Lionfish Invasion: Density-Dependent Population Dynamics

Modeling yearly changes in a lionfish population

We can use mathematical models to study how populations of lionfish, and other species, change over time. You can learn more about these types of models in BioInteractive’s Population Dynamics Click & Learn.

One common population model is the logistic model, which describes how a population changes when its growth rate depends on the population’s current density. This type of growth is called density-dependent.

A discrete-time logistic model describes how a population with density-dependent growth changes over specific time periods. It is often used to model populations of organisms that reproduce at specific times — for example, once per year or season. An equation for this model is:

Δ N Δ t = r max,d N t ( K - N t K )

This equation can also be written as:

Δ N Δ t = r max,d N t ( 1 - N t K )

The table below describes the biological meaning of each symbol in the equations above.

Symbol Meaning
t A specific point in time — in this case, the start of the time period over which we are measuring the population.
N t The size or density of the population at time t .
Δ t The symbol Δ (delta) means “change in,” so Δ t means the change in time for the population — in other words, the length of the time period we are using.
Δ N The change in the population’s size over the time period we are using.
r max,d The maximum per capita growth rate, which is the growth rate of the population per individual, in the discrete-time model. This rate is a constant, and it represents how fast a population can grow when it is not limited by density-dependent factors. Its value depends on the species’ physiology and life history features (rate of development, lifespan, number of offspring, frequency of reproduction, etc.).
K The carrying capacity of the population. K is a constant, and it is determined by factors like the availability of resources in the environment.

The graph below shows how a population would grow according to the logistic model. As the population size ( N t ) changes, the population growth rate ( Δ N Δ t ) also changes. This makes the shape of the growth curve change over time.

Logistic Growth

Population Size ( N t )

Time ( t )

An animated graph with the x axis labeled Time with notation lower case t and the y axis labeled Population Size with the notation capital N subscript t. A dotted horizontal line near the top of the graph is labeled Carrying Capacity. The animation traces a curve starting at the lower left of the graph which rises slowly at first, then rises more rapidly until it approaches the carrying capacity, at which point it levels off. A point just above the first horizontal line on the graph is labeled N subscript t is far from K. A second point near the carrying capacity is labeled N subscript t is close to K.

If population size ( N t ) is far below the carrying capacity ( K ), the ( K - N t ) / K term will be close to 1, meaning it will have only a small impact on population growth. In that case, the population can grow faster and faster. If N t is close to K , the ( K - N t ) / K term will be close to 0, drastically slowing population growth.