R/TwopMMmodel.R

In SoilR: Models of Soil Organic Matter Decomposition

#' Implementation of a two-pool Michaelis-Menten model
#' 
#' This function implements a two-pool Michaelis-Meneten model with a microbial
#' biomass and a substrate pool.
#' 
#' This implementation is similar to the model described in Manzoni and
#' Porporato (2007).
#' 
#' @param t vector of times (in days) to calculate a solution.
#' @param ks a scalar representing SOM decomposition rate (m3 d-1 (gCB)-1)
#' @param kb a scalar representing microbial decay rate (d-1)
#' @param Km a scalar representing the Michaelis constant (g m-3)
#' @param r a scalar representing the respired carbon fraction (unitless)
#' @param Af a scalar representing the Activity factor; i.e. a temperature and
#' moisture modifier (unitless)
#' @param ADD a scalar representing the annual C input to the soil (g m-3 d-1)
#' @param ival a vector of length 2 with the initial values of the SOM pool and
#' the microbial biomass pool (g m-3)
#' @return Microbial biomass over time
#' @seealso There are other \code{\link{predefinedModels}} and also more
#' general functions like \code{\link{Model}}.
#' @references Manzoni, S, A. Porporato (2007). A theoretical analysis of
#' nonlinearities and feedbacks in soil carbon and nitrogen cycles. Soil
#' Biology and Biochemistry 39: 1542-1556.
#' @examples
#' days=seq(0,1000,0.5)
#' MMmodel=TwopMMmodel(t=days,ival=c(100,10))
#' Cpools=getC(MMmodel)
#' matplot(days,Cpools,type="l",ylab="Concentrations",xlab="Days",lty=1,ylim=c(0,max(Cpools)*1.2))
#' legend("topleft",c("SOM-C", "Microbial biomass"),lty=1,col=c(1,2),bty="n")
#' ks=0.000018
#' kb=0.007
#' r=0.6
#' ADD=3.2
#' #Analytical solution of fixed points
#' #Cs_=kb/((1-r)*ks) wrong look at the sympy test print twopMModel.pdf
#' Km=900
#' Af=1
#' Cs=kb*Km/(Af*ks*(1-r)-kb)
#' abline(h=Cs,lty=2)
#' Cb=(ADD*(1-r))/(r*kb)
#' abline(h=Cb,lty=2,col=2)
#' #State-space diagram
#' plot(Cpools[,2],Cpools[,1],type="l",ylab="SOM-C",xlab="Microbial biomass")
#' plot(days,Cpools[,2],type="l",col=2,xlab="Days",ylab="Microbial biomass")
#' 
#' #The default parameterization exhaust the microbial biomass.
#' #A different behavior is obtained by increasing ks and decreasing kb
#' MMmodel=TwopMMmodel(t=days,ival=c(972,304) ,Af=3,kb=0.0000001)
#' Cpools=getC(MMmodel)
#' 
#' matplot(days,Cpools,type="l",ylab="Concentrations",xlab="Days",lty=1,ylim=c(0,max(Cpools)*1.2))
#' legend("topleft",c("SOM-C", "Microbial biomass"),lty=1,col=c(1,2),bty="n")
#' 
#' plot(Cpools[,2],Cpools[,1],type="l",ylab="SOM-C",xlab="Microbial biomass")
#' 
#' plot(days,Cpools[,2],type="l",col=2,xlab="Days",ylab="Microbial biomass")
TwopMMmodel<- function 
  (t, 
   ks=0.000018, 
   kb=0.007, 
   Km=900, 
   r=0.6, 
   Af=1, 
   ADD=3.2, 
   ival 
  )
{
    t_start=min(t)
    t_end=max(t)
    nr=2
    if(length(ival)!=2) stop("The vector of initial values ival must be of length 2")
    f=function(C,t){
      S=C[1] 
      B=C[2] 
      O=matrix(byrow=TRUE,nrow=2,ncol=1,c((Af*ks)*B*(S/(Km+S)),
                                   kb*B))
      return(O)
    }
    alpha=list()
    alpha[["1_to_2"]]=function(C,t){
      1-r
    }
    alpha[["2_to_1"]]=function(C,t){
      1
    }
    Anl=new("TransportDecompositionOperator",t_start,Inf,nr,alpha,f)
    inputrates=BoundInFluxes(
      function(t){
        matrix(
          nrow=nr,
          ncol=1,
          c(ADD,  0)
        )
      },
      t_start,
      t_end
    )
    modnl=GeneralNlModel( t, Anl, ival, inputrates, deSolve.lsoda.wrapper)
    return(modnl)
}