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 ( $Δ t = 1$ ) 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 m².

A scuba diver observes a lionfish on a coral reef.

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.

\frac{Δ N}{Δ t} = r_{max,d} N_{t} (\frac{K - N_{t}}{K})

carrying capacity ( $K$ ) = 500
population size at the start of 2004 ( $N_{2004}$ ) = 20
maximum per capita growth rate ( $r_{max,d}$ ) = 1.15

$t$	$N_{t}$	$r_{max,d} N_{t}$	$\frac{K - N_{t}}{K}$	$\frac{Δ N}{Δ t}$	$N_{t + 1}$
2004
2005	42.08	2005 Equation r max,d Nt Value, cell 1 out of 12 to complete	2005 Equation ((K − Nt) ÷ K) Value, cell 2 out of 12 to complete	2005 Equation Δ N÷Δ t Value, cell 3 out of 12 to complete	2005 Equation N t+1 Value, cell 4 out of 12 to complete
2006		2006 Equation r max,d Nt Value, cell 5 out of 12 to complete	2006 Equation ((K − Nt) ÷ K) Value, cell 6 out of 12 to complete	2006 Equation Δ N÷Δ t Value, cell 7 out of 12 to complete	2006 Equation N t+1 Value, cell 8 out of 12 to complete
2007		2007 Equation r max,d Nt Value, cell 9 out of 12 to complete	2007 Equation ((K − Nt) ÷ K) Value, cell 10 out of 12 to complete	2007 Equation Δ N÷Δ t Value, cell 11 out of 12 to complete	2007 Equation N t+1 Value, cell 12 out of 12 to complete
2008
2009
2010
2011
2012
2013

Lionfish Invasion: Density-Dependent Population Dynamics

Calculating lionfish population growth each year

Lionfish per 10,000 m²

Logistic Growth Equation

t

N_t

r_max,dN_t

K - N_t / K

ΔN/Δt

N_t+1

$N_{t}$

$r_{max,d}$

$\frac{K - N_{t}}{K}$

$\frac{Δ N}{Δ t}$

$N_{t + 1}$

Yearly lionfish population estimates using a discrete-time logistic model

Population Size ( $N_{t}$ )

Year ( $t$ )

Calculating lionfish population growth each year

Logistic Growth Equation

Yearly lionfish population estimates using a discrete-time logistic model

Population Size ( N t )

Year ( t )

Population Size ( $N_{t}$ )

Year ( $t$ )