Problem 2 - 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 isolation, species \(x\) grows while \(y\) decays. Together, \(x\) is prey for \(y\).

Step 1: Analyze Each Species Independently

First, consider the scenario where only one species is present, meaning the interactions with the other species are non-existent. For species \(x\), remove the \(-0.05xy\) term, resulting in \(\frac{dx}{dt} = 0.01x\). This equation suggests exponential growth because the rate of growth is positive. Similarly, for species \(y\), ignoring the \(0.08xy\) term gives \(\frac{dy}{dt} = -0.2y\), indicating an exponential decay because the rate is negative.

Step 2: Consider Interactions Between Species

Next, observe how the species interact as indicated by the terms involving both \(x\) and \(y\). The term \(-0.05xy\) in the equation for \(x\) suggests that the presence of \(y\) negatively impacts the growth of \(x\). Similarly, the term \(0.08xy\) in the equation for \(y\) implies that \(x\)'s presence positively impacts \(y\). Thus, this interaction can be interpreted as \(x\) being prey to \(y\), hinting at a predator-prey dynamic where \(x\) increases \(y\)'s growth while \(y\) inhibits \(x\)\.

Step 3: Apply Contextual Knowledge to Interpret Species Interaction

Given the mathematical reflection of interactions similar to a predator-prey model, realistic pairs such as rabbits (as \(x\)) and wolves (as \(y\)) can illustrate this relationship. Rabbits naturally tend to grow exponentially in numbers when undisturbed, while wolves depend on preying on rabbits for survival and growth. The negative impact of wolves on rabbits and the positive influence of rabbits on wolves are consistent with the terms in the equations.

Key Concepts

Population Dynamics

Population dynamics is a way to describe how populations of species change over time. This involves understanding the factors that drive the increase or decrease of population sizes. In our scenario with species \(x\) and \(y\), the dynamics are driven by their growth rates as well as their interactions with each other.
On their own, each species behaves differently. Without the presence of another species, \(x\) grows exponentially as indicated by the positive growth rate (0.01), while \(y\) experiences exponential decay due to its negative growth rate (-0.2).

• Positive growth indicates an increase in population over time.
• Negative growth indicates a decrease in population over time.

These dynamics can become quite complex once species interact, reflecting real-world ecosystems where the presence or absence of certain species can deeply affect the population sizes of others.

Exponential Growth

Exponential growth describes a situation where a population grows by a constant proportion each period. Mathematically, it is characterized by a rate of change that is directly proportional to the current size of the population. The presence of a positive growth rate contributes to exponential growth.
For species \(x\), the equation \(\frac{dx}{dt} = 0.01x\) when not considering interactions represents a classic case of exponential growth. This means that as time increases, the size of the population \(x\) increases multiplicatively.
The process of exponential growth can be observed in many natural populations where resources are ample, leading to rapid increase in numbers. However, this type of growth is often unsustainable in the long term as resources become limited or other conditions change.

Exponential Decay

Exponential decay describes a situation where a population decreases by a constant proportion over time. This is often represented by a negative rate of growth. For species \(y\), the equation \(\frac{dy}{dt} = -0.2y\) in the absence of interaction with species \(x\) demonstrates this concept.
In exponential decay, the rate of population decline is significant as it is proportional to the current population size. This could reflect situations where resources are scarce or where there are high mortality rates due to environmental pressures.

• The population decreases faster when it is larger.
• Over time, the decrease slows as the population size reduces.

Real-world examples include species that experience high levels of predation or disease, leading to a shrinkage in size without positive interaction or intervention.

Species Interaction

Species interaction is a central theme in understanding how populations impact each other's growth and survival. In the given equations, the interaction terms \(-0.05xy\) and \(0.08xy\) provide insight into this dynamic between species \(x\) and \(y\).
The interaction term \(-0.05xy\) indicates that the presence of species \(y\) negatively affects the growth of \(x\). Meanwhile, \(0.08xy\) implies a positive effect on \(y\) due to \(x\)'s presence. This suggests a predator-prey model where species \(x\) acts as prey and species \(y\) acts as a predator.

• Positive interactions increase growth or survival of a species.
• Negative interactions decrease growth or survival.

Classic examples of predator-prey relationships include rabbits and wolves, where wolf populations grow by preying on rabbits, and inversely, rabbit populations can be limited by wolf predation.