Join thousands of students who trust us to help them ace their exams!Watch the first video
Multiple Choice
Assuming that r has a positive value, in the formula dN/dt = rmaxN(K – N)/K, the factor rN tends to cause the population to do what?
A
Remain stable at the carrying capacity
B
Decrease in size
C
Grow at a slower rate than the (K – N/K) factor
D
None of the listed responses is correct.
E
Grow increasingly rapidly
Verified step by step guidance
1
Understand the formula: The equation \( \frac{dN}{dt} = r_{max}N \frac{(K - N)}{K} \) is a logistic growth model, where \( dN/dt \) represents the rate of change of the population size \( N \) over time \( t \).
Identify the components: \( r_{max} \) is the maximum per capita rate of increase, \( N \) is the population size, \( K \) is the carrying capacity, and \( (K - N)/K \) is the factor that reduces growth as the population approaches the carrying capacity.
Analyze the role of \( rN \): The term \( rN \) represents the intrinsic growth rate of the population. When \( r \) is positive, \( rN \) contributes to the population's growth.
Consider the effect of \( (K - N)/K \): This factor decreases as \( N \) approaches \( K \), slowing the growth rate. However, when \( N \) is much smaller than \( K \), \( (K - N)/K \) is close to 1, allowing \( rN \) to drive rapid growth.
Conclude the impact: Since \( rN \) is a positive factor, it tends to cause the population to grow increasingly rapidly, especially when \( N \) is far from \( K \).