The model for exponential growth coupled to exponential death is d𝑆(𝑡)d𝑡=𝑟𝑆(𝑡)−𝑐𝑆(𝑡). Which of the following statements about the value of r/c is CORRECT?Group of answer choicesIf it is smaller than one, the population size will increaseIf it is smaller than zero, the population size will decreaseIf it is greater than zero, the population size will decreaseIf it is greater than one, the population size will increase
Question
The model for exponential growth coupled to exponential death is d𝑆(𝑡)d𝑡=𝑟𝑆(𝑡)−𝑐𝑆(𝑡). Which of the following statements about the value of r/c is CORRECT?Group of answer choicesIf it is smaller than one, the population size will increaseIf it is smaller than zero, the population size will decreaseIf it is greater than zero, the population size will decreaseIf it is greater than one, the population size will increase
Solution
The correct statement about the value of r/c in the given model for exponential growth coupled to exponential death is: "If it is greater than one, the population size will increase".
Here's why:
The equation dS(t)/dt = rS(t) - cS(t) can be rewritten as dS(t)/dt = (r - c)S(t). This means the rate of change of the population size S(t) depends on the difference between the growth rate r and the death rate c.
If r/c > 1, it means r > c, i.e., the growth rate is greater than the death rate. Therefore, the population size will increase.
On the other hand, if r/c < 1, it means r < c, i.e., the growth rate is less than the death rate. Therefore, the population size will decrease.
So, the correct answer is "If it is greater than one, the population size will increase".
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