Nonlinear mixed effects model - object not found

I want to create nonlinear mixed effect model in R using this formula: h=1.30+(dbh^2/(b0+b1*dbh)^2).

in this equation b0 is a random factor and is computed as: a0+a1*BA+a2*VH+a3*QM+a4*NH where a0, a1, a2, a3, a4 are linear regression coefficients.

I have consulted with different chatbots. They recommend different packages and syntaxes for doing so. None of them run successfully.

Your recommendation appreciated.

look at this code which was created by bing chatbot:

library(lme4)
# Define the nonlinear function
h <- function(dbh, b0, b1) {
  1.30 + (dbh^2 / (b0 + b1 * dbh)^2)
}
# Define the random factor
b0 <- function(a0, a1, BA, a2, VH, a3, QM, a4, NH) {
  a0 + a1 * BA + a2 * VH + a3 * QM + a4 * NH
}
# Define some starting values
start_list <- list(
  # Nonlinear parameters
  nlpars = c(b1 = 0.5),
  # Linear parameters
  theta = c(a0 = 0.1, a1 = 0.2, a2 = 0.3, a3 = 0.4, a4 = 0.5)
)
# Fit the model using nlmer with start argument
model <- nlmer(h(dbh) ~ h(dbh, b0(a0, a1, BA, a2, VH, a3, QM, a4, NH), b1) ~ (a0 | ID), data = your_data, start = start_list)

Running this code gave me this error:

> Error in h(dbh) : argument "b0" is missing, with no default

Edit: @GregorThomas
I edited the left hand side of the model formula as: h(dbh, b0, b1). R gave me this error:

> Error in eval(predvars, data, env) : object 'b1' not found.

here is the output of dput command:

structure(list(dbh = c(22L, 87L, 110L, 27L, 69L, 34L, 49L, 89L, 
49L, 87L, 46L, 66L, 74L, 48L, 81L, 26L, 89L, 66L, 100L, 50L, 
74L, 29L, 82L, 58L, 75L, 63L, 100L, 54L, 54L, 64L), h = c(11.4, 
32.1, 31.3, 35.3, 40, 23.6, 34, 35, 30, 43, 36, 41.3, 40, 33.3, 
40.6, 21.4, 39.2, 42, 44, 36.5, 39.6, 24, 29, 39.5, 37, 42, 41.3, 
34.5, 37, 40), BA = c(27.71, 27.71, 43.35, 43.35, 26.48, 26.48, 
34.33, 34.33, 35.29, 35.29, 28.61, 28.61, 39.86, 18.1, 18.1, 
29.8, 29.8, 30.2, 30.2, 47.1, 47.1, 30.69, 30.69, 31.78, 31.78, 
45.49, 45.49, 9.34, 9.34, 34.65), VH = c(393.76, 393.76, 672.33, 
672.33, 388.41, 388.41, 526.63, 526.63, 538.45, 538.45, 396.61, 
396.61, 592.49, 249.89, 249.89, 445.99, 445.99, 466.5, 466.5, 
710.59, 710.59, 447, 447, 477.61, 477.61, 726.05, 726.05, 122.4, 
122.4, 553.31), QMD = c(36.8, 36.8, 53.9, 53.9, 43.3, 43.3, 54, 
54, 51.4, 51.4, 36.1, 36.1, 62.5, 27.2, 27.2, 43.5, 43.5, 46.2, 
46.2, 48, 48, 37.3, 37.3, 47.4, 47.4, 60.2, 60.2, 25.7, 25.7, 
60.6), NH = c(260L, 260L, 190L, 190L, 180L, 180L, 150L, 150L, 
170L, 170L, 280L, 280L, 130L, 310L, 310L, 200L, 200L, 180L, 180L, 
260L, 260L, 280L, 280L, 180L, 180L, 160L, 160L, 180L, 180L, 120L
)), row.names = c(NA, 30L), class = "data.frame")

There are several issues here.

This is not a "mixed model" by the standard definition; mixed models always involve a grouping variable, and allow parameters of the top-level (linear or nonlinear model) to vary across the levels of the grouping variable. What you have is a sort of hierarchical model, but the second-level model (i.e. the model for b0 as a function of BA, VH, QMD, NH) is a linear model, not a sample from a joint distribution.

You can get this done using gnls ("generalized nonlinear least squares"), from the nlme package:

library(nlme)
gnls(h ~ 1.30 + (dbh^2/(b0 + b1*dbh^2)),
     params = list(b0 ~ BA + VH + QMD + NH, b1 ~ 1),
     data = dd,
     start = list (b0 = rep(1,5), b1 = 1))

Plot fitted (predicted) vs observed:

par(las = 1, bty = "l")
plot(predict(m1), dd$h)
abline(a=0, b= 1, col = 2, lwd = 2)

You should also probably be aware that, for better or worse, chatbot-generated code is highly controversial on Stack Overflow at present ...