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```
#' @title Grazing model
#'
#' @description A spatially-explicit model of grazing pressure in drylands.
#'
#' @usage ca(l, grazing)
#'
#' @param del local seed dispersal
#' @param b environmental quality
#' @param c_ global competition
#' @param m0 intrinsic mortality
#' @param g grazing pressure
#' @param r regeneration rate of degraded cells
#' @param f local facilitation
#' @param d intrinsic degradation rate
#' @param p associational resistance against grazing
#'
#' @author Florian D. Schneider and Sonia Kefi (2015, in revision)
#'
#' @details The model builds upon a published model by Kefi et al. 2007. Spatial
#' models of vegetation cover so far have considered grazing mortality a
#' rather constant pressure, affecting all plants equally, regardless of their
#' position in space. In the known models it usually adds as a constant to the
#' individual plant risk (Kefi et al 2007 TPB). However, grazing has a strong
#' spatial component: Many plants in rangelands invest in protective
#' structures such as thorns or spines, or develop growth forms that reduce
#' their vulnerability to grazing. Therefore, plants growing next to each
#' other benefit from the protection of their neighbors.
#'
#' Such \strong{associational resistance} is widely acknowledged in vegetation
#' ecology but hardly integrated in models as a cause for spatially
#' heterogenous grazing pressure. It also renders the plant mortality density
#' dependent, which has important impacts on the bistability of the system.
#'
#' The model investigates how the assumption of spatially heterogeneous
#' pressure alters the bistability properties and the response of spatial
#' indicators of catastrophic shifts.
#'
#' The model knows three different cell states: occupied by vegetation
#' \code{"+"}, empty but fertile \code{"0"} and degraded \code{"-"}.
#' Transitions between cell states are only possible between vegetated and
#' empty (by the processes of plant 'death' and 'recolonization') and between
#' empty and degraded (by 'degradation' and 'regeneration').
#'
#' To account for the spatially heterogeneous impacts of grazing due to
#' associational resistance, we assumed that a plant's vulnerability to
#' grazers decreases with the proportion of occupied neighbors, $q_{+|+}$.
#' The individual probability of dying is therefore defined as
#'
#' \deqn{ w_{ \left\{ +,0 \right\} } = m_0 + g_0 \left( 1 - q_{+|+} \right)}
#'
#' where the additional mortality due to grazing is maximized to \eqn{g_0} if
#' a plant has no vegetated neighbor (i.e., \eqn{q_{+|+} = 0}) and gradually
#' reduces to 0 with an increasing fraction of occupied neighbors,
#' \eqn{q_{+|+}}.
#'
#' @family models
#'
#' @seealso \href{https://github.com/cascade-wp6/2015_schneider_kefi}{project on GitHub}
#'
#' @examples
#' l <- init_landscape(c("+","0","-"), c(0.6,0.2,0.2), width = 50) # create initial landscape
#' p <- list(del = 0.9, b = 0.4, c_ = 0.2, m0 = 0.05, g = 0.25, r = 0.01, f = 0.9, d = 0.1, p = 1) # set parameters
#' r <- ca(l, model = grazing, parms = p, t_max = 100) # run simulation
#'
#' @export
"grazing"
grazing <- list()
class(grazing) <- "ca_model"
grazing$name <- "Spatial Grazing Model"
grazing$ref <- "Schneider and Kefi 2015, in review"
grazing$states <- c("+", "0", "-")
grazing$cols <- grayscale(3)
grazing$parms <- list(
del = 0.9, # local seed dispersal
b = 0.5, # environmental quality
c_ = 0.2, # global competition
m0 = 0.05, # intrinsic mortality
g = 0.2, # grazing pressure
r = 0.01, # regeneration rate of degraded cells
f = 0.9, # local facilitation
d = 0.1, # intrinsic degradation rate
p = 1 # associational resistance against grazing
)
grazing$update <- function(x_old, parms_temp, subs = 10, timestep = NA) {
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$Q_plus <- neighbors(x_old, "+")/4 # count local density of occupied fields for each cell:
# 2 - drawing random numbers
rnum <- runif(x_old$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, (del*rho_plus+(1-del)*Q_plus)*(b-c_*rho_plus)*1/subs)
death <- with(parms_temp, (m0+g*(1-p*Q_plus))*1/subs)
death[death > 1] <- 1
} else {
recolonisation <- 0
death <- 1
}
regeneration <- with(parms_temp, (r + f*Q_plus)*1/subs)
degradation <- with(parms_temp, (d*1/subs))
# check for sum of probabilities to be inferior 1 and superior 0
if(any(c(recolonisation+degradation, death, 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(degradation < 0)) warning(paste("degradation falls below 0 in time step",timestep, "! balance parameters!!!"))
if(any( death < 0)) warning(paste("death 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 <= death)] <- "0"
x_new$cells[which(x_old$cells == "0" & rnum > recolonisation & rnum <= recolonisation+degradation)] <- "-"
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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