select the correct representation of density of population (N) which can be expressed after a period of time `t+1`.

Options

A. `N_(t+1) = N_t+[(B+I) - (D+E)]`
B. `N_(t+1) = N_t[(B+I)+(D+E)]`
C. `N_t = N_(t+1) + [(B+I)-(D+E)]`
D. `N_(t+1) = N_t[(B+I) + (D-E)]`

Answer

To solve the question regarding the correct representation of the density of population (N) after a period of time \( t + 1 \), we need to consider the factors that affect population change over time. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Variables
1. **N**: Represents the density of the population at time \( t \).
2. **b**: Represents the number of births in the population.
3. **d**: Represents the number of deaths in the population.
4. **i**: Represents immigration, or the number of individuals coming into the population.
5. **e**: Represents emigration, or the number of individuals leaving the population.

### Step 2: Establish the Initial Population
At time \( t \), the population density is represented as \( N_t \).

### Step 3: Determine Changes in Population
To find the population density at time \( t + 1 \) (denoted as \( N_{t+1} \)), we need to account for the changes in population due to births, deaths, immigration, and emigration.

### Step 4: Formulate the Equation
The change in population can be expressed as:
\[ N_{t+1} = N_t + b + i - d - e \]
Where:
- \( N_t \) is the population density at time \( t \).
- \( b \) is added to account for births.
- \( i \) is added to account for immigration.
- \( d \) is subtracted to account for deaths.
- \( e \) is subtracted to account for emigration.

### Step 5: Rearranging the Equation
We can rearrange the equation to clearly show the relationship:
\[ N_{t+1} = N_t + (b + i - d - e) \]

### Step 6: Identify the Correct Representation
From the rearranged equation, we can see that the correct representation of the density of population after a period of time \( t + 1 \) is:
\[ N_{t+1} = N_t + b + i - d - e \]
This matches with the first option provided in the question.

### Conclusion
Thus, the correct representation of the density of population \( N \) after a period of time \( t + 1 \) is:
\[ N_{t+1} = N_t + b + i - d - e \]

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