Cobweb diagram
Definition
The cobweb diagram is a graphical method used to analyze the dynamical behavior of iterated functions, particularly in the study of the logistic map. This application is described by the equation:
$$ P_{n + 1} = \left( r P_{n} \left( 1 - P_{n} \right)\right)^b $$
where \( P_{n} \in [0,1] \) represents the normalized population at generation \(n\), and \(r\) is a growth parameter.
Purpose of the diagram
The cobweb diagram provides a visual representation of the iterative process of a function. In the context of the logistic map, it enables the examination of:
- The existence and stability of fixed points.
- The emergence of periodic cycles.
- The transition to chaotic behavior as the parameter \(r\) increases.
Diagram construction
- Plot the function \( f(P) = rP(1 - P)\) alongside the identity line \(y = x\).
- Choose an initial value \(x_{0}\) in the interval \([0,1]\).
- Draw a vertical line from the point \((x_{0},0)\) to the curve \(f(x)\).
- From there, draw a horizontal line to the identity line \(y = x\), obtaining the next iteration \(x_{1} = f(x_{0})\).
- Repeat this alternating process (vertical to the curve, horizontal to the line), generating a visual pattern resembling a cobweb.
Behavior according to parameter \(r\)
- For \(0 < r < 1\), the population tends toward extinction: \(P_{n} \rightarrow 0\).
- For \(1 < r < 3\), the iterations converge to a stable fixed point other than zero.
- For \(3 < r < 3.57 \), the system exhibits bifurcation into periodic cycles of increasing order.
- For \(3.57 < r\), the system demonstrates chaotic dynamics, characterized by high sensitivity to initial conditions.
Simulation
Move black slider \(n\) to add or remove points, the violet slider \(x_{0}\) to change the initial value, the green sider \(r\) 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.
Educational applications
The cobweb diagram serves as a valuable visual tool in the teaching of topics such as:
- Stability in dynamical systems
- Bifurcation theory
- Transition to chaos
- Nonlinear modeling
Its use facilitates the intuitive introduction of these concepts, even at an introductory educational level.