Something must be done to prevent uncontrolled growth at long times. Let the growth rate depend upon the population, and ask the growth rate decrease when the population becomes too large

X_n+1=R(X_n) X_n

_{The simplest possible decreasing function is:}

R(X_n)=R₀[1-X_n/X_max)

where xmax is a population at which population drops immediately to zero so now It is

X_n+1=R₀[1-X_n/X_max)X_n

a) define a new variable Y_nso that dependence upon Xmax drops out, R0 is the only free parameter of the model

b) Express the meaning of Yn in words