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Q2:
If a population growing exponentially double in size in 3 years, what is the intrinsic rate of increase (\(r\)) of the population?
Solution :
Given Variables & Initial Setup
In population ecology, when resources are unlimited, a population exhibits exponential growth. The integral form of the exponential growth equation is given by:
$$ N_t = N_0 e^{rt} $$
Where:
- $N_t$ = Population density/size after time $t$
- $N_0$ = Population density/size at time $t = 0$ (initial population)
- $r$ = Intrinsic rate of natural increase (per capita growth rate)
- $t$ = Time period (in years)
- $e$ = The base of natural logarithms (Euler's number, $\approx 2.71828$)
Based on the problem statement, the population doubles in size in 3 years. Therefore, we can establish the following parameters:
- $t = 3 \text{ years}$
- $N_t = 2N_0$ [Because the population is exactly twice its original size]
Step 1: Setting up the Equation
We substitute the known parameters into the standard exponential growth model to solve for the unknown variable, $r$:
$$ 2N_0 = N_0 e^{r(3)} $$
By dividing both sides of the equation by $N_0$, we isolate the exponential term [Per the division property of equality]:
$$ 2 = e^{3r} $$
Graphical Representation of Exponential Doubling
The following dynamically calculated SVG illustrates the exponential curve of the population over time, highlighting the exact doubling point at $t = 3$.
Step 2: Logarithmic Evaluation
To solve for $r$, which is located in the exponent, we apply the natural logarithm ($\ln$, log base $e$) to both sides of the equation. This utilizes the inverse property of exponential functions, where $\ln(e^x) = x$:
$$ \ln(2) = \ln(e^{3r}) $$
$$ \ln(2) = 3r $$
Now, isolate $r$ algebraically by dividing both sides by 3:
$$ r = \frac{\ln(2)}{3} $$
Step 3: Numerical Calculation
The natural logarithm of 2 is a fundamental constant in population genetics and radioactive decay modeling:
$$ \ln(2) \approx 0.6931 $$
Substitute this constant back into the equation:
$$ r = \frac{0.6931}{3} $$
$$ r \approx 0.23103... $$
Rounding to three decimal places (the standard precision in demographic ecology), we get:
$$ r = 0.231 \text{ year}^{-1} $$
If expressed as a percentage, the population is growing at an intrinsic rate of approximately $23.1\%$ per year.
Final Solution: The intrinsic rate of increase ($r$) of the population is approximately $0.231$ (or $23.1\%$) per year.
More Questions from Class 12 Biology Organisms and Populations EXERCISES
- Q1: List the attributes that populations possess but not individuals.
- Q1: List the attributes that populations possess but not individuals.
- Q2: If a population growing exponentially double in size in 3 years, what is the intrinsic rate of increase ((r)) of the population?
- Q3: Name important defence mechanisms in plants against herbivory.
- Q3: Name important defence mechanisms in plants against herbivory.
- Q4: An orchid plant is growing on the branch of mango tree. How do you describe this interaction between the orchid and the mango tree?
- Q4: An orchid plant is growing on the branch of mango tree. How do you describe this interaction between the orchid and the mango tree?
- Q5: What is the ecological principle behind the biological control method of managing with pest insects?
- Q5: What is the ecological principle behind the biological control method of managing with pest insects?
- Q6: Define population and community.
- Q6: Define population and community.
- Q7(a): Define the following terms and give one example for each: (a) Commensalism
- Q7(a): Define the following term and give one example for it: Commensalism
- Q7(b): Define the following terms and give one example for each: (b) Parasitism
- Q7(b): Define the following term and give one example for it: Parasitism
- Q7(c): Define the following terms and give one example for each: (c) Camouflage
- Q7(c): Define the following term and give one example for it: Camouflage
- Q7(d): Define the following terms and give one example for each: (d) Mutualism
- Q7(d): Define the following term and give one example for it: Mutualism
- Q7(e): Define the following terms and give one example for each: (e) Interspecific competition
- Q7(e): Define the following term and give one example for it: Interspecific competition
- Q8: With the help of suitable diagram describe the logistic population growth curve.
- Q8: With the help of suitable diagram describe the logistic population growth curve.
- Q9: Select the statement which explains best parasitism.
- Q9: Select the statement which explains best parasitism.
- Q10: List any three important characteristics of a population and explain.
- Q10: List any three important characteristics of a population and explain.
CBSE Solutions for Class 12 Biology Organisms and Populations
Chapters in CBSE - Class 12 Biology
- Reproduction in Organisms
- Human Reproduction
- Reproductive Health
- Principles of Inheritance and Variation
- Molecular Basis of Inheritance
- Evolution
- Human Health and Disease
- Strategies for Enhancement in Food Production
- Microbes in Human Welfare
- Biotechnology: Principles and Processes
- Biotechnology and Its Applications
- Ecosystem
- Biodiversity and Conservation
- Environmental Issues
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