When you compiled and loaded the code in the three examples before (ODEs in compiled languages - definition in R, ODEs in compiled languages - definition in C and ODEs in compiled languages - definition in fortran) you are able to run a benchmark test.
library(microbenchmark)
R <- function(){
  out <- ode(y = yini, times = times, func = caraxis_R,
             parms = parameter)
}
C <- function(){
  out <- ode(y = yini, times = times, func = "caraxis_C",
             initfunc = "init_C", parms = parameter,
             dllname = dllname_C)
}
fortran <- function(){
  out <- ode(y = yini, times = times, func = "caraxis_fortran",
             initfunc = "init_fortran", parms = parameter, 
             dllname = dllname_fortran)
}
Check if results are equal:
all.equal(tail(R()), tail(fortran()))
all.equal(R()[,2], fortran()[,2])
all.equal(R()[,2], C()[,2])
Make a benchmark (Note: On your machine the times are, of course, different):
bench <- microbenchmark::microbenchmark(
  R(), 
  fortran(),
  C(),
  times = 1000
)
summary(bench)
     expr         min        lq       mean     median         uq        max neval cld
      R()   31508.928 33651.541 36747.8733 36062.2475 37546.8025 132996.564  1000   b
fortran()     570.674   596.700   686.1084   637.4605   730.1775   4256.555  1000  a 
      C()     562.163   590.377   673.6124   625.0700   723.8460   5914.347  1000  a 
We see clearly, that R is slow in contrast to the definition in C and fortran. For big models it's worth to translate the problem in a compiled language.
The package cOde is one possibility to translate ODEs from R to C.