MAT3150 Differential Equations and Numerical Methods
A Biologist introduced a new fish species to a lake. They stocked the lake with an initial population of 500 fish and estimated that the carrying capacity of the lake is 15,000 fish. Over the first year the number of fish approximately tripled. The biologist used the logistic equation to model the number of fish in the lake t years after the inroduction of the initial population as
dP = 1.2P 1 − P , P (0) = 500
dt 15000
where P denotes the number of fish in the lake.
Solve the initial value problem to determine an explicit expression for the displacement of the population P (t). You must explain and demonstrate the working required to solve the IVP.
Use Matlab to plot the population P over the time interval 0 ≤ t ≤??. Be sure to label your axes and provide a figure caption. Comment on what happens to the rate of population growth over the time interval plotted.
In the graph of population growth over the 10-year period, we see that the initial period of rapid population increase. This rapid expansion because of the abundance of resources, beginnings with a quite small population size, and the prevalence of the reproductive survival. With the passing of time, population growth rate gradually declines. The populations growth level slows down as it gets near the environment's carrying capacity, which the habitat cannot carry its inhabitants sustainably.
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Figure 1 Population Growth Over Time
With the resources for MBA assignment expert starting to get scarce, competition for the remaining resources becomes fiercer, environments shift and stress conditions such as climate change kick in. These factors all which collectively makes a negative impact on the rate of population growth. In the end, after the population stabilizes and asymptotically approaches the carrying capacity, the growth rate slows to almost zero because the population starts to be in balance with the environment.
Strang, G., & Herman, E. (2016, July 11). 8.4: The Logistic Equation. Mathematics LibreTexts. https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/08%3A_Introduction_to_Differential_
Equations/8.04%3A_The_Logistic_Equation
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