Population Growth. A simple (though approximate) model of population growth is the
Malthusian growth model. The preferred population growth model is the
logistic function.
Model of a particle in a potential-field. In this model we consider a particle as being a point of mass m which describes a trajectory which is modeled by a function x : R → R3 given its coordinates in space as a function of time. The potential field is given by a function V:R3 → R and the trajectory is a solution of the
differential equationNote this model assumes the particle is a point mass, which is certainly known to be false in many cases we use this model, for example, as a model of planetary motion.
Model of rational behavior for a consumer. In this model we assume a consumer faces a choice of n commodities labeled 1,2,...,n each with a market price p1, p2,..., pn. The consumer is assumed to have a cardinal utility function U (cardinal in the sense that it assigns numerical values to utilities), depending on the amounts of commodities x1, x2,..., xn consumed. The model further assumes that the consumer has a budget M which she uses to purchase a vector x1, x2,..., xn in such a way as to maximize U(x1, x2,..., xn). The problem of rational behavior in this model then becomes an
optimization problem, that is:
subject to:
This model has been used in
general equilibrium theory, particularly to show existence and
Pareto optimality of economic equilibria. However, the fact that this particular formulation assigns numerical values to levels of satisfaction is the source of criticism (and even ridicule). However, it is not an essential ingredient of the theory and again this is an idealization.
Neighbour-sensing model explains the
mushroom formation from the initially chaotic
fungal network.
