# 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 ($dN$) over an instantaneous time interval ($dt$), which is written as . In a continuous-time logistic model:

## $\frac{dN}{dt}={r}_{\mathrm{m}\mathrm{a}\mathrm{x}}N\left(\frac{K-N}{K}\right)$

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.

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.