Editing Logistic map (section)

=== Demographic model example ===
Taking the biological [[Lotka–Volterra equations|population model]] as an example {{mvar|x<sub>n</sub>}} is a number between zero and one, which represents the ratio of existing [[population]] to the [[carrying capacity|maximum possible population]]. <ref name=":1" group="May, Robert M. (1976)" >{{harvnb|May|1976|loc=formula 2 and 3}}</ref>
This nonlinear difference equation is intended to capture two effects:
* ''reproduction'', where the population will increase at a rate [[Proportionality (mathematics)|proportional]] to the current population when the population size is small,
* ''starvation'' (density-dependent mortality), where the [[Population growth|growth rate]] will decrease at a rate proportional to the value obtained by taking the theoretical "carrying capacity" of the environment less the current population.

The usual values of interest for the parameter {{mvar|r}} are those in the interval {{math|[0,&nbsp;4]}}, so that {{mvar|x<sub>n</sub>}} remains bounded on {{math|[0,&nbsp;1]}}. The {{math|''r'' {{=}} 4}} case of the logistic map is a nonlinear transformation of both the [[bit-shift map]] and the {{math|''μ'' {{=}} 2}} case of the [[tent map]]. If {{math|''r'' > 4}}, this leads to negative population sizes. (This problem does not appear in the older [[Ricker model]], which also exhibits chaotic dynamics.) One can also consider values of {{mvar|r}} in the interval {{math|[−2,&nbsp;0]}}, so that {{mvar|x<sub>n</sub>}} remains bounded on {{math|[−0.5,&nbsp;1.5]}}.<ref name="Takashi Tsuchiya, Daisuke Yamagishi, 1997"/>

{{expand Japanese}}
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