Question

Make a sketch of the population function (as a function of time) that results from the following growth rate functions. Assume the population at time \(t=0\) begins at some positive value.

Short answer

Answer: The general form of the population function is \(P(t) = \int g(t) dt + P_0\), where \(g(t)\) is the growth rate function, \(P_0\) is the initial population, and \(t\) is time.

Step-by-step solution

Express the Growth rate functions.

In this exercise, we are not given specific growth rate functions to work with. So, let's denote the growth rate function as \(g(t)\), which represents the rate of change in population with respect to time, i.e., \(\frac{dP(t)}{dt=}=g(t)\).

Find the Population function.

To find the population function \(P(t)\), integrate the growth rate function with respect to time. In other words, \(P(t) = \int g(t) dt\). After finding the integral, we get the population function including a constant of integration, \(C\).

Determine the initial condition.

As per the exercise's instruction, the population at time \(t = 0\) begins at some positive value. To obtain a specific value, we're going to assume the population at \(t=0\) is \(P_0\), which represents a general initial population. Therefore, the initial condition is given as \(P(0) = P_0\).

Solve for the constant of integration.

Using the initial condition obtained in Step 3, substitute it into the population function equation. So, \(P(0) = \int g(0) dt + C\). This will give us the value of the constant of integration, \(C\), in terms of \(P_0\).

Finalize the Population function.

Now, by substituting the solved value of \(C\) into the population function, we get the desired population function \(P(t)\), considering \(P_0\) and the growth rate function \(g(t)\).

Sketch the Population function.

Finally, sketch the population function in the form \(P(t) = \int g(t) dt + P_0\) over time. Take some example values for \(P_0\) and different forms of growth rate functions \(g(t)\) (like exponential, linear, logistic growth) to get insights into how the population function will behave over time.