. Tong, this data is not exhaustive, nor is it necessarily reliable given the rapidity of changes brought about by war and resulting partition of China into competing political zones. It does, however, represent at least a starting point for visualizing what China’s wartime states looked like “on the ground,” viewed through the lens of communications and other institutional infrastructure.

Below the level of national boundaries, modern China is divided into numerous separate administrative units known as provinces. However, the number of provinces has changed with time and successive governments, which poses a challenge for those seeking to visualize data at the province level for eras during which the number of these units was larger than it is today—as was the case during the latter half of the Republican Era, which witnessed a proliferation of efforts to tame China’s restive and geopolitically fragile borders through the process of province-building. A key part of Bonnie’s contribution, then—the results of which will hopefully be used and refined by other researchers working at the intersection of geographic information systems (GIS) and modern Chinese history—was the creation of new shapefiles corresponding to each province that existed during the 1937-1945 period. The resulting maps are thus entirely new creations, and will hopefully serve to help bridge the current gap which lies between geospatial research on imperial China and research on contemporary China after Mao.  The shapefiles are available for download in DASIL’s Downloadable Data section.

For the map:

    • The Contents button(contentsbutton) will display all layers. Unclick the checkbox next to the layer name to hide the layer. To view the legend, click on the “Show Legend” icon (contentsbutton) below the layer name.
    • To examine other variables, find the “Change Style” button (contentsbutton) below the layer name you wish to view, then select the desired variable from the “Choose an attribute to show” drop-down menu.  You may alter the map with colors, symbols or size. You may also alter variables (e.g. normalize variables by population).
    • Click on an individual Chinese province to see available data.
    • The shapefiles featured in the map are available for download on the DASIL website. Click here for the download.

 

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New Data Visualizations this Summer

We are pleased to introduce two new data visualizations.  Both are map visualizations.

First, there is the map of Boko Haram Conflicts in Nigeria.  It is a kernel density map showing the concentration of events.

Kernel density map of Boko Haram conflicts in Nigeria 2012

Second, we have a Cartogram of U.S. Population by State & Year.  This map distorts the size and shape of each state to reflect its population relative to the other states.

Cartogram of US Population by State and Year for 2010

Click on either of the maps to go to the visualizations to explore!

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Kick Off Summer Vacation With Tourism Data!

Commencement is over at Grinnell College, and here at DASIL we’re settling into our summer routine. What better way to start the summer than with a look at tourism data?

* Interested in knowing how many people visit U.S. national parks annually? The National Parks Service has 111 years of visitor data for national parks.

Total Visitors to Yellowstone Nat'l Park from 1903 to 2014

* Can you guess which country sends the most visitors to the United States annually? When you have your guess, visit the site of the Office of Travel and Tourism Industries of the U.S. International Trade Administration to find out the answer!

* Thinking about vacationing somewhere chilly this year? The International Association of Antarctica Tour Operators has fifteen years of historical data about visitors to Antarctica.


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Gender Inequality Visualizations

In honor of International Women’s Day on March 8, we at DASIL have found some great visualizations on the web that speak to gender inequality. At DASIL, we do host a few visualizations that speak to economic inequality in the US, but we’d like to highlight some other areas of inequality here.

The New York Times found that more men named John are C.E.O.s than all female C.E.O.s combined. They also explore the breakdown of gender in Congress.

UN’s Global Pulse has a great map showing the number of tweets about various topics—including gender inequality, education, and discrimination. It’s a great way to look at global opinion on many issues.

The Food and Agriculture Organization of the United Nations put together an interactive map of various statistics from the Gender Landrights Database. The link here will show you the percent of female agricultural holders in various countries.

Finally, at DASIL we have several visualizations that point to other factors of gender inequality. Two striking ones are Mean Income by Age, Race, and Gender, and Hourly Wages by Education and Gender. Each of these interactive graphs will let you select which combinations of variables you’d like to compare.

Wages by Education and Gender comparing females with a high school education and females with no high school degree

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