Problem 3 - Applied Calculus

Question

Give the rates of growth of two populations, \(x\) and \(y,\) measured in thousands. (a) Describe in words what happens to the population of each species in the absence of the other. (b) Describe in words how the species interact with one another. Give reasons why the populations might behave as described by the equations. Suggest species that might interact in that way. $$\begin{aligned} &\frac{d x}{d t}=0.01 x-0.05 x y\\\ &\frac{d y}{d t}=0.2 y-0.08 x y \end{aligned}$$

Step-by-Step Solution

Answer

In absence of each other, both species grow exponentially, \( x \) at a slow rate and \( y \) at a faster rate. Interaction terms indicate they compete, possibly predator-prey. Wolves and rabbits are an example.

Step 1: Understanding Growth in Absence of Interaction

To find what happens to each population in the absence of the other, consider setting the interaction terms (those with both x and y) to zero. For \( \frac{d x}{d t} = 0.01 x - 0.05 xy \), if \( y = 0 \), the equation simplifies to \( \frac{d x}{d t} = 0.01 x \). Similarly, for \( \frac{d y}{d t} = 0.2 y - 0.08 xy \), if \( x = 0 \), the equation simplifies to \( \frac{d y}{d t} = 0.2 y \). Thus, in absence of interaction, \( x \) shows an exponential growth with a small rate, while \( y \) grows with a larger rate.

Step 2: Describing Interaction Effects

The interaction terms \( -0.05xy \) and \( -0.08xy \) are negative for both equations, suggesting that as the population of one species increases, it has a negative effect on the growth rate of the other species. This indicates a competitive relationship or predatory-prey-like interaction, where species \( x \) and \( y \) inhibit each other's growth.

Step 3: Identifying Possible Real-Life Interactions

The equation form resembles a Lotka-Volterra model, often used to describe predator-prey interactions. Given the interaction described, species \( x \) and \( y \) could represent a predator (like wolves) and prey (like rabbits). The prey has a higher growth rate when alone, while the predator's existence reduces this growth.

Key Concepts

Predator-Prey Interactions

In predator-prey relationships, two species are interconnected, each influencing the other's growth and survival. These interactions are crucial for understanding how ecosystems maintain balance.
Predators are species that hunt other species, prey, for their survival. Prey populations provide food resources, impacting predator abundance.
On the other hand, the presence of predators places pressure on prey populations, influencing their growth and survival. The Lotka-Volterra model is a great way to understand these dynamics, showcasing how changes in one population directly affect the other.
When prey is abundant, predators thrive, increasing their population. However, as predators increase, they consume more prey, decreasing prey numbers. Eventually, fewer prey causes a decline in the predator population, allowing prey numbers to recover. This cyclical interaction is common in nature.
Examples of predator-prey pairs include:

• Lions and zebras
• Wolves and rabbits
• Foxes and chickens

Population Dynamics

Population dynamics study how populations change over time and the factors influencing these changes. It encompasses birth and death rates, immigration, and emigration, as well as interactions like competition and predation.
In predator-prey dynamics, the birth rate and death rate of each species can vary based on the interaction between the species. Predators may have a high birth rate when prey is abundant. Conversely, prey species may die off quickly in the presence of many predators.
These dynamics can be observed in a boom-and-bust cycle:

• "Boom" occurs when prey species rapidly increase in favorable conditions, leading to predator population growth.
• "Bust" happens when increased predation depletes prey numbers, causing predator populations to decrease due to a food shortage.

Understanding population dynamics helps wildlife managers and ecologists devise strategies for conservation and manage human impacts on wildlife habitats.

Differential Equations

Differential equations describe how a variable changes concerning another.In the Lotka-Volterra predator-prey model, differential equations are used to express changes in populations over time. These equations can reveal patterns and predict future changes:

• For the prey population: The change over time is modeled by the equation: \[ \frac{d x}{d t}=0.01 x-0.05 x y \].
  This equation states that prey population grows by a factor of 0.01 when without predators.
• For the predator population: The corresponding change is given by: \[ \frac{d y}{d t}=0.2 y-0.08 x y \].
  This equation shows that predators rely on the availability of prey for growth.

Understanding these equations helps in grasping how interactive forces between species affect their survival and existence within an environment. By solving these equations, one can predict potential impacts of changes, such as an increase or decrease in one of the populations.