Logistic Growth

Applied Population Dynamics
WILD 5700/7700

Today in APD

The equation for logistic growth in discrete time.

The definition of density-dependent growth.

Basic properties of the model.

Strange behavior of the (discrete time) model, such as damped oscillations and chaos.

From Geometric to Logistic Growth

Geometric growth

\[N_{t+1} = N_t + N_tr\]

Logistic growth

\[N_{t+1} = N_t + N_t r_{max}\left(1 - \frac{N_t}{K}\right)\]

where

• \(r_{max}\) is the growth rate when \(N_t\) is close to 0

• \(K\) is the carrying capacity

Density-dependent growth

Logistic growth is an example of density-dependent growth.

Definition: Population growth rate is affected by population size (\(N\)).

Implications: Resources are limited and there is a carrying capacity.

Graphical depiction

Growth rate

\[\lambda_t = N_{t+1}/N_t\]

Growth

\[\Delta_t=N_{t+1}-N_t\]

Growth as a function of \(N\)

What happens when we change \(r_{max}\)?

Definitions

Overcompensation

Density-dependent response in which populations over- or under-shoot carrying capacity rather than approach it gradually.

Chaos

Highly variable deterministic dynamics that are extremely sensitive to small changes in parameters.

Assumptions of basic model

• \(K\) and \(r_{max}\) are constant

• No sex or age effects or other sources of individual heterogeneity

• No time lags

• No stochasticity

Does logistic growth occur in nature?

Fruit fly (Drosophila melanogaster) in the lab¹

1. Data from Pearl (1927). Figure from Krebs (2014)

Does logistic growth occur in nature?

Ibex (Capra ibex) in Switzerland¹

1. Data from Sæther et al. (2002). Figure from Krebs (2014)

Summary

• Logistic growth is a form of density-dependent growth

• Growth rate (\(\lambda_t=N_{t+1}/N_t\)) declines as \(N\) approaches \(K\)

• Growth (\(\Delta_t=N_{t+1}-N_t\)) peaks at \(K/2\) (the inflection point)

• The model doesn’t include birth, mortality, and movement processes, but it can be derived from density-dependent birth and death rates.

• But it does allow for complex dynamics that resemble patterns seen in nature.

Assignment

Read pages 32–36 in Conroy and Carroll.