The logistic model describes population growth in an environment with limited resources, making it more realistic than the exponential model. It is represented by the equation , where r is the growth rate and K is the carrying capacity meaning the maximum population the environment can support. At the beginning, when the population N is small, growth is rapid and resembles exponential growth. However, as the population increases and gets closer to K, resources such as food, space, and oxygen become limited, causing the growth rate to slow down. Eventually, the population stabilizes and levels off at the carrying capacity. This pattern is shown in the graph, where the curve rises quickly at first but then flattens near K. The trout population example demonstrates this behavior: the number of trout increases rapidly in the early years but gradually slows as it approaches around 4000, showing how environmental limits control long-term population size.
![The logistic model describes population growth in an environment with limited resources, making it more realistic than the exponential model. It is represented by the equation [math]\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)[/math], where [i]r[/i] is the growth rate and [i]K[/i] is the carrying capacity meaning the maximum population the environment can support. At the beginning, when the population [i]N[/i] is small, growth is rapid and resembles exponential growth. However, as the population increases and gets closer to [i]K[/i], resources such as food, space, and oxygen become limited, causing the growth rate to slow down. Eventually, the population stabilizes and levels off at the carrying capacity. This pattern is shown in the graph, where the curve rises quickly at first but then flattens near [i]K[/i]. The trout population example demonstrates this behavior: the number of trout increases rapidly in the early years but gradually slows as it approaches around 4000, showing how environmental limits control long-term population size.](https://www.geogebra.org/resource/ra78amfb/eJyCSadFcCi48pnR/material-ra78amfb.png)