Question

What does the population growth model equation mean? \[\dfrac{{dN}}{{dt}} = rN\]

Answer 1

Hint: To use a differential equation to model population growth, we must first add some variables and terminology. In this equation N denotes the population density at a given time and r denotes the intrinsic rate of natural increase. K stands for carrying power.

Complete answer:
The equation \[\dfrac{{dN}}{{dt}} = rN\] denotes that the population's rate of change is proportional to its size, with r denoting the proportionality constant.
Since it represents exponential population growth, this is a very simple and impractical equation. If you're familiar with the Future Value of a Compounded Interest Rate, \[FV = PV{\left( {1 + r} \right)^n}\]
\[\dfrac{{dN}}{{dt}} = rN\] is a differential equation describing the population growth and where N is the population size, r is the growth rate, and t is time.
\[N\left( t \right) = {N_0}{e^{rt}}\] This is the solution to the exponential growth differential equation. Since the equation grows exponentially, and since population does not grow exponentially, a more rational model called "The Logistic Equation" has been created. The Logistic model imposes a growth limit.
\[\left( {d\dfrac{N}{{dt}}} \right) = rN\left( {1 - \dfrac{N}{K}} \right)\]: N is the population number, r is the growth rate, and K is the carrying capacity in the logistic differential equation.
Populations are forced to converge to the carrying capacity by this equation. The growth rate r is related to the rate at which populations approach K.

Note: A population model is a type of mathematical model used to study the dynamics of populations. They make it easier to comprehend how complicated relationships and processes function. Modeling complex interactions in nature can help you understand how numbers shift over time or in relation to each other in a more manageable way. Using population modelling as a technique, many trends can be discovered.