Problem 4 - Applied Calculus

Question

The following system of differential equations represents the interaction between two populations, \(x\) and \(y\). $$ \begin{array}{l} \frac{d x}{d t}=-3 x+2 x y \\ \frac{d y}{d t}=-y+5 x y \end{array} $$ (a) Describe how the species interact. How would each species do in the absence of the other? Are they helpful or harmful to each other? (b) If \(x=2\) and \(y=1\), does \(x\) increase or decrease? Does \(y\) increase or decrease? Justify your answers. (c) Write a differential equation involving \(d y / d x\). (d) Use a computer or calculator to draw the slope field for the differential equation in part (c). (e) Draw the trajectory starting at point \(x=2, y=1\) on your slope field, and describe how the populations change as time increases.

Step-by-Step Solution

Answer

The species interact mutually; at \(x=2\), \(y=1\), \(x\) decreases, \(y\) increases; \(dy/dx = \frac{-y + 5xy}{-3x + 2xy}\).

Step 1: Understanding the interaction between species

To understand how the species interact, we examine the system of differential equations. The equation \( \frac{d x}{d t} = -3x + 2xy \) suggests that the species \( x \) decreases due to the term \(-3x \) but increases in the presence of \( y \) with term \(2xy\). This implies \( y \) helps \( x \). Conversely, from \( \frac{d y}{d t} = -y + 5xy \), the species \( y \) decreases with term \(-y\) but increases due to interaction with \( x \) (term \(5xy\)), meaning \( x \) helps \( y \). Thus, both species benefit from the presence of each other, indicating a mutualistic interaction. Without the other, the species would decline over time.

Step 2: Determine the change in populations at specific values

Given \(x = 2\) and \(y = 1\), we substitute these values into the system of equations. For \( \frac{dx}{dt} = -3(2)+2(2)(1) = -6+4 = -2\), \(x\) decreases. For \( \frac{dy}{dt} = -(1) + 5(2)(1) = -1 + 10 = 9\), \(y\) increases. Thus, \(x\) decreases while \(y\) increases at these values.

Step 3: Formulate the differential equation involving \(dy/dx\)

To find \( \frac{dy}{dx} \), use the chain rule which yields \( \frac{dy}{dx} = \/ \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \). Substituting from the differential equations, we get \( \frac{dy}{dx} = \frac{-y + 5xy}{-3x + 2xy} \). This expression allows us to relate changes in \(y\) to changes in \(x\).

Step 4: Constructing the slope field (using computational tools)

To draw the slope field for the equation from Step 3, input \( \frac{dy}{dx} = \frac{-y + 5xy}{-3x + 2xy} \) into graphing software. This will plot the directional fields that visually depict the gradients of change for various points \((x, y)\).

Step 5: Analyzing the trajectory from a starting point

Beginning at point \((x = 2, y = 1)\), follow the slope field. As previously calculated, \(y\) increases while \(x\) decreases at this point; thus, the trajectory will show \(y\) moving upwards and \(x\) moving downward initially. Over time, the interaction may stabilize if reaching an equilibrium point, often depicted by a steady state within the slope field.

Key Concepts

Population Dynamics

Population dynamics is a key topic in the study of biological systems. It involves understanding how and why the number of individuals in a population changes over time. This can include birth rates, death rates, immigration, and emigration. Differential equations are a common tool used to model such dynamics as they can provide insights into how populations interact with each other and their environment.

In this exercise, we are using a system of differential equations to model the population dynamics between two species, represented by variables \(x\) and \(y\). These populations could represent, for example, two species within an ecosystem interacting with each other. The equations indicate how each species' population changes based on their current size as well as the size of the other population.

• The first equation \( \frac{d x}{d t} = -3x + 2xy \) indicates that population \(x\) has a natural decline rate (due to the \(-3x\) term) but also grows due to the interaction term \(2xy\).
• The second equation \( \frac{d y}{d t} = -y + 5xy \) shows that population \(y\) also naturally declines (\(-y\) term) but grows significantly with the term \(5xy\).

Understanding these equations helps explain the bigger picture of how certain species might thrive or diminish over time based on internal and external interactions, showcasing their importance in ecology and conservation strategies.

Mutualistic Interaction

Mutualistic interaction occurs when two species interact in a manner that is beneficial to both parties involved. In the context of this system of differential equations, both species \(x\) and \(y\) appear to help each other thrive.

The interaction terms \(2xy\) and \(5xy\) in their respective differential equations suggest mutualistic interaction. This type of interaction benefits both species, ensuring that each population supports the growth of the other:

• For \(x\), the presence of \(y\) through the \(2xy\) term counters its natural decline by a boosting effect that can lead to better survival and reproduction rates.
• Similarly, for \(y\), the presence of \(x\) provides a strong positive influence (as seen with the \(5xy\) term), contributing to \(y\)'s population growth.

This mutualistic relationship highlights a balanced interdependent ecosystem. It shows how in nature, certain species can develop a relationship that is equally beneficial, resulting in both populations improving over time. Such analysis is critical in predicting how ecosystems manage stressors and change.

Slope Field Analysis

Slope field analysis is a visual tool used in differential equations to understand how solutions behave without actually solving the equation analytically. It provides an interpretative viewpoint by sketching a "field" of small line segments, each aligning with the slope of the solution curve at that point.

To perform slope field analysis for the differential equation derived from this exercise, we use \( \frac{dy}{dx} = \frac{-y + 5xy}{-3x + 2xy} \). By plotting this in a graphing software:

• The slopes give insight into how the populations \(y\) and \(x\) change with respect to each other.
• At each point on the plot, a small line segment represents the direction in which the populations change, revealing trends over time.

For our specific exercise starting at \(x = 2, y = 1\):

The slope field would allow students to visually track how the population trajectory moves. Initially, as calculated, \(y\) increases and \(x\) decreases, which is visually observable as a part of the trajectory on the slope field. With further analysis, we can see how these populations may balance towards an equilibrium or observe dynamic behaviors such as oscillations. This intuitive graphical technique provides a powerful tool for understanding complex dynamic systems modeled by differential equations.