From complex spatial dynamics to simple Markov chain models: do predators and prey leave footprints?

In this paper we present a concept for using presence absence data to recover information on the population dynamics of predator prey systems. We use a highly complex and spatially explicit simulation model of a predator prey mite system to generate simple presence absence data: the number of patches with both prey and predators, with prey only, with predators only, and with neither species, along with the number of patches that change from one state to another in each time step. The average number of patches in the four states, as well as the average transition probabilities from one state to another, are then depicted in a state transition diagram, constituting the ‘‘footprints’’ of the underlying population dynamics. We investigate to what extent changes in the population processes modeled in the complex simulation (i.e. the predator’s functional response and the dispersal rates of both species) are reflected by different footprints. The transition probabilities can be used to forecast the expected fate of a system given its current state. However, the transition probabilities in the modeled system depend on the number of patches in each state. We develop a model for the dependence of transition probabilities on state variables, and combine this information in a Markov chain transition matrix model. Finally, we use this extended model to predict the long-term dynamics of the system and to reveal its asymptotic steady state properties.