Problem 1 - 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. $$\begin{aligned} &\frac{d x}{d t}=0.2 x\\\ &\frac{d y}{d t}=0.4 x y-0.1 y \end{aligned}$$

Step-by-Step Solution

Answer

Species \(x\) grows exponentially alone, \(y\) declines alone; \(x\) enhances \(y\)'s growth, suggesting mutualism.

Step 1: Analyzing the Growth of Population X

The rate of change of population \(x\) is given by the differential equation \(\frac{d x}{d t} = 0.2 x\). This is a simple exponential growth equation, indicating that in the absence of species \(y\), the population \(x\) grows continuously at an exponential rate of 0.2 per unit time. This means population \(x\) will grow without bound as long as no other factors affect it.

Step 2: Analyzing the Growth of Population Y

The rate of change of population \(y\) is given by \(\frac{d y}{d t} = 0.4 x y - 0.1 y\). In the absence of species \(x\), the term \(0.4 x y\) drops to zero, leaving \(\frac{d y}{d t} = -0.1 y\). This indicates that population \(y\) decreases exponentially due to a natural decay rate of 0.1 per unit time.

Step 3: Understanding the Interaction between X and Y

The interaction between the populations is described by the term \(0.4 x y\) in the equation for \(y\). This implies that population \(x\) positively affects the growth of population \(y\). Specifically, as the population of \(x\) increases, it contributes to the growth of population \(y\) by providing more of some beneficial effect or resource, suggesting a mutualistic or commensal relationship.

Step 4: Suggesting Rational Species Interactions

Given the equations, one could suggest that species \(x\) and \(y\) might reflect a relationship like that of a flowering plant (species \(x\)) and a pollinator insect (species \(y\)). Here, as the number of plants \(x\) increases, they provide more food or pollination opportunities, hence boosting the growth of the insect population \(y\). Meanwhile, the plant itself grows independently of the insect presence.

Key Concepts

Differential Equations

Differential equations are mathematical tools used to describe how things change. In population dynamics, they help us understand how populations grow or shrink over time. A differential equation like \( \frac{d x}{d t} = 0.2 x \) tells us that the population \( x \) changes at a rate proportional to itself. This equation is simple because it models the change in the population over time without including other factors. It shows pure exponential growth for population \( x \) because the rate of change depends solely on the population size at that moment. Understanding these equations is like knowing the rules of a game. Knowing these rules allows us to predict how a population might behave in an unchanging environment. Similarly, for \( y \)'s rate of change, the equation \( \frac{d y}{d t} = 0.4 x y - 0.1 y \) means that the change in \( y \) is influenced by both its current size and other factors, like population \( x \). This equation helps us see that interactions between species can majorly affect population dynamics.

Exponential Growth

Exponential growth describes how populations can increase rapidly when each member contributes to a constant rate of increase. For population \( x \), characterized by the equation \( \frac{d x}{d t} = 0.2 x \), this means that it doubles over regular intervals when left unchecked. Each individual in the population leads to more growth, making the number of individuals grow faster over time.

• In exponential growth, the bigger the population, the faster it grows.
• The growth is unlimited unless other factors, like scarcity or predators, limit it.

Exponential growth is common in ecosystems when resources are abundant, and there are no significant threats. However, unchecked exponential growth is rare in nature because resources eventually run out or conditions change, leading to stability or decline. That's why understanding the beginning phases of population growth with exponential models is crucial for understanding potential future scenarios.

Species Interaction

Species interaction refers to how different species in an ecosystem affect one another. In our exercise, the interaction comes from the term \( 0.4 x y \) in the differential equation for \( y \), suggesting that species \( x \) positively impacts species \( y \). This term models a mutualistic interaction where the increase in species \( x \) leads to beneficial effects on species \( y \).

• Positive interaction: Species \( x \) positively influences the growth of species \( y \).
• Mutualism: Both parties benefit from the interaction.
• Dependency: Species \( y \) relies on species \( x \) for its positive growth rate.

Such interactions can occur between a flowering plant and its pollinator. As the plant population increases, more resources become available for the pollinator, boosting its population. This mutual benefit promotes biodiversity and stability in ecosystems. Understanding these interactions helps us realize the interconnectedness of life and the importance of balancing ecosystems.