Logistic map
Definition
The logistic map is a simple yet rich mathematical function studied extensively in the field of dynamical systems and chaos theory. It was popularized by biologist Robert McCredie May in the 1970s as a model to describe nonlinear population growth with a limited carrying capacity.
Its general form is:
$$ P_{n + 1} = \left( r P_{n} \left( 1 - P_{n} \right)\right) $$
Where:
- \(P_{n} \in [0,1]\): represents the proportion of the population at generation \(n\) (e.g., as a fraction of the environment's maximum capacity)
- \(r \in [0,4]\) is a parameter representing the growth rate.
The equation shows how a population grows rapidly at first when it is small, but its growth slows as it gets bigger.
Behavior depending on the value of \(r\)
- For \(0 < r < 1\): The population decreases to zero over time.
- For \(1 < r < 3\): The population converges to a stable fixed point.
- For \(3 < r < 3.449 \ldots\): Bifurcations appear, and the system oscillates between two values (period-2 cycle), then 4, 8, etc. (period doubling).
- For \( r > 3.57 \ldots \): The system enters chaos, showing long-term behavior that is extremely sensitive to initial conditions.
- For certain values within the chaotic range: There are windows of periodicity, where the system temporarily returns to periodic cycles.
Interactive chart
Move black slider \(n\) to add or remove points, the violet slider \(P_{1}\) to change the initial value, the green slider \(\mu\) for the relationship between reproduction and starvation, and the orange slider \( b \) to change the exponent value. Press the checkbox to remove the connecting lines.
Key properties:
- Nonlinearity: Arises from the term \(P_{n} \left( 1 - P_{n} \right)\), which models resource competition.
- Bifurcations: Qualitative changes in system behavior when \(r\) is varied.
- Deterministic chaos: For some values of \(r\), the system behaves unpredictably over time, yet is fully determined by the equation.
Importance
- It is a simple model that exhibits complex behavior, useful for illustrating concepts such as stability, bifurcation, and chaos.
- It serves as a canonical example in applied mathematics, physics, biology, and chaos theory courses.
- It has been studied in relation to fractals, entropy, and bifurcation diagrams.
Representation in the Bifurcation Diagram
The Bifurcation Diagram is a graphical representation of the values taken by \(P_n\) for different values of \(r\), allowing the visualization of bifurcations and transitions to chaos. This map is useful for analyzing dynamic systems and complex processes.
In this interactive plot, you can:
- Start the plot
- Change the range of \(r\)
- Modify the number of iterations
- Adjust the (discarded) transient iterations
Cobweb diagram
To visualize the dynamic behavior of the logistic map, the cobweb diagram is used— a graphical tool that shows how a population evolves over time from an initial value. This diagram iteratively represents the logistic function on the plane, illustrating how the values approach an equilibrium point, enter periodic cycles, or exhibit chaotic behavior depending on the parameter \( P \). Through this representation, users can visually explore the nonlinear dynamics of the model and gain a better understanding of phenomena such as stability, bifurcations, and chaos.
Application in Biology and Ecology
In ecology, the logistic equation helps explain how populations stabilize in an environment with limited resources, such as food, space, or nutrients. At first, with few individuals, there are sufficient resources, and the population grows exponentially. As the population increases, resources per individual decrease, which reduces the growth rate until an equilibrium point is reached.
Other Applications
Beyond biology, the logistic equation also applies in:
- Economics: to model the growth of companies in a market where resources (customers, capital) are limited.
- Epidemiology: to represent the initial growth of diseases in a population until the number of susceptible individuals decreases.
- Innovation Models: in technology adoption, where a new technology grows in popularity, reaches saturation, and stabilizes its adoption.