Non-autonomous properties of decoupled systems: implications for the use of simple autonomous models

The parameters estimated for a population model depend on another component of its ecosystem. One way they may do so is considered in order to evaluate the long-term dynamical effects of cycling in one ecosystem component, such as whales, on other components, such as euphasiids preyed on by whales, that operate on time scales much shorter than the period of the cycle. It was found that even though the parameter that cycled over long time scales was effectively constant on the time scale over which the population operated, the subtle changes that did occur in the parameter on short time scales were sufficient to radically alter both short and long-term behaviour of the population: e.g. changing chaotic behaviour for an autonomous model with parameters equal to those of the non-autonomous model at the time of observation into constant, or periodic, behaviour. The model is dynamically stable, the trajectory followed is asymptotically stable, but structurally unstable for significant periods of time. Although an autonomous system may be decoupled into subsystems for the purpose of study, these subsystems are non-autonomous when isolated because of changes in their parameters through time due to changes in the system outside of the subsystem which is modelled. The effect of components of the system which are not included in the subsystem studied must be evaluated by simulating their influence, making the parameters through which they exert their influence functions of time. Systems studied in isolation may create an illusion of instability, an illusion of overexploitation or some other deleterious phenomenon often attributed to human activity.