Editing Logistic map (section)

=== Discrete population model ===
While Lorenz used the logistic map in 1964,<ref name=":2" /> it gained widespread popularity from the research of British mathematical biologist Robert May and became widely known as a formula for considering changes in populations of organisms. <!--[ 271 ]--> In such a logistic map for organism populations, the variable <math>x_n</math> represents the number of organisms living in a certain environment (more technically, the population size). <!--[ 272 ]--> Furthermore, it is assumed that no organisms leave the environment and no external organisms enter the environment (or that there is no substantial impact even if there is immigration), and the mathematical model for considering the increase or decrease in population in such a situation is the logistic map in mathematical biology . <!--[ 273 ]-->

There are two types of mathematical models for the growth of populations of organisms: discrete-time models using difference equations and continuous-time models using differential equations . <!--[ 274 ]--> For example, in the case of a type of insect that dies soon after laying eggs, the population of the insect is counted for each generation, i.e., the number of individuals in the first generation, the number of individuals in the second generation, and so on . <!--[ 275 ]--> Such examples fit the former discrete-time model . <!--[ 276 ]--> On the other hand, when the generations are continuously overlapping, it is compatible with the continuous-time model . <!--[ 277 ]--> The logistic map corresponds to such a discrete or generation-separated population model . <!--[ 278 ]-->

Let N denote the number of individuals of a single species in an environment . The simplest model for population growth is one in which the population continues to grow at a constant rate relative to the number of individuals. This type of population growth model is called the Malthusian model, and can be expressed as follows <!--[ 279 ]-->:

{{NumBlk|:|<math>{\displaystyle N_{n+1}=\alpha N_{n}}</math>|{{EquationRef|5-1}}}}

Here, N n is the number of individuals in the nth generation, and α is the population growth rate, a positive constant <!--[ 280 ]-->. However, in model ( 5-1 ), the population continues to grow indefinitely, making it an unrealistic model for most real-world phenomena <!--[ 281 ]-->. Since there is a limit to the number of individuals that an environment can support, it seems natural that the growth rate α decreases as the population N n increases <!--[ 282 ]-->. This change in growth rate due to changes in population density is called the density effect <!--[ 283 ]-->. The following difference equation is the simplest improvement model that reflects the density effect in model ( 5-1 )<!--[ 284 ]-->.

{{NumBlk|:|<math>{\displaystyle N_{n+1}=(a-bN_{n})N_{n}}</math>|{{EquationRef|5-2}}}}

Here, a is the maximum growth rate possible in the environment, and b is the strength of the influence of density effects <!--[ 284 ]-->. Model ( 5-2 ) assumes that the growth rate declines simply in proportion to the number of individuals <!--[ 285 ]--> . Let N n in equation ( 5-2 ) be

{{NumBlk|:|<math>{\displaystyle x_{n}={\frac {b}{a}}N_{n}}</math>|{{EquationRef|5-3}}}}

After performing the variable transformation, the following logistic map is derived <!--[ 286 ]-->:

{{NumBlk|:|<math>{\displaystyle x_{n+1}=a(1-x_{n})x_{n}}</math>|{{EquationRef|5-4}}}}

When using equation ( 5-2 ) or equation ( 5-4 ) as the population size of an organism, if Nn or xn becomes negative, it becomes meaningless as a population size . <!--[ 287 ]--> To prevent this, the condition 0 ≤ x0 ≤ 1 for the initial value x0 and the condition 0 ≤ r ≤ 4 for the parameter a are required . <!--[ 286 ]-->

Alternatively, we can assume a maximum population size K that the environment can support, and use this to

{{NumBlk|:|<math>{\displaystyle N_{n+1}=a\left(1-{\frac {N_{n}}{K}}\right)N_{n}}</math>|{{EquationRef|5-5}}}}

The logistic map can be derived by considering a difference equation that incorporates density effects in the form <math>x_n = N_n / K </math> <!--[ 5 ]--><!--[ 288 ]-->, where the variable <math>x_n</math> represents the ratio of the number of individuals <math>N_n</math> to the maximum number of individuals K <!--[ 5 ]--><!-- [ 288 ] -->.