Integrating ecological insight derived from individual-based simulations and physiologically structured population models

Two contrasting approaches are widely used to derive population dynamics as an emergent property deriving from the physiology and behavior of individual organisms. “Individual-based models” (IBMs) are computer simulations where the “state” (e.g., age, size) of each individual in a population is followed explicitly along with changes in its environment. Population properties (e.g., density, age- or size-structure) emerge from simple bookkeeping and descriptive statistics. Physiologically structured population models (PPSMs) have an identical philosophy, but assume a very large (formally infinite) population and that all individuals in a given state have an identical response to any given environment. These assumptions allow the bookkeeping to proceed through a series of mathematical steps that lead to partial differential or integral equations describing the population dynamics. There is software for both approaches that handles the bookkeeping, with the modeler specifying solely the individual model using stylized files, thereby eliminating the need for technical expertise in either complex computer simulations or advanced calculus. Each approach has its advantages and disadvantages. IBMs are easier to formulate and to explain to people with limited mathematical experience than PSPMs, but PSPMs allow for more extensive mapping of possible dynamic attractors. IBMs alone can reveal the population level effects of demographic stochasticity and of differences among individuals. Formal equilibrium analysis of PSPMs show possible stable states (size distributions) of the populations that include unstable steady states from which slightly perturbed populations may start cycling. The equilibrium size structure at these unstable states can serve as an initial condition for IBMs, thereby facilitating study of the cycles. We illustrated the interconnections and contrasting insights from the two approaches using a food-chain model for which the PSPM was previously studied by De Roos and Persson (Proc. Nat. Acad. Sci. USA: 99, 12907-12912, 2002). Future general population ecology theory requires work with model populations that are both physiologically structured and distributed in space. We describe concepts from spatially explicit IBMs with identical individuals that, in combination with the results in this paper, may point to a way forward.