Question

Population Growth When predicting population growth, demographers must consider birth and death rates as well as the net change caused by the difference between the rates of immigration and emigration. Let \(P\) be the population at time \(t\) and let \(N\) be the net increase per unit time resulting from the difference between immigration and emigration. So, the rate of growth of the population is given by $$ \frac{d P}{d t}=k P+N $$ where \(N\) is constant. Solve this differential equation to find \(P\) as a function of time, when at time \(t=0\) the size of the population is \(P_{0} .\)

Short answer

The solution to the differential equation given is \(P(t) = (P_0 + \frac{N}{k})e^{kt}-\frac{N}{k}\)

Step-by-step solution

Separate the Variables

We rewrite equation \(\frac{d P}{d t}=k P+N\) to separate variables into the form \(\frac{1}{P+N/k} dP = k dt\). This is done by subtracting \(N\) from both sides, then dividing by \(P + \frac{N}{k}\), separating \(P\) terms and \(t\) terms.

Integrate Both Sides

Now integrate both sides to get the antiderivatives. \[\int \frac{1}{P+N/k} dP = \int k dt\] After carrying out the integrations, we get: \[\ln | P + \frac{N}{k} | = kt + C \] where \(C\) is the constant of integration.

Solve for \(P\)

We now solve for \(P\), which involves getting rid of the natural logarithm (using the property that \(e^{\ln a} = a\)) and then isolating \(P\): \[e^{kt+C} = P + \frac{N}{k} \] \[P = e^{kt+C} - \frac{N}{k} \] We can absorb a part of this constant \(C\) into a new constant, say \(A\), because \(e^C\) is also a constant. So, the equation becomes: \[P = Ae^{kt} - \frac{N}{k} \]

Apply Initial Condition

We apply the given initial condition, \(P(0) = P_0\), to solve for the constant \(A\): \[P_0 = Ae^{k*0} - \frac{N}{k} \] This simplifies to \[P_0 = A - \frac{N}{k} \] Solving for \(A\) gives \[A = P_0 + \frac{N}{k} \]

Write Down the Final Solution

Now we substitute \(A\) back to the earlier equation of \(P\), we obtain the solution of the first order differential equation: \[P(t) = (P_0 + \frac{N}{k})e^{kt}-\frac{N}{k}\]

Key concepts

Population Growth

Population growth is a critical concept that demographers often analyze to predict how a population will change over time. This concept takes into account several factors:

• Birth rates: The number of new individuals born in a particular period.
• Death rates: The number of individuals who die in a particular period.
• Net migration: This is determined by the difference between the rates of immigration (people moving in) and emigration (people moving out). This net change is often denoted by a constant, in this case, represented as \(N\).

When we compute the rate of population growth, we see how these factors influence the number of individuals in a population. The differential equation \(\frac{dP}{dt} = kP + N\) shows how population size \(P\) changes over time \(t\), with \(k\) being the proportional growth rate and \(N\) accounting for net change due to migration. Understanding these aspects helps us make predictions and policy decisions related to human demographics.

Integration Techniques

Integration techniques are essential in solving differential equations, especially when we need to find an antiderivative or a function whose derivative is known. In our population growth problem, we meet a differential equation: \(\frac{dP}{dt} = kP + N\). To find \(P\), we need to integrate the separated variables.Here's what happens:

• We first rewrite the equation in terms of integrable parts: \(\int \frac{1}{P+N/k} \mathrm{d}P = \int k \mathrm{d}t\).
• Integrating \(\frac{1}{P+N/k}\) leads us to a natural log expression: \(\ln |P + \frac{N}{k}| = kt + C\), where \(C\) is the constant of integration.

Remember, the antiderivative helps find a general solution, and constants play a crucial role in adjusting the solution to fit specific starting points or conditions.

Separation of Variables

Separation of variables is a clever mathematical technique used to solve differential equations by isolating different variables on different sides of the equation. In the population growth example, this technique is employed to solve \(\frac{dP}{dt} = kP + N\).The process involves:

• Rearranging the equation: Subtract \(N\) to isolate terms including \(P\) on one side, giving \(\frac{1}{P+N/k} dP = k dt\).
• Separating the forms: This involves manipulating the equation so \(P\) terms are exclusively in one integral and \(t\) terms in another.

By successfully isolating \(P\) from \(t\), we can integrate each side accordingly and eventually solve for \(P\) without any implicit functions involving \(t\). This technique is most effective for first-order differential equations, which are prevalent in real-world applications.

Initial Conditions

Initial conditions are pivotal in the application of mathematical models to real-world situations. They are the given values at a starting point that allow us to find specific solutions within a family of general solutions. In our example of population growth, the exercise provides an initial condition: the population size at time \(t = 0\), labeled \(P_0\).Here's how it works:

• We integrate our separated equation to get a solution involving a constant \(A\).
• To determine this constant, we need to substitute \(P(0) = P_0\) into our general equation \(P(t) = (P_0 + \frac{N}{k})e^{kt}-\frac{N}{k}\).
• This substitution allows us to solve for \(A\), refining our solution to match the initial population condition.

By applying these initial conditions, we transition from a broad general solution to a precise answer that accurately reflects the initial state of the population.