Logistic map
Instructions:
Move black slider \(n\) to add or remove points, the violet slider \(x_{0}\) to change the initial value, the green sider \(\mu\) for the relationship between reproduction and starvation, and the green slider \( b \) to change the exponent value. Press the checkbox to remove the connecting lines. This animation is related to the cobweb diagram.
The logistic equation is a mathematical model used to describe the growth of populations under certain environmental and resource constraints. It is widely used in biology, ecology, and other fields to represent phenomena where the growth of a system is limited by reaching a threshold capacity.
Equation and General Form
The basic logistic equation is:
$$ P_{n + 1} = \left( r P_{n} \left( 1 - P_{n} \right)\right) $$
Where:
- \(P_{n}\): population at time \(n\),
- \(r\): intrinsic growth rate.
The equation shows how a population grows rapidly at first when it is small, but its growth slows as it gets bigger.
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.
Chaotic Behavior
The logistic model can exhibit chaotic behavior under certain values of \(r\). As \(r\) increases above \(3\), the system begins to behave unpredictably, and small variations in \(r\) generate significant differences in \(P_n\). This phenomenon is an example of how a simple equation can generate a complex dynamic system.
Representation in the Logistic Map
The logistic map 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.
The logistic equation and its map are fundamental tools for studying and visualizing growth behaviors in populations and dynamic processes that show equilibrium, fluctuations, and chaos, depending on the parameters involved.