Land use, macroecology, and the accuracy of the Maximum Entropy Theory of Ecology: A case study of Azorean arthropods

Human activity and land management practices, in particular land use change, have resulted in the global loss of biodiversity. These types of disturbances affect the shape of macroecological patterns, and analyzing these patterns can provide insights into how ecosystems are affected by land use change. The Maximum Entropy Theory of Ecology (METE) simultaneously predicts many of these patterns using a set of ecological state variables: the number of species, the number of individuals, and the total metabolic rate. The theory’s predictions have been shown to be successful across habitats and taxa in undisturbed natural ecosystems, although previous tests of METE in relation to disturbance have focused primarily on systems where the state variables are changing relatively quickly. Here, we assess predictions of METE applied to a different type of disturbance: land use change. We use METE to simultaneously predict the species abundance distribution (SAD), the metabolic rate distribution of individuals (MRDI), and the species–area relationship (SAR) and compare these predictions to arthropod data from 96 sites at Terceira Island in the Azores archipelago across four different land uses of increasing management intensity: 1. native forest, 2. exotic forest, 3. semi-natural pasture, and 4. intensive pasture. Across these patterns, we find that the forest habitats are the best fit by METE predictions, while the semi-natural pasture consistently provided the worst fit. The intensive pasture is intermediately well fit for the SAD and MRDI, and comparatively well fit for the SAR, though the residuals are not normally distributed. The direction of failure of the METE predictions at the pasture sites is likely due to the hyperdominance of introduced spider species present there. We hypothesize that the particularly poor fit for the semi-natural pasture is due to the mix of arthropod communities out of equilibrium and the changing management practices throughout the year, leading to greater heterogeneity in composition and complex dynamics that violate METE’s assumption of static state variables. The comparative better fit for the intensive pasture could then result from more homogeneous arthropod communities that are well adapted to intensive management, and thus whose state variables are less in flux.

Derivations of the Core Functions of the Maximum Entropy Theory of Ecology

The Maximum Entropy Theory of Ecology (METE), is a theoretical framework of macroecology that makes a variety of realistic ecological predictions about how species richness, abundance of species, metabolic rate distributions, and spatial aggregation of species interrelate in a given region. In the METE framework, “ecological state variables” (representing total area, total species richness, total abundance, and total metabolic energy) describe macroecological properties of an ecosystem. METE incorporates these state variables into constraints on underlying probability distributions. The method of Lagrange multipliers and maximization of information entropy (MaxEnt) lead to predicted functional forms of distributions of interest. We demonstrate how information entropy is maximized for the general case of a distribution, which has empirical information that provides constraints on the overall predictions. We then show how METE’s two core functions are derived. These functions, called the “Spatial Structure Function” and the “Ecosystem Structure Function” are the core pieces of the theory, from which all the predictions of METE follow (including the Species Area Relationship, the Species Abundance Distribution, and various metabolic distributions). Primarily, we consider the discrete distributions predicted by METE. We also explore the parameter space defined by the METE’s state variables and Lagrange multipliers. We aim to provide a comprehensive resource for ecologists who want to understand the derivations and assumptions of the basic mathematical structure of METE.

Derivations of the Core Functions of the Maximum Entropy Theory of Ecology

The Maximum Entropy Theory of Ecology, or METE, is a theoretical framework of macroecology that makes a variety of realistic ecological predictions about how species richness, abundance of species, metabolic rate distributions, and spatial aggregation of species interrelate in a given region. In the METE framework, "ecological state variables" (representing total area, total species richness, total abundance, and total metabolic energy) describe macroecological properties of an ecosystem. METE incorporates these state variables into constraints on underlying probability distributions. The method of Lagrange multipliers and maximization of information entropy (MaxEnt) lead to predicted functional forms of distributions of interest. We demonstrate how information entropy is maximized for the general case of a distribution, which has empirical information that provides constraints on the overall predictions. We then show how METE’s two core functions are derived. These functions, called the "Spatial Structure Function" and the "Ecosystem Structure Function" are the core pieces of the theory, from which all the predictions of METE follow (including the Species Area Distribution, the Species Abundance Distribution, and various metabolic distributions). Primarily, we consider the discrete distributions predicted by METE.We also explore the parameter space defined by the METE’s state variables and Lagrange multipliers. We aim to provide a comprehensive resource for ecologists who want to understand the derivations and assumptions of basic mathematical structure of METE.

Fatigue reliability analysis of crack growth life using maximum entropy method

In this article, a new method for fatigue reliability analysis of crack growth life based on the maximum entropy theory and a long crack propagation model is proposed. A modified generalized passivation-lancet model for long fatigue crack propagation rate is presented with explicit physical meaning. Experimental results for turbine disk alloy ZSGH4169 under different strain ratios and temperatures (at 650°C and room temperature) are used to verify the applicability of the new model. Results show that predictions by the proposed model are almost identical to the experimental data. The presented model is better than the other three models to reflect the rapid propagation characteristics of the crack. In order to perform fatigue reliability estimation, the probabilities of failure are calculated using the maximum entropy theory based on the fatigue crack growth life that derived from the proposed modified crack propagation model and the above existing three models. Results have shown that maximum entropy theory is very apt for fatigue reliability analysis of turbine disk under different loading conditions with a limited number of samples because it does not need any distribution assumptions for random variables. The effectiveness and accuracy of the combination of fatigue crack propagation models and maximum entropy method for fatigue reliability analysis are demonstrated with examples.