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. $$ \begin{array}{l} \frac{d x}{d t}=0.01 x-0.05 x y \\ \frac{d y}{d t}=-0.2 y+0.08 x y \end{array} $$

Step-by-Step Solution

Answer

Species \(x\) grows alone, \(y\) declines alone; together, \(y\)'s predation hinders \(x\), aids \(y\). Potential species: rabbits (\(x\)) and foxes (\(y\)).

Step 1: Understanding independent growth of each population

In the absence of the other population, each population would grow according to only its internal factors. For species \(x\), if there were no \(y\), its rate of change would be determined by \(\frac{dx}{dt} = 0.01x\). This indicates that \(x\) grows by 1% every unit of time, assuming \(y = 0\). For species \(y\), without \(x\), the equation \(\frac{dy}{dt} = -0.2y\) implies that \(y\) is decreasing or dying out by 20% every unit of time as \(x = 0\). Thus, \(x\) is increasing, and \(y\) is decreasing.

Step 2: Understanding the interaction term

To understand how the species interact, examine the terms involving both \(x\) and \(y\) in the equations. For \(x\), \(-0.05xy\) indicates that the presence of \(y\) negatively impacts \(x\) at a rate proportional to their populations. This suggests a predatory or competitive interaction where \(y\) consumes or competes with \(x\), reducing \(x\)'s growth rate compared to when \(y\) is absent. For \(y\), the term \(0.08xy\) suggests the presence of \(x\) provides a benefit, possibly food or a positive interaction, enhancing \(y\)'s growth rate.

Step 3: Suggesting species interactions

Based on the analysis, species \(x\) could be a prey species and \(y\) a predator, such as rabbits and foxes. Without foxes, rabbits multiply, but the presence of foxes reduces their growth rate through predation. Conversely, without rabbits, foxes would decline due to lack of food, but with rabbits, foxes increase as they have more food to sustain their population. Alternatively, mice and owls fit this model where an increase in mice leads to more food for owls, allowing their population to grow.

Key Concepts

Population Dynamics

Population dynamics refers to how populations of species change over time, particularly in response to environmental conditions and interactions with other species. This concept involves understanding patterns such as growth, decay, and fluctuations in population sizes. These changes can be modeled using differential equations, which mathematically describe the rates at which these populations grow or shrink.

In the given differential equations, each species' population size changes according to not only their individual growth or death rates but also their interactions with each other.

• Species like population \( x \) might increase due to natural birth rates or sufficient resources.
• Conversely, population \( y \) might decrease due to higher mortality or limited resources.

Understanding these dynamics helps in predicting future population sizes and managing ecological systems effectively.

Predator-Prey Model

The predator-prey model is a fundamental concept in population dynamics and describes interactions between two species where one, the predator, feeds on the other, the prey. This relationship is characterized using the famous Lotka-Volterra equations, which help to capture the cyclical nature of such interactions.

In our example, the given equations show:

• Prey population \( x \), in the absence of predators \( y \), would naturally grow over time due to reproduction and resource availability.
• However, the presence of predator \( y \) decreases \( x \) through predation, which is shown by a negative term \(-0.05xy\).
• Meanwhile, the predator population \( y \), which declines without prey, gains growth from consuming prey, shown by a positive term \(0.08xy\).

The interaction between predators and prey results in oscillations in their population sizes. When prey is abundant, predators increase in number, which in turn reduces prey numbers, leading to a decline in predators. This model is crucial for understanding and predicting the dynamics of ecological food chains.

Rate of Change

The rate of change in differential equations quantifies how quickly a variable, such as population size, changes over time. This is typically represented by derivatives, such as \( \frac{dx}{dt} \) for the rate of change of population \( x \).

In our exercise:

• The term \(0.01x\) shows that population \( x \) increases at a rate of 1% per unit time if alone.
• Conversely, the term \(-0.2y\) indicates that population \( y \) decreases by 20% per unit time without \( x \).

By incorporating terms such as \( -0.05xy \) and \(0.08xy\), these equations show how interaction rates imply that species affect each other's growth rates.

This information allows scientists to predict population trends and make informed decisions about wildlife conservation or ecosystem management. Understanding rate of change is essential in exploring how populations respond faster or slower to changes in their environment or due to interspecies interactions.