Fortran logistic

logostic.f90

subroutine f(u, r)
implicit none
  real(kind(0.d0)), intent(out) :: r
  real(kind(0.d0)), intent(in) :: u
  r = u*(1.d0-u)
end subroutine f

subroutine  fortran_logistic(N, dt, u)
  integer, intent(in) :: N
  real(kind(0.d0)), intent(in) :: dt
  real(kind(0.d0)), intent(out) :: u(N)
  
  integer :: i
  real(kind(0.d0)) :: temp
  
  u(1) = 1.d-5
  do i=1, N-1
    call f(u(i), temp)
    u(i+1) = u(i) + dt*temp
  end do
end subroutine  fortran_logistic

I would recommend to use functions instead of subroutines:

function fortran_logistic(N, dt) result(u)
  implicit none

  integer, intent(in) :: N
  real(kind(0.d0)), intent(in) :: dt
  real(kind(0.d0)) :: u(N)

  integer :: i

  u(1) = 1.d-5
  do i=1, N-1
    u(i+1) = u(i) + dt*f(u(i))
  end do

  contains
  function f(u)
    real(kind(0.d0)), intent(in) :: u
    real(kind(0.d0)) :: f

    f = u*(1.d0-u)

  end function f

end function fortran_logistic