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Inverse problems are ubiquitous in science and engineering. Two categories of inverse problems concerning a physical system are (1) estimate parameters in a model of the system from observed input–output pairs and (2) given a model of the system, reconstruct the input to it that caused some observed output. Applied inverse problems are challenging because a solution may (i) not exist, (ii) not be unique, or (iii) be sensitive to measurement noise contaminating the data. Bayesian statistical inversion (BSI) is an approach to tackle ill-posed and/or ill-conditioned inverse problems. Advantageously, BSI provides a “solution” that (i) quantifies uncertainty by assigning a probability to each possible value of the unknown parameter/input and (ii) incorporates prior information and beliefs about the parameter/input. Herein, we provide a tutorial of BSI for inverse problems by way of illustrative examples dealing with heat transfer from ambient air to a cold lime fruit. First, we use BSI to infer a parameter in a dynamic model of the lime temperature from measurements of the lime temperature over time. Second, we use BSI to reconstruct the initial condition of the lime from a measurement of its temperature later in time. We demonstrate the incorporation of prior information, visualize the posterior distributions of the parameter/initial condition, and show posterior samples of lime temperature trajectories from the model. Our Tutorial aims to reach a wide range of scientists and engineers. 

We develop a method to identify how ecological, evolutionary, and eco-evolutionary feedbacks influence system stability. We apply our method to nine empirically parametrized eco-evolutionary models of exploiter–victim systems from the literature and identify which particular feedbacks cause some systems to converge to a steady state or to exhibit sustained oscillations. We find that ecological feedbacks involving the interactions between all species and evolutionary and eco-evolutionary feedbacks involving only the interactions between exploiter species (predators or pathogens) are typically stabilizing. In contrast, evolutionary and eco-evolutionary feedbacks involving the interactions between victim species (prey or hosts) are destabilizing more often than not. We also find that while eco-evolutionary feedbacks rarely altered system stability from what would be predicted from just ecological and evolutionary feedbacks, eco-evolutionary feedbacks have the potential to alter system stability at faster or slower speeds of evolution. As the number of empirical studies demonstrating eco-evolutionary feedbacks increases, we can continue to apply these methods to determine whether the patterns we observe are common in other empirical communities.