## =======================================================================
## example 1
## a predator and its prey diffusing on a flat surface
## in concentric circles
## 1-D model with using cylindrical coordinates
## Lotka-Volterra type biology
## =======================================================================
## ================
## Model equations
## ================
lvmod <- function (time, state, parms, N, rr, ri, dr, dri) {
with (as.list(parms), {
PREY <- state[1:N]
PRED <- state[(N+1):(2*N)]
## Fluxes due to diffusion
## at internal and external boundaries: zero gradient
FluxPrey <- -Da * diff(c(PREY[1], PREY, PREY[N]))/dri
FluxPred <- -Da * diff(c(PRED[1], PRED, PRED[N]))/dri
## Biology: Lotka-Volterra model
Ingestion <- rIng * PREY * PRED
GrowthPrey <- rGrow * PREY * (1-PREY/cap)
MortPredator <- rMort * PRED
## Rate of change = Flux gradient + Biology
dPREY <- -diff(ri * FluxPrey)/rr/dr +
GrowthPrey - Ingestion
dPRED <- -diff(ri * FluxPred)/rr/dr +
Ingestion * assEff - MortPredator
return (list(c(dPREY, dPRED)))
})
}
## ==================
## Model application
## ==================
## model parameters:
R <- 20 # total radius of surface, m
N <- 100 # 100 concentric circles
dr <- R/N # thickness of each layer
r <- seq(dr/2,by = dr,len = N) # distance of center to mid-layer
ri <- seq(0,by = dr,len = N+1) # distance to layer interface
dri <- dr # dispersion distances
parms <- c(Da = 0.05, # m2/d, dispersion coefficient
rIng = 0.2, # /day, rate of ingestion
rGrow = 1.0, # /day, growth rate of prey
rMort = 0.2 , # /day, mortality rate of pred
assEff = 0.5, # -, assimilation efficiency
cap = 10) # density, carrying capacity
## Initial conditions: both present in central circle (box 1) only
state <- rep(0, 2 * N)
state[1] <- state[N + 1] <- 10
## RUNNING the model:
times <- seq(0, 200, by = 1) # output wanted at these time intervals
## the model is solved by the two implemented methods:
## 1. Default: banded reformulation
print(system.time(
out <- ode.1D(y = state, times = times, func = lvmod, parms = parms,
nspec = 2, names = c("PREY", "PRED"),
N = N, rr = r, ri = ri, dr = dr, dri = dri)
))
## 2. Using sparse method
print(system.time(
out2 <- ode.1D(y = state, times = times, func = lvmod, parms = parms,
nspec = 2, names = c("PREY","PRED"),
N = N, rr = r, ri = ri, dr = dr, dri = dri,
method = "lsodes")
))
## ================
## Plotting output
## ================
# the data in 'out' consist of: 1st col times, 2-N+1: the prey
# N+2:2*N+1: predators
PREY <- out[, 2:(N + 1)]
filled.contour(x = times, y = r, PREY, color = topo.colors,
xlab = "time, days", ylab = "Distance, m",
main = "Prey density")
# similar:
image(out, which = "PREY", grid = r, xlab = "time, days",
ylab = "Distance, m", main = "Prey density")
image(out2, grid = r)
## =======================================================================
## Example 2.
## Biochemical Oxygen Demand (BOD) and oxygen (O2) dynamics
## in a river
## =======================================================================
## ================
## Model equations
## ================
O2BOD <- function(t, state, pars) {
BOD <- state[1:N]
O2 <- state[(N+1):(2*N)]
## BOD dynamics
FluxBOD <- v * c(BOD_0, BOD) # fluxes due to water transport
FluxO2 <- v * c(O2_0, O2)
BODrate <- r * BOD # 1-st order consumption
## rate of change = flux gradient - consumption + reaeration (O2)
dBOD <- -diff(FluxBOD)/dx - BODrate
dO2 <- -diff(FluxO2)/dx - BODrate + p * (O2sat-O2)
return(list(c(dBOD = dBOD, dO2 = dO2)))
}
## ==================
## Model application
## ==================
## parameters
dx <- 25 # grid size of 25 meters
v <- 1e3 # velocity, m/day
x <- seq(dx/2, 5000, by = dx) # m, distance from river
N <- length(x)
r <- 0.05 # /day, first-order decay of BOD
p <- 0.5 # /day, air-sea exchange rate
O2sat <- 300 # mmol/m3 saturated oxygen conc
O2_0 <- 200 # mmol/m3 riverine oxygen conc
BOD_0 <- 1000 # mmol/m3 riverine BOD concentration
## initial conditions:
state <- c(rep(200, N), rep(200, N))
times <- seq(0, 20, by = 0.1)
## running the model
## step 1 : model spinup
out <- ode.1D(y = state, times, O2BOD, parms = NULL,
nspec = 2, names = c("BOD", "O2"))
## ================
## Plotting output
## ================
## select oxygen (first column of out:time, then BOD, then O2
O2 <- out[, (N + 2):(2 * N + 1)]
color = topo.colors
filled.contour(x = times, y = x, O2, color = color, nlevels = 50,
xlab = "time, days", ylab = "Distance from river, m",
main = "Oxygen")
## or quicker plotting:
image(out, grid = x, xlab = "time, days", ylab = "Distance from river, m")Run the code above in your browser using DataLab