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Simulation models are commonly used to understand and predict the development of ecological systems, for instance to study the occurrence of tipping points and their possible ecological effects. Sensitivity analysis is a key tool in the study of model responses to changes in conditions. The applicability of available methodologies for sensitivity analysis can be problematic if tipping points are involved. In this paper we demonstrate that not considering these tipping points may result in misleading statistics on model behaviour. In turn, this limits the applicability of simulation models in ecological research. Tipping points are best revealed when asymptotic model behaviour is considered, i.e. by applying bifurcation analysis. Bifurcation analysis, however, is limited to deterministic dynamic models, whereas many ecological simulation models are nondeterministic and can only be analysed using sensitivity analysis methodologies. In this paper we explore the possibilities for applying methodologies of sensitivity analysis to analyse models with tipping points. The Bazykin-Berezovskaya model, a deterministic ecological model of which the structure regarding tipping points is known a priori, is used as case study. We conclude that important clues about the occurrence of tippings points can be revealed from different sensitivity analysis methodologies, if proper statistical and graphical measures are used. The results raise awareness about how tipping points affect temporal model responses in ecological simulation models, and may also be more generally applicable for nondeterministic models that cannot be analysed using bifurcation analysis. [ABSTRACT FROM AUTHOR]

Since Rosenzweig showed the destabilisation of exploited ecosystems, the so called Paradox of enrichment, several mechanisms have been proposed to resolve this paradox. In this paper we will show that a feeding threshold in the functional response for predators feeding on a prey population stabilizes the system and that there exists a minimum threshold value above which the predator-prey system is unconditionally stable with respect to enrichment. Two models are analysed, the first being the classical Rosenzweig-MacArthur (RM) model with an adapted Holling type-II functional response to include a feeding threshold. This mathematical model can be studied using analytical tools, which gives insight into the mathematical properties of the two dimensional ODE system and reveals underlying stabilisation mechanisms. The second model is a mass-balance (MB) model for a predator-prey-nutrient system with complete recycling of the nutrient in a closed environment. In this model a feeding threshold is also taken into account for the predator-prey trophic interaction. Numerical bifurcation analysis is performed on both models. Analysis results are compared between models and are discussed in relation to the analytical analysis of the classical RM model. Experimental data from the literature of a closed system with ciliates, algae and a limiting nutrient are used to estimate parameters for the MB model. This microbial system forms the bottom trophic levels of aquatic ecosystems and therefore a complete overview of its dynamics is essential for understanding aquatic ecosystem dynamics. [ABSTRACT FROM AUTHOR]

In Part I of this paper we have discussed new methods for the numerical continuation of point-to-cycle connecting orbits in three-dimensional autonomous ODE's using projection boundary conditions. In this second part we extend the method to the numerical continuation of cycle-to-cycle connecting orbits. In our approach, the projection boundary conditions near the cycles are formulated using eigenfunctions of the associated adjoint variational equations, avoiding costly and numerically unstable computations of the monodromy matrices. The equations for the eigenfunctions are included in the defining boundary-value problem, allowing a straightforward implementation in AUTO, in which only the standard features of the software are employed. Homotopy methods to find the connecting orbits are discussed in general and illustrated with an example from population dynamics. Complete AUTO demos, which can be easily adapted to any autonomous three-dimensional ODE system, are freely available. [ABSTRACT FROM AUTHOR]

We propose new methods for the numerical continuation of point-to-cycle connecting orbits in three-dimensional autonomous ODE's using projection boundary conditions. In our approach, the projection boundary conditions near the cycle are formulated using an eigenfunction of the associated adjoint variational equation, avoiding costly and numerically unstable computations of the monodromy matrix. The equations for the eigenfunction are included in the defining boundary-value problem, allowing a straightforward implementation in AUTO, in which only the standard features of the software are employed. Homotopy methods to find connecting orbits are discussed in general and illustrated with several examples, including the Lorenz equations. Complete AUTO demos, which can be easily adapted to any autonomous three-dimensional ODE system, are freely available. [ABSTRACT FROM AUTHOR]