Problem 9 - Thomas Calculus

Question

In 1925 Lotka and Volterra introduced the predator-prey equations, a system of equations that models the populations of two species, one of which preys on the other. Let \(x(t)\) represent the number of rabbits living in a region at time \(t,\) and \(y(t)\) the number of foxes in the same region. As time passes, the number of rabbits increases at a rate proportional to their population, and decreases at a rate proportional to the number of encounters between rabbits and foxes. The foxes, which compete for food, increase in number at a rate proportional to the number of encounters with rabbits but decrease at a rate proportional to the number of foxes. The number of encounters between rabbits and foxes is assumed to be proportional to the product of the two populations. These assumptions lead to the autonomous system $$ \begin{array}{l}{\frac{d x}{d t}=(a-b y) x} \\ {\frac{d y}{d t}=(-c+d x) y}\end{array} $$ where \(a, b, c, d\) are positive constants. The values of these constants vary according to the specific situation being modeled. We can study the nature of the population changes without setting these constants to specific values. What happens to the rabbit population if there are no foxes present?

Step-by-Step Solution

Answer

Without foxes, the rabbit population grows exponentially as \(x(t) = x(0)e^{at}\).

Step 1: Understand the given equations

We are given the system of equations: \( \frac{d x}{d t} = (a - b y) x \) and \( \frac{d y}{d t} = (-c + d x) y \). These equations describe the rate of change of rabbit and fox populations, where \(x(t)\) is the number of rabbits and \(y(t)\) is the number of foxes. The constants \(a, b, c, d\) are all positive.

Step 2: Set the fox population to zero

To find out what happens to the rabbit population when there are no foxes, we set \(y = 0\). By substituting \(y = 0\) into the first equation, we get: \(\frac{d x}{d t} = a x\).

Step 3: Solve the simplified equation for rabbits

The equation \(\frac{d x}{d t} = a x\) is a first-order linear differential equation. The solution to this equation is given by exponential growth: \(x(t) = x(0)e^{at}\), where \(x(0)\) is the initial population of rabbits. As \(a\) is positive, the rabbit population will grow exponentially over time.

Key Concepts

Lotka-Volterra model

The Lotka-Volterra model, often referred to as the predator-prey model, is a pair of first-order, non-linear, differential equations. These equations describe the dynamics of biological systems in which two species interact, where one is a predator and the other is its prey. This model highlights critical ecological interactions and is foundational in the study of population biology.

This model can be formulated as follows for two populations: rabbits (prey) and foxes (predators):

• Prey population, represented by \(x(t)\), changes according to: \(\frac{d x}{d t} = (a - b y) x\)
• Predator population, represented by \(y(t)\), alters as: \(\frac{d y}{d t} = (-c + d x) y\)

The interaction between species is modeled based on the rates of reproduction and consumption between the prey and predators. While it's an idealized model, it's pivotal in understanding ecological stability and interaction dynamics.

Differential equations

Differential equations are mathematical equations that involve derivatives, representing rates of change. They are essential for modeling real-world phenomena where changing values are interconnected, such as in physics, engineering, and biology.

The predator-prey equations are examples of such differential equations that model the rate of change of populations over time. Specifically, the Lotka-Volterra model uses first-order differential equations to describe the growth of prey and changes in predator populations. These equations are:

• \(\frac{d x}{d t} = (a - b y) x\) - Describes the dynamic changes in the prey population.
• \(\frac{d y}{d t} = (-c + d x) y\) - Shows the changes in the predator population.

By solving these differential equations, one can predict population sizes over time and understand the interactions between different species.

Exponential growth

Exponential growth is a process that increases quantity over time, and it's characterized by the rate of growth being proportional to the current size, leading to faster and faster growth as time goes on.

In the context of the Lotka-Volterra model without predators, the rabbit population exhibits exponential growth. When there are no foxes present (i.e., predators are removed), the equation simplifies to:

• \(\frac{d x}{d t} = a x\)

This can be solved to show that the population growth of rabbits follows:

• \(x(t) = x(0)e^{at}\)

Where \(x(0)\) is the initial rabbit population. Because "a" is positive, the population grows exponentially, doubling at regular intervals, illustrating unchecked growth due to the absence of predators.

Autonomous systems

Autonomous systems in the context of differential equations refer to systems where the change in a variable over time doesn't explicitly depend on time itself. Instead, these systems rely on other variables within the system.

The Lotka-Volterra predator-prey model is an example of an autonomous system. Here, the differential equations that describe the system do not have explicit time dependence:

• \(\frac{d x}{d t} = (a - b y) x\)
• \(\frac{d y}{d t} = (-c + d x) y\)

Because of this structure, the system's future behavior is determined entirely by its current state, not by how long it has been evolving. Autonomous systems are advantageous for their simplicity and are widely used to model natural processes where events depend on interactions and current states rather than external time factors.