Analyse the Lotka-Volterra Equations
The Lotka-Volterra equations describe a predator and its prey’s population development and go back to Lotka (1925) and Volterra (1926). The prey’s population at time \(t\) (in days) will be denoted with \(P(t)\) and the predator’s (or rather consumer’s) population with \(C(t)\). \(P(t)\) and \(C(t)\) are called state variables. This ODE model is two-dimensional, but it should be noted that ODE models of arbitrary dimensions (including one-dimensional ODE models) can be analyzed with ODEsensitivity.
Model Definition
Now we define the model according to the definition in deSolve::ode().
LVmod = function(Time, State, Pars) {
with(as.list(c(State, Pars)), {
Ingestion <- rIng * Prey * Predator
GrowthPrey <- rGrow * Prey * (1 - Prey/K)
MortPredator <- rMort * Predator
dPrey <- GrowthPrey - Ingestion
dPredator <- Ingestion * assEff - MortPredator
return(list(c(dPrey, dPredator)))
})
}Each of the five parameter names, their lower and upper boundaries, the initial values for the state variables and the timepoints of interest are saved in separate vectors:
Sensitivity Analysis
The sensitivity analysis of a general ODE model can be performed by using the generic function ODEsensitivity::ODEmorris().
set.seed(1618)
LVres_morris = ODEmorris(mod = LVmod, pars = LVpars, state_init = LVinit
, times = LVtimes, binf = LVbinf, bsup = LVbsup
)Let’s take a look at the output LVres_morris.
str(LVres_morris, give.attr = FALSE)
#> List of 2
#> $ Prey : num [1:16, 1:51] 1.00e-02 -1.90e-02 2.42e-02 4.78e-05 -3.24e-05 ...
#> $ Predator: num [1:16, 1:51] 0.01 0.009441 0.000063 -0.01796 0.009611 ...The first row of each state variable contains a copy of all timepoints. The other rows contain the Morris sensitivity indices \(\mu\), \(\mu^\star\), and \(\sigma\) for all 5 parameters and all 51 timepoints.
Plotting
ODEsensitivity provides a plot() method for objects of class ODEmorris:
plot.ODEmorris() has two important arguments: pars_plot and state_plot. Using pars_plot, a subset of the parameters included in the sensitivity analysis can be selected for plotting (the default is to use all parameters). state_plot gives the name of the state variable for which the sensitivity indices shall be plotted (the default being the first state variable):
