  A generalization of the economic model of logistic growth, which takes into
account the effects of memory and crises, is suggested. Memory effect means
that the economic factors and parameters at any given time depend not only on
their values at that time, but also on their values at previous times. For the
mathematical description of the memory effects, we use the theory of
derivatives of non-integer order. Crises are considered as sharp splashes
(bursts) of the price, which are mathematically described by the
delta-functions. Using the equivalence of fractional differential equations and
the Volterra integral equations, we obtain discrete maps with memory that are
exact discrete analogs of fractional differential equations of economic
processes. We derive logistic map with memory, its generalizations, and
"economic" discrete maps with memory from the fractional differential
equations, which describe the economic natural growth with competition,
power-law memory and crises.

Economics