Estimate mortality with GenEst when only one search has been performed during the search efficiency trial

Question

I am trying to estimate mortality from carcasses counts using the GenEst package. This allows observed mortality to be corrected using models fitted on carcass persistence and searcher efficiency trials data.

In my case, searcher efficiency trials consist of a single search. However, the pkm model used to model searcher efficiency requires a parameter k governing the decrease in searcher efficiency in subsequent searches. The pkm model can be fitted without the k parameter, but then mortality cannot be estimated from this model. I can replace the second search with 0 instead of NAs, but it would mean that carcasses not found on the first search would not be discovered in later searches, which has not been tested here in reality.

Is there a better way to proceed than to replace Search2 with 0 ? Can I trust the results from the model fitted with 0 on the second search or is it biased?

Here is a reproducible example using the data present in the GenEst package :

require(GenEst)

data(wind_RPbat)
data_CP <- wind_RPbat$CP
data_SS <- wind_RPbat$SS
data_DWP <- wind_RPbat$DWP
data_CO <- wind_RPbat$CO

# fit carcass persistence model
model_CP <- cpm(l ~ Season, s ~ Season, data = data_CP, dist = "weibull",
                left = "LastPresent", right = "FirstAbsent")

# simulate searcher efficiency trial data with a unique search
data_SE_sim <- data.frame(
  seID = paste0("se", seq(1, 20, 1)),
  Search1 = rbinom(n = 20, 1, prob = 0.5) ,
  Search2 = NA
)

# fit searcher efficiency model without a k parameter
model_SE_sim <-
  pkm(formula_p = p ~ 1,
      formula_k = NULL,
      data = data_SE_sim)`

# try to fit mortality estimation model -> does not fit
Mhat <- estM(nsim = 1000, data_CO = data_CO, data_SS = data_SS,
             data_DWP = data_DWP, model_SE = model_SE_sim, model_CP = model_CP,
             unitCol = "Turbine", COdate = "DateFound")


# simulate a second dataset with 2 searches, with the second search 
data_SE_sim_2 <- data.frame(
  seID = paste0("se", seq(1, 20, 1)),
  Search1 = rbinom(n = 20, 1, prob = 0.5) 
)
data_SE_sim_2$Search2 = ifelse(data_SE_sim_2$Search1 == 1, NA, 0)

# fit searcher efficiency model with a k parameter
model_SE_sim_2 <-
  pkm(formula_p = p ~ 1,
      formula_k = k ~ 1 ,
      data = data_SE_sim_2)
plot(model_SE_sim_2)

# fit mortality estimation model -> works 
Mhat <- estM(nsim = 1000, data_CO = data_CO, data_SS = data_SS,
             data_DWP = data_DWP, model_SE = model_SE_sim_2, model_CP = model_CP,
             unitCol = "Turbine", COdate = "DateFound")