What is the definition of carrying capacity \((K) ?\) How is it used in the framework of the logistic model to introduce a mechanism of density-dependent population regulation?

Carrying Capacity (K) is the maximum number of individuals in a population that a specific environment can support sustainably. That is, it's the population size at which the population growth rate becomes zero due to limited resources or other factors.

The logistic model is a mathematical model that describes population growth in a limited environment. The logistic model can be represented by the following equation: \(\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right)\) where \(N\) represents the population size, \(r\) is the intrinsic growth rate, \(t\) is time, and \(K\) is the carrying capacity.

In the logistic model, carrying capacity (K) is used to introduce a mechanism of density-dependent population regulation. The term \(\left(1 - \frac{N}{K}\right)\) in the logistic equation makes the population growth rate dependent on the population density. When the population size is small compared to the carrying capacity, i.e., \(\frac{N}{K} << 1\), the growth rate is approximately equal to the intrinsic growth rate, \(rN\), giving exponential growth. As the population size (N) approaches the carrying capacity (K), the term \(\left(1 - \frac{N}{K}\right)\) approaches zero, meaning the growth rate will also approach zero, preventing the population from growing indefinitely. This results in an "S"-shaped curve called a logistic growth curve.

Carrying capacity is an example of density-dependent population regulation because it ties the growth rate of a population to its density. It results in negative feedback, where growth rate decreases as population size approaches the carrying capacity (K). This prevents populations from spiraling out of control and leading to overpopulation. The logistic model is a more realistic representation of population growth than exponential growth because it incorporates the limitation of resources and density-dependent regulation that exist in nature.