Problem 75 - Calculus Single Variable

Question

Suppose that, in a certain ecosystem, there is one type of predator and one type of prey. Let \(x \cdot 100\) denote the number of predators and \(y \cdot 1000\) denote the number of prey. The Austrian mathematician A.J. Lotka \((1880-1949)\) and the Italian mathematician Vito Volterra \((1860-1940)\) proposed the following relationship, the so-called Lotka-Volterra Equation, between the two population sizes: $$ \frac{d y}{d x}=\frac{y \cdot(a-b x)}{x \cdot(c y-d)} $$ Separate variables and solve this equation. What is the predator-prey relationship if the initial prey population is \(1500,\) the initial predator population is \(200,\) and if \(a=6,\) \(b=2, c=4,\) and \(d=7 ?\) Figure 11 illustrates the solution curve of this Lotka-Volterra equation. Its potato- pancake shape is typical.

Step-by-Step Solution

Answer

The predator-prey relationship is described by the equation \(2y^2 - 7y = 6 \ln|x| - 2x + C\), with \(C\) determined by initial conditions.

Step 1: Understand the Lotka-Volterra Equation

The given differential equation represents the relationship between predator and prey populations through their respective growth rates. For our problem, we have: \[\frac{d y}{d x} = \frac{y(a-b x)}{x(c y-d)}\]where the parameters \(a\), \(b\), \(c\), and \(d\) affect the growth rates.

Step 2: Substitute Given Values

Substitute the given values into the equation: \(a=6\), \(b=2\), \(c=4\), and \(d=7\). The equation becomes:\[\frac{d y}{d x} = \frac{y(6-2x)}{x(4y-7)}\]

Step 3: Separate Variables

Rearrange the equation for separation of variables. Multiply both sides by \( x(4y-7)\) and divide by \( y(6-2x)\):\[4y - 7\, dy = \frac{y}{x}(6 - 2x)\, dx\]

Step 4: Integrate Both Sides

We integrate separately:- Left side with respect to \( y \): \( \int (4y - 7) \, dy = 2y^2 - 7y + C_1 \)- Right side with respect to \( x \): \( \int \frac{6 - 2x}{x} \, dx = 6 \ln|x| - 2x + C_2 \)

Step 5: Combine Integrals and Solve for Constant

Set the two integrals equal to each other (including constants):\[2y^2 - 7y = 6 \ln|x| - 2x + C\]Apply initial conditions \( x=2, y=1.5 \) to find \( C \). Substitute and solve:\[2(1.5)^2 - 7(1.5) = 6 \ln(2) - 2(2) + C\]Calculate to find \( C \).

Step 6: Final Relationship Between Predators and Prey

With the calculated constant \( C \), the relationship between prey \( y \) and predators \( x \) is complete:\[2y^2 - 7y = 6 \ln|x| - 2x + C \]This equation describes how the populations of predators and prey interact over time under the given conditions.

Key Concepts

Differential Equations

Differential equations are fundamental mathematical tools used to describe the relationship between quantities and their rates of change. In the field of population dynamics, they help us understand how populations change over time. The Lotka-Volterra Equation, which involves the rates of change of predator and prey populations, provides a prominent example. This particular equation is expressed as:
\[ \frac{d y}{d x} = \frac{y(a-b x)}{x(c y-d)} \]
Here, \( \frac{d y}{d x} \) represents the rate at which the prey population changes with respect to the predator population. The constants \( a, b, c, \) and \( d \) influence how these changes occur. Substituting values for these constants helps solve for specific condition dynamics in a real-world situation.

• Purpose: To model the interactions between two species - predators and prey.
• Application: Useful in ecosystem management, biology, and conservation efforts.

Understanding and solving differential equations like these enable predictions and insights into how ecosystems can stabilize or fluctuate, guiding practical decision-making.

Predator-Prey Model

The predator-prey model is a classical model that describes biological interactions between two species: one as a predator and the other as prey. In the predator-prey cycle, the size of each population affects and is affected by the other.

The Lotka-Volterra Equations offer a mathematical framework to represent these dynamics. The basic assumptions include:

• Prey population growth is exponential in the absence of predators.
• Predators reduce prey through consumption, affecting their growth.
• Predator population growth relies on prey availability.
• Both populations have their own natural death rates.

These assumptions help explain why populations fluctuate over time, often represented as cycling through phases of abundance and scarcity. The equations can produce graphically distinctive "potato-pancake" shaped curves. These curves illustrate how predator and prey populations rise and fall in response to each other, creating a continuous balancing act that can be seen in real-world ecosystems.

Population Dynamics

Population dynamics is the study of changes in population sizes and compositions over time and space, often influenced by birth rates, death rates, immigration, and emigration.

In the context of the Lotka-Volterra predator-prey model, these dynamics illustrate how two interdependent species influence one another's growth and decline over time. For example:

• When the prey population increases, predators have more food available, potentially increasing their population size as well.
• Conversely, a large predator population can decrease the prey population due to higher consumption.
• If prey becomes scarce, the predator population may decline due to starvation.

These interactions can lead to cycles of boom and bust within ecosystems. Understanding population dynamics can help determine sustainable population sizes and predict how external factors like environment changes will impact the ecosystem's balance. By applying this knowledge, resource management and conservation strategies can be formulated effectively.