Editing Logistic map (section)

=== Discretization of the logistic equation ===
The logistic map can also be derived from the discretization of the logistic equation for continuous-time population models . <!--[ 289 ]--> The name of the logistic map comes from Robert May's introduction of the logistic map from the discretization of the logistic equation. <!--[ 71 ]--><!--[ 290 ]--> The logistic equation is an ordinary differential equation that describes the time evolution of a population as follows : <!--[ 291 ]-->

{{NumBlk|:|<math>{\displaystyle {\frac {dN}{dt}}\ =rN\left(1-{\frac {N}{K}}\right)}</math>|{{EquationRef|5-6}}}}

Here, N is the number or population density of an organism, t is continuous time, and K and r are parameters. K is the carrying capacity, and r is the intrinsic rate of natural increase, which is usually positive . <!--[ 292 ]--> The left-hand side of this equation 
dN/dt
denotes the rate of change of the population size at time t <!--[ 293 ]-->.
[[File:Logistic curve examples.png|class=skin-invert-image|thumb|An example of a solution to the logistic equation. After time t, the population size N converges to the carrying capacity K regardless of the initial value.]]
The logistic equation ( 5-6 ) appears similar to the logistic map ( 5-4 ), but the behavior of the solutions is quite different from that of the logistic map <!--[ 277 ]-->. As long as the initial value N 0 is positive, the population size N of the logistic equation always converges monotonically to K <!--[ 294 ]--> .

The logistic map can be derived by applying the Euler method, which is a method for numerically solving first-order ordinary differential equations, to this logistic equation . [ Note 2 ] The Euler method uses a time interval (time step size) Δt to approximate the growth rate dN/dt
is approximated as follows <!--[ 296 ]-->:

{{NumBlk|:|<math>{\displaystyle {\frac {dN}{dt}}\approx {\frac {N(t+\Delta t)-N(t)}{\Delta t}}}</math>|{{EquationRef|5-7}}}}

This approximation leads to the following logistic map <!--[ 297 ]-->:

{{NumBlk|:|<math>{\displaystyle x_{n+1}=ax_{n}(1-x_{n})}</math>|{{EquationRef|5-8}}}}

where <math>x_n</math> and a in this equation are related to the original parameters, variables, and time step size as follows <!--[ 297 ]-->:

{{NumBlk|:|<math>{\displaystyle x_{n}={\frac {r\Delta t}{K(1+r\Delta t)}}N(n\Delta t)}</math>|{{EquationRef|5-9}}}}

{{NumBlk|:|<math>{\displaystyle a=1+r\Delta t}</math>|{{EquationRef|5-10}}}}

If Δt is small enough, equation ( 5-8 ) serves as a valid approximation to the original equation ( 5-6 ), and coincides with the solution of the original equation as Δt → 0 <!--[ 298 ]-->. On the other hand, as Δt becomes large, the solution deviates from the original solution <!--[ 298 ]-->. Furthermore, due to the relationship in equation ( 5-10 ), increasing Δt is equivalent to increasing the parameter a <!--[ 299 ]-->. Thus, increasing Δt not only increases the error from the original equation but also produces chaotic behavior in the solution <!--[ 300 ]-->.