Calculating lionfish population growth each year
Using what is known about lionfish in the Bahamas, let’s model their population growth each year from 2004 to 2014. To do this, we’ll use a discrete-time logistic model to project changes in the population over time periods of one year ($\mathrm{\Delta}t=1$) each. We can write the population size at the start of a year as ${N}_{t}$. The population size at the end of that year, or start of the next year, will be ${N}_{t+1}$.
Over each year, our model will estimate the change in population size, in terms of the number of lionfish per 10,000 m^{2}.
The table below lists the values calculated from the model for each year. The first row, for the year 2004, is completed for you. Click on any cell in the first row with data to see how the values were calculated.
Fill in the next three rows of this table to determine the population size (${N}_{t}$) after each year. Enter your calculated values in order from left to right starting with the first empty cell in the row for 2005. Round your calculations to 2 decimal places and hit enter to move to the next cell. If the calculation is incorrect, you will receive an alert and you can modify your answer. If the answer is correct, you will not be able to change your response. As you complete each row, the population sizes you calculated will appear on the graph below.
Use the following values for the lionfish population to complete your calculations.
- carrying capacity ($K$) = 500
- population size at the start of 2004 (${N}_{\mathrm{2004}}$) = 20
- maximum per capita growth rate (${r}_{\text{max,d}}$) = 1.15
The x axis of the graph is labeled Year using notation lowercase t shows the years 2004 to 2012 in increments of 2. The y axis is labeled Population Size using the notation uppercase N subscript t and ranges from 0 to 500 in increments of 100. The first segment of the curve is drawn between two points: the y value of 20 in 2004 increasing to the y value of 42.08 in 2005. The remaining segments are added to the graph as users complete the calculations in the table.
The next line segment is drawn between the next two points showing that from 2005 to 2006, the population size increased from 42.08 to 86.60.
The next line segment is drawn between the next two points showing that from 2006 to 2007, the population size increased from 86.60 to 169.26.
With the completion of the third row of calculations in the table, the remaining rows are filled in automatically and the remaining points of the graph are placed and the rest of the line segments are drawn to show the curve. The population continued to increase in 2008 to 297.73, and in 2009 to 434.69. The population then remained at 499.68 for the years 2010 through 2013.