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By Teppi
As humans, we constantly encounter examples of growth in our everyday life. Whether that may be the cost of electrical fees, the natural phenomenon of fire, or the plants in your very home, we are constantly being exposed to growth. Some types of growth can have positive effects, such as the development of the economy, while some, like the exponential growth of the COVID-19 pandemic, can leave a lasting impact on history (StudiousGuy).
An Introduction to Exponential Growth:
Exponential growth is a specific form of growth that demonstrates greater increase as time progresses, resulting in growth that escalates over time. Many aspects of our lives revolve around exponential growth, such as the popularity of media on the internet, or more recently (and unfortunately) the spread of Covid-19. In exponential growth, the value of y scales upward as time increases, while linear growth forms a straight line, as the increase between values remains constant across all points in the graph.
Exponential Growth in population size of invasive species:
One application of exponential growth is through the population of an invasive species, especially species without a consistent predator. In invasive species, as more members of the species reproduce, the children of that set will continue to reproduce at an expansive rate (“Invasive Species”). We can think of these sets as our x-axis, where more reproductive cycles will result in more growth. Although exponential growth does occur naturally in native species, it becomes more apparent when looking at an invasive species due to the lack of natural predators or adaptations made to counter them, meaning that these species have a higher chance of reproduction.
In the equation, r is the rate of offspring per female, while a is the initial population. The value of a can be thought of as how a larger starting population would result in more instances of reproduction, therefore creating more offspring and leading to increased population growth.
In standard exponential growth, the growth rate r is not halved under the assumption that all members of the starting population contribute to the exponential growth, such as how all money in a bank account would contribute to interest, or how all citizens infected with COVID have the ability to spread it. When measuring population growth, we have to take into account that only the females of our population will be able to reproduce, meaning the growth rate of the population must be halved to eliminate the inclusion of males, who do not contribute to growth.
Figure 1: How a difference in the starting population impacts the curve of an equation
Figure 2: Map of Burmese Python presence in Florida
(“Map of Burmese Python Presences”)
The exponential growth of Burmese python in Florida:
To understand this phenomenon, we can examine real-life examples of invasive species and identify the exponential growth of their populations. One example of an invasive species is the Burmese python, which saw large population growth in Florida during the 2000s, endangering many of the native wildlife species due to the lack of a natural predator (Figure 2). Not only did the growth of the Burmese python impact the habitats of native wildlife, but saw an even larger development in protected regions such as the Everglades national park, where hunting was prohibited (“Burmese Pythons”).
In an invasive environment, female Burmese pythons typically lay clutches every two years of up to 36 eggs, although most lay approximately 12 eggs. Although adult Burmese pythons have very few natural predators in the Everglades, newborn Burmese pythons, which are only 20 inches in length and an inch in diameter, have a relatively low rate of surviving to adulthood due to their frail nature. Through analysing data from recorded python sightings, a clutch of Burmese python hatchlings which are laid every 2 years will produce an average of 3 adult Burmese pythons, or 1.5 hatchlings per year (Riverview Park and Zoo). Using this information, we can substitute the reproductive rate into our base equation in order to measure the exponential growth in populations of Burmese python:
In order to verify that our equation is an accurate representation of Burmese python population growth in the wild, we can compare our results to real-world data in isolated environments, such as the reported sightings of python in the Everglades from 2000 to 2010. In the year 2000, a population of 4 Burmese pythons was introduced into the Everglades national park through unknown means, and over a span of 10 years, were able to develop into a stable population of 350. When analysing the sightings of Burmese python in the Everglades, we can see a trendline resembling exponential growth, as seen in Figure 3. It is worth noting that in the graph shown below, the slight decrease in population in 2010 was likely caused by a severe freeze in South Florida (“January 2010 Cold Spell”).
Figure 3: Sightings of Burmese python in Everglades national (Dormas. E)
Comparing equation to real-world data:
In order to compare our population growth formula to the set of real-world data, we must substitute our unknown variables to the data gathered about the population in the Everglades:
Using the equation f(P) = 4(1.75)x, we can overlay the path formed by the function over the population data to gauge the effectiveness of the equation in representing the population growth of Burmese pythons in the Everglades, shown in Figure 3.
Figure 4: Comparison of population estimate and real world data
Through this comparison, we can see a clear correlation between the two models, although the equation does begin to fail in certain areas of the data. In the diagram, there are key points in the model where the equation deviates from the real life data, such as the timeframe between 2004 to 2007, and how the equation begins to diverge towards infinity while the data remains finite. These inconsistencies can be attributed to two main factors, the invalidity of the data and a concept known as habitat capacity.
Flaws of the equation in a practical setting:
In the comparison, data based on the sightings of Burmese pythons in the park, which serves as a decent approximation or representation of the population, is not perfect at describing the size of a population. Due to the nature of exponential growth, a small difference in the starting population of a species will have a much larger impact on the population after long periods of time. Since our data merely accounts for the sightings of python, which is likely to be less than the exact value, the lower estimates from the year 2004 to 2007 are likely caused by inaccuracies in the initial population.
The second limitation of our model, known as habitat capacity, occurs when the population of a species becomes so large that the habitat is incapable of supporting it’s development. In our specific model, as the number of Burmese pythons reached the upper threshold of the Everglades capacity, the population was clamped below a certain amount, as more reproduction would lead to overuse of resources, which would decrease the population to a stable level. Because our equation assumes the snakes have the resources to continually reproduce, while the Everglades forces the population below a certain level, the equation was much more accurate during the beginning of the population growth when the population was still far from the habitat capacity.
Figure 4: Curve of a logistic model equation (“9: Population Growth and Regulation”)
Although the exponential equation does succeed at modeling population growth to some extent, a more effective model would be a logistics model. A logistic model or equation is a function used in statistics and probability, where the likelihood of an event will increase or decrease as time progresses. From this description, it is clear how the logistics model shares similarities with an exponential equation, as the probability and difference in value both increase or decrease in a similar fashion, seen in Figure 4. When looking at a logistic function, the curve of the equation exhibits the same curve as the population data in our set, since as time progresses, the “probability” of reproduction approaches 0, effectively clamping the population at a certain value.
Parting Note:
By understanding how the population growth of invasive species follows the curve of exponential growth, we can understand the impact of these species on the habitats and native wildlife. In knowing the threat that invasive species pose to different ecosystems, we can more effectively predict and mitigate their impact. Overall, understanding how exponential equations have links to not only population growth but other aspects of our daily lives can give us a better understanding of the systems and workings of different global issues, helping us more effectively plan for the future.
Works Cited:
“9: Population Growth and Regulation.” 9 | Population Growth and Regulation, http://www.zo.utexas.edu/courses/bio373/chapters/Chapter9/Chapter9.html.
“Burmese Python.” Burmese Python | National Invasive Species Information Center, https://www.invasivespeciesinfo.gov/terrestrial/vertebrates/burmese-python.
Dorcas, Michael E., et al. “Severe Mammal Declines Coincide with Proliferation of Invasive Burmese Pythons in Everglades National Park.” PNAS, National Academy of Sciences, 14 Feb. 2012, https://www.pnas.org/content/109/7/2418.
“Invasive Species.” National Wildlife Federation, https://www.nwf.org/Educational-Resources/Wildlife-Guide/Threats-to-Wildlife/Invasive-Species.
Riverview Park and Zoo, “Burmese Python.” Riverview Park and Zoo, https://www.riverviewparkandzoo.ca/en/zoo/burmese-python.aspx.
January 2010 Cold Spell - National Weather Service. https://www.weather.gov/media/tbw/2010/ColdJanuary2010.pdf.
Map of Burmese Python Presences Used in Maxent Modeling ... https://researchgate.net/figure/Map-of-Burmese-python-presences-used-in-MaxEnt-modeling-scenarios-in-relation-to-roads_fig1_321675812.
StudiousGuy. “10 Real Life Examples of Exponential Growth.” StudiousGuy, StudiousGuy, 8 July 2019, https://studiousguy.com/real-life-examples-exponential-growth/.
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