Landscape Dynamics Concepts
Landscape dynamics uses simple concepts
of agent based systems
and of evolutionary game theory.
- We use concepts of agent based models (ABM).
- Intelligent agents move about in a geometric space.
- There is a fundamental geometric space, the strategy space.
- This is a finite-dimensional Euclidean space or smooth manifold,
every point of which, x, represents a strategy that may be chosen
by an agent.
- The agents positions are summarized in a normalized histogram.
- The strategy space is discretized into bins, the number of agents
in each bin counted, and the count divided by the total number
- The function obtained is a probability density, f.
- The cumulative distribution to this density is denoted, F.
- Thus F is the integral of f.
- The agents move according to a dynamical rule depending on F.
- As they move, F changes, so the dynamical rule changes as well.
- Such a system may be represented by a partial differential equation.
- Discretizing time into small increments, the agents move in steps.
- After each step, F is recomputed, determining the rule for the next step.
- We use concepts of evolutionary game theory (EGT).
- Agents play a game as they move.
- A landscape dynamical system also has a payoff function, P.
- Each agent earns a payoff for each step of play.
- The payoff for one agent depends on the distribution
of all other agents:
P(x, F), a real number.
- The dynamical rule in landscape dynamics is the gradient rule.
- Each agent moves up the slope of the payoff function.
- The landscape dynamical system is a flow on a function space.
- Think of the initial distribution as the initial point of a trajectory.
- The motion of agents according to the dynamical rule moves F.
- We may observe the action of the landscape dynamical system
by watching the histogram as an animated movie.
Revised 03 October 2004 by Ralph Abraham