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The number of trout will be limited by the available resources such as food supply as well as by spawning habitat.A small population of fish might grow exponentially if the pond is large and food is abundant, but the growth rate will decline as the population increases and the availability of resources declines.Think of a dashpot as that small cylinder that keeps your screen door from slamming shut.
Suppose that we wish to study how a population \(P(t)\) grows with time \(t\text\) We might make the assumption that a constant fraction of population is having offspring at any particular time.
We might use a system of differential equations to model two interacting species, say where one species preys on the other.).
We have good historical data for the populations of the lynx and snowshoe hare from the Hudson Bay Company.
For example, Japan has experienced negative growth in recent years.
The equation \(d P/dt = k P\) can also be used to model phenomena such as radioactive decay and compound interest—topics which we will explore later.
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First, we must consider the restorative force on the spring.