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As ecologists, understanding variation and the mechanisms that create variation, are one of the focuses for our research. What mechanisms drive different patterns and variation away from patterns. To understand these complex processes in nature we simplify mechanisms and trade off complexities with assumptions. This process of simplification has enabled us to expand our knowledge and even make predictions about ecosystems. However, understanding variation and the underlying processes that generate variation within processes and traits is fundamental to move forward and make reliable predictions in a changing world, where rare climatic events become ever more frequent and rates of deforestation and urbanization are at its highest.
To begin with, I would like to focus on the concept of Jensen's inequality and it's role in ecological systems. Jensen's inequality is a mathematical concept from the early twentieth century, for which the mathematical proof was developed by Danish mathematician Johan Jensen. Overall, the concept of Jensen's inequality states that the convex transformation of a mean is less than or equal to the mean applied after the convex transformation. In ecology, we can understand this in various scenarios, where the average of a population is less than or equal to the average of the averages of individuals. This is also what has been termed the 'fallacy of the average' in the ecological context.
To provide an example of how Jensen's inequality plays into ecological systems, I will focus on describing a simple simulation model for animal movements using an uncorrelated random walk. Although we know that more complex models are better able to capture real animal movement patterns, an uncorrelated random walk has been used for decades to simplify animal movement and is sufficient for this example.

Select a movement model and then incorporate variation.