Lionfish Invasion: Density-Dependent Population Dynamics

Revising the Model

All models are based on assumptions. Our model of lionfish population growth so far assumes that any major changes in population size, such as reproduction and death, occur in separate yearly steps. In reality, lionfish reproduce all throughout the year — every four days for some fish! To include these more frequent changes in our model, we can use time periods shorter than one year. If the time periods are so short that changes happen instantly, our discrete-time model turns into a continuous-time model.

Continuous-time models assume that changes happen instantly and describe how a population is changing at any time. They are used to model populations of organisms that reproduce year-round, like lionfish do. In these models, the population growth rate is the change in population size ( d N ) over an instantaneous time interval ( d t ), which is written as d N / d t . In a continuous-time logistic model:

d N d t = r m a x N K - N K

This equation is similar to the one that we used for the discrete-time logistic model, and the symbols have similar meanings. To learn more about the continuous-time logistic model, visit the “Logistic” section of the Population Dynamics Click & Learn.

The graph below shows estimates of the lionfish population size over time, using the three different approaches: estimates based on realistic data collected by divers, the discrete-time logistic model, and the continuous-time logistic model.

Discrete vs. Continuous vs. Data

Population Size


The x axis of this graph is labeled Year and shows the years from 2004 to 2014 in increments of 2 years. The y axis is labeled Population Size and is marked from 0 to 800 in increments of 100. The graph automatically populates three lines which are labeled in a legend below the graph: the white dotted line represents the continuous model, the dashed green line represents the discrete model, and the solid orange line represents the actual data. The line for the continuous model is fairly smooth, showing a gradual increase between 2004 and 2006, then a more rapid increase until 2010, at which point it levels off. The line for the the discrete model shows the same gradual increase from 2004 to 2006 as in the continuous model and also levels off after 2010, but it shows higher numbers between 2006 and 2014. The line for the actual data shows a similar gradual increase from 2004 to 2006, then rises steeply to a maximum of nearly 800 by 2008, then drops to under 400 in 2010, then rises again to nearly 600 in 2011, drops slightly to around 550 in 2012, drops more sharply to just above 400 in 2013, then rises again to around 650 in 2014.


  • dotted white line Continuous
  • dashed green line Discrete
  • solid orange line Actual

Lionfish population estimates based on diver data (orange line, solid), the discrete-time logistic model (green line, dashed), and the continuous-time logistic model (white line, dotted). Population size is given as the number of lionfish per 10,000 m2.

The graphs for the discrete-time and continuous-time models look slightly different but have the same overall shape. In both cases, the population size goes to, and stays stable at, the same carrying capacity.