Question

Which of the following formula is correct to calculate "the average rate of change in the number of organisms per time, per organisms", in population growth rate ?
Where, N represent the number of organisms and t is the time.

• A. $\Delta N$
• B. $\frac{\Delta N}{N \Delta t}$
• C. $\frac{\Delta N}{\Delta t}$
• D. $\frac{\Delta t}{N \Delta N}$

Hint:

Dimensional check for demographic rates:
"Per unit time" requires $\Delta t$ in the denominator.
"Per organism" requires $N$ in the denominator.
Net change $\Delta N$ is in the numerator, yielding $\frac{\Delta N}{N \Delta t}$ with units of $\text{time}^{-1}$.

Answer 1

The Correct Option is B

Concept:
Population dynamics involves quantifying changes in population size over space and time.
In population ecology, demographic rates can be expressed either as absolute changes across the whole population or as specific per capita rates normalized per individual organism.

Step 1: Derivation of Absolute Population Growth Rate:
Let $N$ denote the initial population size (number of organisms).
Let $\Delta N$ denote the change in the total number of individuals over a specified time interval $\Delta t$.
The absolute rate of population growth describes the number of individuals added or lost per unit of time:
\[ \text{Absolute Growth Rate} = \frac{\Delta N}{\Delta t} \]

Step 2: Derivation of Per Capita Growth Rate:
The problem specifies the rate of change:
- "per time" $\implies$ divided by $\Delta t$
- "per organisms" $\implies$ divided by the number of individuals $N$
Normalizing the absolute growth rate to a single individual basis yields the average per capita rate of change:
\[ \text{Per Capita Growth Rate} = \frac{1}{N} \left( \frac{\Delta N}{\Delta t} \right) = \frac{\Delta N}{N \Delta t} \] In continuous differential notation, this corresponds to the specific growth rate:
\[ r = \frac{1}{N} \frac{dN}{dt} \] where $r$ is the intrinsic rate of natural increase.

Step 3: Examination of Incorrect Formulations:
- Option (A), $\Delta N$, is simply the absolute net increment in individual counts.
- Option (C), $\frac{\Delta N}{\Delta t}$, represents the absolute population growth rate per unit time, but is not normalized per organism.
- Option (D), $\frac{\Delta t}{N \Delta N}$, is dimensionally inverted.
Final Answer:
The correct mathematical formulation is $\frac{\Delta N}{N \Delta t}$, corresponding to option (B).