These models are simplifications of natural population behaviors. Each highlights particular features of the growth response. These models are so commonly used that the patterns of behaviors have become metaphors for other systems and even the parameters of specific models (r & K in the Logistic Model) have taken on meaning of their own. For example, people speak of a "carrying capacity" for a population and attempt to apply this concept to other situations, such as human population.

List of patterns and models:

• constant intrisic rate of increase, "exponential"
• resource limited and the "logistic" model
• oscillation
• boom and bust
• predatory & prey

 

Constant intrinsic rate of increase

the population increases by the same fixed ratio each time period

for example an increase of 1% of the population per year

over a longer period of time (40 time intervals) this population will show a curve upwards in which each new period has a increasing arithmetic increment because the population is bigger each period

 

this behavior is often called "exponential" because it can be fit with an exponential function

N(t) = N(0)* e ^r*t

where

• N(t) is the population at any time
• N(0) is the population at the beginning
• e is the Naperian constant (2.7128)
• r is the intrinsic growth rate

this curve is also called the "J curve" because if you look at populations over a very long period on an arithmetic scale, it looks like a "J" on it's side and backwards.

 

Resource limitation

populations don't continue to growth at a constant rate indefinitely, they run out of resources

when then start running out, the growth rate slows down

This relationship is sometimes modeled with the "Logistic" equation.

where:

r = the intrisic growth rate that controls the initial exponential growth phase

K = the population level that has used the resources and results in zero net growth

even though the general behavior, or pattern of growth, is similar it is not always a valid model to apply

 

Oscillation

it is possible to have negative growth rates above the "carrying capacity"

damped oscillation can be a behavior that "homes in" on the carrying capacity

 

Boom and Bust

Oscillations may not be damped and the population may crash

Predator & Prey

A closely related model to population oscillation is the community predator&prey interaction. Sometimes the crash or oscillation mechanism inferred is a predation (predator decreasing the prey) or resource depletion (prey depletion results in predator decrease)

this pulsing behavior may actually be the most general behavior and the other examples are specific cases