Problem 38

Question

Population Model The differential equation in Example 3 is a well-known population model. Suppose the DE is changed to $$ \frac{d P}{d t}=P(a P-b) $$ where $a$ and $b$ are positive constants. Discuss what happens to the population $P$ as time $t$ increases.

Step-by-Step Solution

Verified

Answer

The population stabilizes at $ P = \frac{b}{a} $ as time increases.

1Step 1: Identify Equilibrium Points

To find the equilibrium points, set the differential equation equal to zero: $ \frac{dP}{dt} = P(aP - b) = 0 $. This implies that either $ P = 0 $ or $ aP - b = 0 $. Solving $ aP - b = 0 $ gives $ P = \frac{b}{a} $. Thus, the equilibrium points are $ P = 0 $ and $ P = \frac{b}{a} $.

2Step 2: Analyzing Stability of Equilibrium Points

Examine the stability of each equilibrium point. For $ P = 0 $, consider the sign of $ aP - b $ when $ P > 0 $. Since $ aP - b < 0 $ when $ P < \frac{b}{a} $, the population decreases, indicating $ P = 0 $ is an unstable equilibrium. For $ P = \frac{b}{a} $, when $ P > \frac{b}{a} $, $ aP - b > 0 $, making the population increase towards the equilibrium, indicating $ P = \frac{b}{a} $ is stable.

3Step 3: Long-term Behavior of the Population

From the stability analysis, if the initial population $ P(0) > 0 $, it will increase toward $ P = \frac{b}{a} $ if $ P(0) < \frac{b}{a} $, or decrease towards $ P = \frac{b}{a} $ if $ P(0) > \frac{b}{a} $. Therefore, over time, the population will stabilize to $ P = \frac{b}{a} $, irrespective of whether it starts above or below this value. This indicates that $ P = \frac{b}{a} $ acts as an attractor for the population.

Key Concepts

Equilibrium PointsStability AnalysisDifferential Equations

Equilibrium Points

In the context of population dynamics, equilibrium points represent states where the population size remains constant over time. These are the values of population, $ P $, for which the growth rate $ \frac{dP}{dt} $ is zero. To identify equilibrium points, we set the differential equation equal to zero:

$ \frac{dP}{dt} = P(aP - b) = 0 $
This gives two potential equilibrium points: $ P = 0 $ and $ aP - b = 0 $.

Solving $ aP - b = 0 $ yields $ P = \frac{b}{a} $. Thus, our equilibrium points are $ P = 0 $ and $ P = \frac{b}{a} $.
The equilibrium point $ P = 0 $ represents a scenario where the population is nonexistent. In contrast, $ P = \frac{b}{a} $ suggests a stable population size that could exist in balance with resources and conditions given by $ a $ and $ b $. Understanding these points is vital for predicting long-term population trends and ensuring sustainable management of populations.

Stability Analysis

Stability analysis helps determine whether a population will return to an equilibrium point after small disturbances. For each equilibrium, we assess how the population tends to move around it.

At $ P = 0 $: We examine the sign of $ aP - b $ for $ P > 0 $. Here, $ aP - b<0 $ when $ P < \frac{b}{a} $, which implies that the population decreases, indicating instability. Thus, if population ever grows to be non-zero, it will not return to $ P = 0 $.
At $ P = \frac{b}{a} $: Analyze what happens when $ P $ is slightly greater or lesser than $ \frac{b}{a} $. If $ P > \frac{b}{a} $, then $ aP - b > 0 $, making the population shrink towards $ \frac{b}{a} $, and vice-versa for $ P < \frac{b}{a} $. Hence, this point is considered stable as populations tend to converge to this equilibrium.

Through stability analysis, we understand that $ P = \frac{b}{a} $ serves as a stable attractor for the population, guiding it toward a state of balance even if it starts at a different level.

Differential Equations

Differential equations, such as $ \frac{dP}{dt} = P(aP - b) $, are mathematical formulations that describe the rate at which quantities change. In population dynamics, they model how populations evolve over time based on current population size and interactions like birth, death, and limits like food or space.

In this population model, the equation incorporates a quadratic factor $ aP^2 $ which represents population growth, while the linear term $-bP$ might indicate some form of limiting factor.
These factors combined give insight into cooperative interactions within the population, such as resource limitation when $ P $ goes beyond a certain level.

The beauty of differential equations lies in their ability to predict how populations behave and reach equilibrium points under various scenarios — offering invaluable insight for researchers, ecologists, and policy-makers to address environmental and societal challenges.

Problem 38

Problem 39

Other exercises in this chapter

Problem 38

Often a radical change in the form of the solution of a differential equation corresponds to a very small change in either the initial condition or the equation

View solution

Problem 38

The differential equation in Example 3 is a well-known population model. Suppose the DE is changed to $$ \frac{d P}{d t}=P(a P-b) $$ where $a$ and $b$ are p

View solution

Problem 39

Express the solution of the initial-value problem $y^{\prime}-2 x y=1$, $y(1)=1$, in terms of $\operatorname{erf}(x)$

View solution

Problem 39

(a) Show that a one-parameter family of solutions of the equation $$ \left(4 x y+3 x^{2}\right) d x+\left(2 y+2 x^{2}\right) d y=0 $$ is \(x^{3}+2 x^{2} y+y^{2}

View solution