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 of agents.

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.