Suppose further that the population’s rate of change is governed by the differential equation
dP/dt = f(P)
where f (P) is the function graphed.
For which values of the population P does the population increase?
(b) For which values of the population P does the population decrease?
(c) If P(0) = 3, how will the population change in time?
(d) If the initial population satisÆes 0 < P(0) < 1, what will happen to the
population after a very long time?
(e) If the initial population satisfies 1 < P(0) < 3, what will happen to the
population after a very long time?
(f) If the initial population satisfies 3 < P(0), what will happen to the population
after a very long time?
(g) This model for a population’s growth is sometimes called “growth with a
threshold.” Explain why this is an appropriate name.
Sorry, no graphs here.
But you know that since f(P) is dP/dt, when f is positive, P is increasing.
As for the other items, no idea, since no picture is available.
Sounds like P is some kind of exponential or logistic model. Better review those topics.
To determine the behavior of the population as described by the differential equation dP/dt = f(P), where f(P) is the given function, we need to analyze the sign of f(P) at different values of P. Here's how you can find the answers to the given questions:
(a) For which values of the population P does the population increase?
To identify where the population increases, we need to look for values of P where f(P) > 0. This indicates that the rate of change (derivative) of P with respect to time is positive, meaning the population is growing.
(b) For which values of the population P does the population decrease?
To find where the population decreases, we need to look for values of P where f(P) < 0. This indicates that the rate of change (derivative) of P with respect to time is negative, meaning the population is shrinking.
(c) If P(0) = 3, how will the population change in time?
When P(0) = 3, it means the initial population is 3. To determine how the population changes in time, you can substitute this value into the differential equation and solve it to find the corresponding behavior. You can integrate the differential equation to obtain the resulting population function.
(d) If the initial population satisfies 0 < P(0) < 1, what will happen to the population after a very long time?
Based on the given graph, it appears that for values of P between 0 and 1, the function f(P) is positive. Therefore, the population will increase after a long time and approach some higher value.
(e) If the initial population satisfies 1 < P(0) < 3, what will happen to the population after a very long time?
For values of P between 1 and 3, the graph of f(P) is negative. This means the population will decrease after a long time and approach some lower value.
(f) If the initial population satisfies 3 < P(0), what will happen to the population after a very long time?
When the initial population is greater than 3, the graph of f(P) indicates that it is positive. Thus, the population will increase after a long time and approach some higher value.
(g) This model for a population's growth is sometimes called "growth with a threshold." Explain why this is an appropriate name.
The name "growth with a threshold" is appropriate because as we observe from the graph of f(P), there is a range of population values (threshold) where the growth behavior changes. Below the threshold, the population decreases, and above the threshold, the population increases. This threshold acts as a boundary, causing a shift in the population's growth pattern.