The ideas presented in the previous sections have depicted population growth in terms of deterministic models because these are the most convenient tools we have for conceptualizing and exploring biological mechanisms. Population growth in nature is not so neatly defined. Even deterministic models can have unpredictable behavior and extreme sensitivity to initial conditions, leading to "chaotic dynamics" (Hastings et al., 1993). Stochastic processes and other processes that are not density-dependent can have profound impacts on populations. Examples of density-independent factors include weather, physical disturbance (e.g., avalanche, tillage), or the application of broad-spectrum toxins (e.g., antibiotics, insecticides).
The term "population regulation" has been equated to "a long-term stationary probability distribution of population density," and it is now recognized that such population regulation is necessarily density-dependent (Turchin, 1995). Over the long term, the absence of population regulation leads to either extinction or infinite population growth. Hence, the majority of populations display density-dependent patterns when tested by experimental manipulations or time-series analysis.
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