Combining perturbations and parameter variation to influence mean first passage times.

Perturbations are relatively large shocks to state variables that can drive transitions between stable states, while drift in parameter values gradually alters equilibrium magnitudes. This latter effect can lead to equilibrium bifurcation, the generation, or annihilation of equilibria. Equilibrium annihilations reduce the number of equilibria and so are associated with catastrophic population collapse. We study the combination of perturbations and parameter drift, using a two-species intraguild predation (IGP) model. For example, we use bifurcation analysis to understand how parameter drift affects equilibrium number, showing that both competition and predation rates in this model are bifurcating parameters. We then introduce a stochastic process to model the effects of population perturbations. We demonstrate how to evaluate the joint effects of perturbations and drift using the common currency of mean first passage time to transitions between stable states. Our methods and results are quite general, and for example, can relate to issues in both pest control and sustainable harvest. Our results show that parameter drift (1) does not importantly change the expected time to reach target points within a basin of attraction, but (2) can dramatically change the expected time to shift between basins of attraction, through its effects on equilibrium resilience.

Managing ecological thresholds in coupled environmental-human systems.

Many ecosystems appear subject to regime shifts--abrupt changes from one state to another after crossing a threshold or tipping point. Thresholds and their associated stability landscapes are determined within a coupled socioeconomic-ecological system (SES) where human choices, including those of managers, are feedback responses. Prior work has made one of two assumptions about managers: that they face no institutional constraints, in which case the SES may be managed to be fairly robust to shocks and tipping points are of little importance, or that managers are rigidly constrained with no flexibility to adapt, in which case the inferred thresholds may poorly reflect actual managerial flexibility. We model a multidimensional SES to investigate how alternative institutions affect SES stability landscapes and alter tipping points. With institutionally dependent human feedbacks, the stability landscape depends on institutional arrangements. Strong institutions that account for feedback responses create the possibility for desirable states of the world and can cause undesirable states to cease to exist. Intermediate institutions interact with ecological relationships to determine the existence and nature of tipping points. Finally, weak institutions can eliminate tipping points so that only undesirable states of the world remain.

Shot noise perturbations and mean first passage times between stable states.

Predicting crossings between stable states is a central issue in population biology. Crossings from low-density to high-density equilibria are often associated with pest outbreaks, while the opposite crossings are often associated with population collapse of harvested species. Here I use a simple, bistable model to demonstrate a technique for estimating mean first passage times (MFPT) of thresholds, including boundaries between stable equilibria. The approach is based on stochastic "shot-noise" perturbations to the population and the MFPTs compare favorably with mean crossing times from Monte Carlo numerical solutions of the stochastically perturbed model. This agreement suggests that MFPT approximations can be used to quantify expected effects of species manipulations, whether the goal is pest control or sustainable harvest.

Combining stochastic models with experiments to understand the dynamics of monarch butterfly colonization.

Stochastic models are of increasing importance in ecology but are usually only applied to observational data. Here we use a stochastic population model to combine experimental and observational data to understand the colonization of old fields by monarch butterflies Danaus plexippus. We experimentally tested for density dependence in oviposition rates when predators were excluded, and we measured predation rates under natural conditions. Significance tests on the resulting data showed that both oviposition and predation were density dependent but could not show how oviposition and mortality combine to determine egg densities in nature. We therefore used our data to calculate the Akaike Information Criterion to choose between a nested suite of stochastic models that differed in their oviposition and mortality terms. When we simply fit the models to the observational data, the best model assumed density independence in both oviposition and predation. When we instead first estimated the oviposition rate at low density from experimental data, however, the best model included density dependence in oviposition, and a model that included density dependence in both oviposition and predation performed nearly as well. This result is consistent with our experiments and suggests that experiments can enhance the usefulness of stochastic models in ecology.