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What is the Formula of dN?

Published in Population Dynamics Formula 3 mins read

Based on the provided reference related to the law of population growth, the formula for the infinitesimal change in population size, denoted as dN, is derived from the rate equation.

The reference provides the equation dN/dt = rmN. This equation, attributed to French mathematician P.F. Verhulst, describes the rate of increase of population per unit time (dN/dt).

From this fundamental equation, we can isolate dN to represent the change in population size over an infinitesimal period of time (dt).

The formula for dN is:

*dN = rmN dt**

This formula indicates that the change in population size (dN) during a very small time interval (dt) is equal to the current rate of population growth (rmN) multiplied by that time interval.

Understanding the Population Growth Formula (dN/dt = rmN)

The reference highlights the formula dN/dt = rmN as the law of population growth. This differential equation describes how a population changes over time under specific conditions.

Let's break down the components of this formula:

Term Description Unit (Example)
dN Infinitesimal change in population size Individuals
dt Infinitesimal change in time Time unit (e.g., Year)
dN/dt Rate of change of population size over time Individuals per time unit
r Intrinsic growth rate (a constant) Per time unit
m Another constant affecting growth rate (as per reference notation) Unitless or per time unit
N Population size at a given time Individuals

The reference states that this rate of increase (dN/dt) depends upon "Population size" and "All of the above" (referencing unspecified options, but confirming dependence on N).

The Meaning of dN

In calculus, dN represents a tiny, infinitesimal change in the variable N (population size). It's not a standalone value in the same way N is, but rather signifies a change or increment. When seen in the context of dN/dt, it's the numerator, representing the amount by which the population size changes during the infinitesimally small time period dt.

Deriving the Formula for dN

Starting with the rate equation given in the reference:

  1. dN/dt = rmN

To find a formula specifically for dN, we can multiply both sides of the equation by dt:

  1. (dN/dt) dt = (rmN) dt

This simplifies to:

  1. *dN = rmN dt**

This derived formula *dN = rmN dt** mathematically represents the infinitesimal change in population size as the product of the growth rate parameters (r and m), the current population size (N), and the infinitesimal time interval (dt).

Understanding this relationship is crucial in studying population dynamics, as it forms the basis for models that predict how populations grow or decline over time.