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Journal of Online Mathematics and its Applications
An Introduction to Population Ecology
Introduction to Population Modeling
The ability to predict the population size of a group of individuals is extremely useful to the study of ecology. It allows for the estimation of the various effects imposed upon a group by internal and external forces. We note that the word force has a different meaning in population modeling than in physics. You can think of these forces as factors that impact the population – for example, availability of food, spread of disease, interactions with other species. These forces can then be divided into density-independent forces and density-dependent forces.
With no forces acting upon a population, we expect the population to have simple exponentially increasing growth (Vandermeer & Goldberg, 2004). Mathematically, we expect a population function whose rate of growth increases with the population’s size, that is,
This differential equation says that the rate of change in population size over time (dP/dt) increases by a proportional rate of growth (r) multiplied by the current population size (P). The biological force modeled here is an example of a density-independent force, because it depends only on the population P, not on external forces such as crowding or food supply.
We know from calculus that the solution of this equation is P(t) = P0ert. The graph of this exponential function (Figure 1) shows the behavior of the population over time. Biologists call this a J-shaped curve, and it is the foundation for all of the ecological models that we will discuss in this module (Molles, 2004; Vandermeer & Goldberg, 2004).
Figure 1. The exponential function has a J-shaped curve.