Inverse estimation of integral projection model parameters using time series of population-level data

Integral projection models (IPMs) allow us to describe quantitatively the dynamics of a population structured by a continuous variable. They rely on information gathered at the individual level by recording survival, reproduction and changes in some structuring variable over time. This requires the ability to track individuals over the course of their entire life cycle. When this is not feasible, we would like to use alternative information to infer a population's dynamics. Time series of population-level data are an option. An inverse modelling approach allows inferring the vital rates of a population when only population-level data, in the form of a time series of the size of a population and the distribution of its individuals along a structuring variable, are available. The approach also allows incorporating estimates obtained through individual-level data. Here, we explore how inverse modelling performs with simulated data and a relatively simple demographic model. We explore scenarios of data availability in terms of time-series length, per-year sample size and availability of independent vital-rate estimates. We also test model performance in a real system using a 15-year long data set from a chamaephyte plant, Cryptantha flava. We show that an inverse model can provide accurate reconstructions of the vital rates in a scenario where no individual-level information is available. Better results can be obtained if independent estimates on any vital rate are provided, as was the case for C.flava where high interannual variation is present. Parameter estimation becomes more difficult with shorter time series, but per-year sample size can be greatly reduced without significantly affecting parameter accuracy. Inverse modelling of IPMs allow for the estimation of unobserved vital rates, which is important for systems where any or all of the vital rates are hard to quantify. It also helps to determine whether a forward IPM is capturing the population dynamics: if the inverse version produces incorrect reconstructions of the vital rates, the forward IPM can be considered as inadequately describing the system.