# Lionfish Invasion: Density-Dependent Population Dynamics

## Visualizing a population growth model

Let’s take a closer look at the continuous-time logistic model. We’ll now explore a new variable, ${r}_{\text{realized}}$, which represents the realized per capita growth rate: the actual growth rate of the population per individual. Unlike the maximum per capita growth rate ${r}_{\text{max}}$, which is a constant, ${r}_{\text{realized}}$ will change as the population grows.

Click on the three points highlighted on the lionfish population growth curve in the graph below. For each point you click, you will get more information about the population growth rates (both the overall growth rate $\frac{dN}{dt}$ and the per capita growth rate ${r}_{\text{realized}}$) at that time. The estimated numbers of lionfish at that time will also appear, separated into two groups: adults and juveniles. The adults, shown in white, are fish that were present at the start of that year. The juveniles, shown in blue, are new fish that were added to the population during that year.

### Lionfish population estimates using a continuous-time logistic model The x axis of the graph is labeled Year using notation lowercase t and shows the years 2004 to 2014 in increments of 2 years. On the left hand side of the graph, the y axis is labeled Population Size using notation uppercase N and ranges from 0 to 50 in increments of 10. On the right hand side of the graph, the y axis is labeled lowercase r realized and ranges from 0.0 to 1.2 in increments of 0.1. A legend below the graph indicates that a larger fish with vertical white stripes represents adults and a smaller blue fish represents juveniles. A solid green line indicates Population Size and a dashed orange line indicates r realized. The solid green line rises from 2 in 2004 to 4 in 2005, 24 in 2008, 40 in 2010, and then levels off, ending at just under 50 in 2014. Points are plotted for the years 2005, 2008, and 2010. Clicking on one of the points reveals the number of adults and juveniles for that year. The dashed orange line starts at about 1.1 in 2004, decreases at a fairly steady rate to 0.1 in 2010, and then gradually levels off to 0.0 in 2014. Legend Adult Juvenile solid green line Population Size ( N ) dotted orange line r realized The population size is measured as the number of lionfish per 1,000 m2, which is about the size of two basketball courts, instead of per 10,000 m2.