Bayesian Inference for Stochastic Cusp Catastrophe Model with Partially Observed Data

The purpose of this paper is to develop a data augmentation technique for statistical inference concerning stochastic cusp catastrophe model subject to missing data and partially observed observations. We propose a Bayesian inference solution that naturally treats missing observations as parameters and we validate this novel approach by conducting a series of Monte Carlo simulation studies assuming the cusp catastrophe model as the underlying model. We demonstrate that this Bayesian data augmentation technique can recover and estimate the underlying parameters from the stochastic cusp catastrophe model.

Performance Escapes and Catastrophes

This chapter presents an overview of newer thinking about how suicide risk fluctuates over time using concepts informed by mathematics, which provides a useful model for understanding why and how suicide emerges in different ways for different people at different times. It focuses in particular on the implications of this perspective for understanding suicides that emerge suddenly or “out of the blue” without much advance notice or warning signs. In the world of dynamical systems, sudden and discontinuous change processes are often referred to as “catastrophic” change because they represent a fundamental shift in how a system operates. Catastrophic change can be so dramatic that it defies reason and cannot be easily anticipated. The chapter then considers the cusp catastrophe model, which stands in contrast to the unidimensional suicide-risk continuum model that has dominated thinking about suicide risk for decades.

Comparison of the Fold and Cusp Catastrophe Models for Tensile Cracking and Sliding Rockburst

The failure modes of rockburst in catastrophe theory play an essential role in both theoretical analysis and practical applications. The tensile cracking and sliding rockburst is studied by analyzing the stability of the simplified mechanical model based on the fold catastrophe model. Moreover, the theory of mechanical system stability, together with an engineering example, is introduced to verify the analysis accuracy. Additionally, the results of the fold catastrophe model are compared with that of the cusp catastrophe model, and the applicability of two catastrophe models is discussed. The results show that the analytical results of the fold catastrophe model are consistent with the solutions of the mechanical systems stability theory. Moreover, the critical loads calculated by two catastrophe models are both less than the sliding force, which conforms to the actual situations. Nevertheless, the critical loads calculated from the cusp catastrophe model are bigger than those obtained by the fold catastrophe model. In conclusion, a reasonable result of the critical load can be obtained by the fold catastrophe model rather than the cusp catastrophe model. Moreover, the fold catastrophe model has a much wider application. However, when the potential function of the system is a high-order function of the state variable, the fold catastrophe model can only be used to analyze local parts of the system, and using a more complex catastrophe model such as the cusp catastrophe model is recommended.

Development of a wheat aphid population dynamics model based on cusp catastrophe theory

Aphids are a major global wheat pest that can cause considerable loss of yield. Modeling of aphid population dynamics is an integral part of management strategies to manage or control aphid populations. In this paper, first, a wheat aphid population dynamics model was developed based on a logistic model and the Holling III functional response, which includes three factors: temperature, natural enemies and insecticide. Second, this model fitted with a cusp catastrophe model to describe how abrupt changes in the wheat aphid population were influenced by these factors. Finally, the system was validated with field data from 2016 to 2018. The bifurcation set of the cusp catastrophe model was deemed to be the quantified dynamic control threshold, so an outbreak of aphid’s population can be explained according to the variation of control variables. In short, this aphid population model was successfully validated on survey data, which can be used to guide the prevention and control of aphids.