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Fig. 6.4 Fractal structures
arising during the evolution
of the colonies of the
bacterium Paenibacillus
alvei (a) and distribution of
bacteria in them depending
on the concentrations of
agar and peptone (b)
discrete time scale. Then the number of individuals X n +1 at the moment t n +1 should
be proportional to the number of individuals X n at the previous moment t n :
X 1 ) ʱ
X n :
However, since the habitat is limited, overpopulation arising due to uncontrolled
fertility should lead to a decrease in population, for example, due to lack of food.
Therefore, introducing a restriction on the maximum population size, its evolution
can be described by the so-called logistic equation:
X 1 ¼ ʱ
X n N
ð
X n
Þ:
Using relative values x n ¼
X n /N we obtain a more convenient expression:
x n ¼ ʱ
x n 1
ð
x n
Þ:
Based on the structure of the problem, the first thing that comes into mind is that
for large n the solution of this equation tends to some limit. But the logistic equation
is nonlinear with respect to x n and its solution exhibits typical nonlinear features—
bifurcations. This term refers to splitting the dependence of the solution from a
certain parameter into two (or more) branches. The behavior of solutions of the
logistic equation is determined by the parameter
ʱ
. It is easy to see that with
ʱ<
1
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