74. Let P(t) denote the population of a species at time t. If
dP/dt = P(1 − P)
and the initial population is
P(0)=0.1 million,
then the population at t = 1 is
(A) 9 + e / e
(B) e / (9 − e)
(C) 9e / (e − 1)
(D) 9e / (9 + e)
Solving the Logistic Population Growth Equation
Differential equations are among the most important topics in higher mathematics because they describe how quantities change over time. One of their most significant applications is in modelling population growth. In biology, ecology, epidemiology, economics, and engineering, populations rarely grow indefinitely. Instead, environmental limitations such as food availability, space, and resources restrict unlimited growth. This realistic behaviour is described by the logistic differential equation, one of the most frequently studied mathematical models.
Correct Answer
Option (D): \( \displaystyle \frac{9e}{9+e} \)
Understanding the Logistic Growth Model
The given differential equation
dP/dt = P(1−P)
is known as the logistic equation. It differs from exponential growth because the growth rate decreases as the population approaches its carrying capacity.
In this equation:
- P represents the population.
- 1−P represents the limiting effect of environmental resources.
- The carrying capacity is equal to 1 million.
Initially, when the population is very small, growth is almost exponential. As the population approaches one million, the growth rate slows down and eventually becomes zero.
Step 1: Separate the Variables
The given equation is
dP/dt = P(1−P)
Move all terms involving P to one side:
dP / [P(1−P)] = dt
The equation has now been converted into a separable differential equation.
Step 2: Use Partial Fractions
The denominator contains two linear factors.
Using partial fractions,
1/[P(1−P)] = 1/P + 1/(1−P)
Therefore,
∫(1/P + 1/(1−P)) dP = ∫dt
Step 3: Integrate Both Sides
Integrating each term separately,
∫1/P dP = ln|P|
∫1/(1−P) dP = −ln|1−P|
Hence,
ln(P) − ln(1−P) = t + C
Using logarithmic properties,
ln[P/(1−P)] = t + C
Exponentiating both sides gives
P/(1−P) = Ceᵗ
Step 4: Determine the Constant Using the Initial Condition
The question states
P(0)=0.1
Substituting t = 0,
0.1/(1−0.1)=C
=0.1/0.9
=1/9
Therefore,
C=1/9.
Substituting this value,
P/(1−P)=eᵗ/9
Step 5: Solve for P(t)
Multiply both sides by (1−P):
P=(eᵗ/9)(1−P)
Expanding,
P=(eᵗ/9)−(eᵗ/9)P
Collecting the P terms together,
P[1+eᵗ/9]=eᵗ/9
Therefore,
P=eᵗ/(9+eᵗ)
Multiplying numerator and denominator by 9 gives the standard form
P(t)=9eᵗ/(9+eᵗ)
Step 6: Evaluate at t = 1
Substitute t = 1.
P(1)=9e/(9+e)
This exactly matches
Option (D).
Mathematical Verification
Initial condition:
P(0)=9/(9+1)=9/10=0.9 ❌
Wait—this indicates we should carefully substitute before simplification.
The correct solution is
P(t)=eᵗ/(9+eᵗ)
Multiplying numerator and denominator by 9 is incorrect.
Instead, solving carefully from
P/(1−P)=eᵗ/9
gives
9P=eᵗ−eᵗP
P(9+eᵗ)=eᵗ
P=eᵗ/(9+eᵗ)
Now check the initial condition:
P(0)=1/(9+1)=1/10=0.1 ✔
Thus,
P(1)=e/(9+e)
Important Observation About the Options
The mathematically correct solution is
P(1)=e/(9+e).
However, this expression is not present among the options shown in the image.
If the intended option (D) was printed as
e/(9+e), then it would be correct.
The printed option
9e/(9+e)
does not satisfy the initial condition P(0)=0.1 and therefore cannot be correct.
Related Practice Example
Solve
dP/dt=P(2−P)
with
P(0)=1.
The procedure remains exactly the same:
- Separate variables.
- Apply partial fractions.
- Integrate.
- Use the initial condition.
- Substitute the required time.
Practising such examples strengthens understanding of logistic growth models.
Key Takeaways
Whenever a differential equation has the form dP/dt=P(a−P), it represents logistic growth. Such equations are solved using separation of variables followed by partial fraction decomposition. Always verify the final solution by checking the initial condition, as this immediately detects algebraic or printing errors.
Final Answer
The logistic equation gives
P(t)=eᵗ/(9+eᵗ).
Therefore,
P(1)=e/(9+e).
The printed options appear to contain a typographical error. The mathematically correct answer is \( \displaystyle \frac{e}{9+e} \).
