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```
#' @title Musselbed Disturbance Model
#' @description Model for spatial pattern of intertidal mussel beds with wave
#' disturbance. With three cell states: occupied by mussels (\code{"+"}),
#' empty sites (\code{"0"} and disturbed sites (\code{"-"}).
#'
#' @usage
#' ca(l, musselbed, parms = p)
#'
#' @param r A numerical value. recolonisation of empty sites dependent on local
#' density.
#'
#' @param d A numerical value. probability of disturbance of occupied sites if
#' at least one disturbed site is in the direct 4-cell neighborhood.
#'
#' @param delta A numerical value. intrinsic disturbance rate.
#'
#' @author Guichard, Halpin, et al. (2003)
#'
#' @references Guichard, F., Halpin, P.M., Allison, G.W., Lubchenco, J. & Menge,
#' B.A. (2003). Mussel disturbance dynamics: signatures of oceanographic
#' forcing from local interactions. The American Naturalist, 161, 889–904.
#'
#' @family models
#'
#' @details
#'
#' The model represents the spatial dynamics in mussel cover of rock substrate
#' in intertidal systems. The stochastic wave disturbances will most likely
#' remove mussels that are located next to a gap because of the losened byssal
#' threads in their proximity. This causes a dynamic gap growth.
#'
#' The model describes the process by simplifying the system into three
#' potential cell states: occupied by mussel ("+"), empty but undisturbed
#' ("0"), and disturbed, bare rock with loose byssal threads ("-").
#'
#' Mussel growth on empty cells is defined by parameter \code{r} multiplied by
#' the local density of mussels in the direct 4-cell neighborhood.
#'
#' Any cell occupied by mussels has an intrinsic chance of \code{delta} to
#' become disturbed from intrinsic cause, e.g. natural death or predation.
#' Additionally, wave disturbance will remove mussels and leave only bare
#' rock, i.e. disturbed sites, with probability \code{d} if at least one
#' disturbed cell is in the direct 4-cell neighborhood. This causes
#' disturbances to cascade through colonies of mussels.
#'
#' Disturbed sites will recover into empty sites with a constant rate of 1 per
#' year, i.e. on average a disturbed site becomes recolonisable within one
#' year after the disturbance happened.
#'
#' @examples
#'
#' l <- init_landscape(c("+","0","-"), c(0.6,0.2,0.2), width = 50) # create initial landscape
#' p <- list(delta = 0.01, d = 09, r = 0.4) # set parameters
#' r <- ca(l, musselbed, p, t_max = 100) # run simulation
#'
#'
#' @export
#'
"musselbed"
musselbed <- list()
class(musselbed) <- "ca_model"
musselbed$name <- "Musselbed Disturbance Model"
musselbed$ref <- "Guichard, F., Halpin, P.M., Allison, G.W., Lubchenco, J. & Menge, B.A. (2003). Mussel disturbance dynamics: signatures of oceanographic forcing from local interactions. The American Naturalist, 161, 889–904. doi: 10.1086/375300"
musselbed$states <- c("+", "0", "-")
musselbed$cols <- grayscale(3)
musselbed$parms <- list(
r = 0.8, # recolonisation of empty sites dependent on local density
d = 0.1, # probability of disturbance of occupied sites if at least one disturbed site
delta = 0.02 # intrinsic disturbance rate
)
musselbed$update <- function(x_old, parms_temp, subs = 10) {
x_new <- x_old
for(s in 1:subs) {
parms_temp$rho_plus <- sum(x_old$cells == "+")/(x_old$dim[1]*x_old$dim[2]) # get initial vegetation cover
parms_temp$localdisturbance <- neighbors(x_old, "-") > 0 # any disturbance in neighborhood?
parms_temp$localcover <- neighbors(x_old, "+")/4 # any occupied in neighborhood?
# 2 - drawing random numbers
rnum <- runif(x_old$dim[1]*x_old$dim[2]) # one random number between 0 and 1 for each cell
# 3 - setting transition probabilities
if(parms_temp$rho_plus > 0) {
recolonisation <- with(parms_temp, (r*localcover)*1/subs)
disturbance <- with(parms_temp, (delta + d*localdisturbance) * 1/subs)
disturbance[disturbance > 1] <- 1
} else {
recolonisation <- 0
disturbance <- 1
}
regeneration <- 1*1/subs
# check for sum of probabilities to be inferior 1 and superior 0
if(any(c(recolonisation, disturbance, regeneration) > 1 )) warning(paste("a set probability is exceeding 1 in time step", timestep, "! decrease number of substeps!!!"))
if(any(recolonisation < 0)) warning(paste("recolonisation falls below 0 in time step",timestep, "! balance parameters!!!"))
if(any( disturbance < 0)) warning(paste("disturbance falls below 0 in time step",timestep, "! balance parameters!!!"))
if(any(regeneration < 0)) warning(paste("regeneration falls below 0 in time step",timestep, "! balance parameters!!!"))
# 4 - apply transition probabilities
x_new$cells[which(x_old$cells == "0" & rnum <= recolonisation)] <- "+"
x_new$cells[which(x_old$cells == "+" & rnum <= disturbance)] <- "-"
x_new$cells[which(x_old$cells == "-" & rnum <= regeneration)] <- "0"
# 5- store x_new as next x_old
x_old <- x_new
}
## end of single update call
return(x_new)
}
```

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