Problem 43 - Precalculus: Functions and Graphs

Question

Competition for food A competition model is a collection of equations that specifies how two or more species interact in competition for the food resources of an ecosystem. Let \(x\) and \(y\) denote the numbers (in hundreds) of two competing species, and suppose that the respective rates of growth \(R_{1}\) and \(R_{2}\) are given by $$\begin{array}{l}R_{1}=0.01 x(50-x-y) \\\R_{2}=0.02 y(100-y-0.5 x)\end{array}$$ Determine the population levels \((x, y)\) at which both rates of growth are zero. (Such population levels are called stationary points.)

Step-by-Step Solution

Answer

The stationary points are \((0, 100)\) and \((50, 0)\).

Step 1: Set the Growth Rates to Zero

To find the stationary points, we need to set the growth rates \( R_1 \) and \( R_2 \) to zero. This gives us the system of equations:\[ 0.01 x (50 - x - y) = 0 \]\[ 0.02 y (100 - y - 0.5x) = 0 \]

Step 2: Solve for First Species Population

For the equation \( 0.01 x (50 - x - y) = 0 \), either \( x = 0 \) or \( 50 - x - y = 0 \). Solving for \( 50 - x - y = 0 \), we get:\[ y = 50 - x \]

Step 3: Solve for Second Species Population

For the equation \( 0.02 y (100 - y - 0.5x) = 0 \), either \( y = 0 \) or \( 100 - y - 0.5x = 0 \). Solving for \( 100 - y - 0.5x = 0 \), we find:\[ y = 100 - 0.5x \]

Step 4: Find Common Solutions

Combine the results from steps 2 and 3:1. From \( y = 50 - x \) and \( y = 100 - 0.5x \): - Solving \( 50 - x = 100 - 0.5x \) results in \( x = 100 \), \( y = -50 \), which isn't biologically feasible because population cannot be negative.2. Check scenarios where either \( x = 0 \) or \( y = 0 \). - \( x = 0 \): \( y = 100 \) from \( y = 100 - 0.5(0) \). - \( y = 0 \): \( x = 50 \) from \( 50 - x - 0 = 0 \).

Step 5: Determine Stationary Points

The feasible stationary points are therefore:1. \( (x, y) = (0, 100) \)2. \( (x, y) = (50, 0) \)

Key Concepts

Stationary Points

Stationary points in a mathematical model are critical values where a function, such as a growth rate, equals zero. This means there are no changes occurring at these points.
For species competing in an ecosystem, stationary points represent steady states where the populations do not grow or shrink. They remain constant over time given the set conditions.
In our competition model, we determine stationary points by setting the growth rates of the species to zero. This involves solving the equations derived from the growth rates, like:

• \( 0.01 x (50 - x - y) = 0 \)
• \( 0.02 y (100 - y - 0.5x) = 0 \)

Solving these equations helps identify equilibrium population levels where neither species increases or decreases in number.

Growth Rates

Growth rates reflect the change in population numbers over time. In the context of species interactions within an ecosystem, these rates can indicate how fast one species grows in relation to another.
In the competition model, growth rates are defined by mathematical equations:

• For the first species: \( R_1 = 0.01 x (50 - x - y) \)
• For the second species: \( R_2 = 0.02 y (100 - y - 0.5x) \)

These equations highlight how each species' growth is affected not only by its own population size but also by the presence of a competitor.
The function's coefficients and parameters depict specific ecological factors, such as carrying capacity and interspecies competition, which play crucial roles in shaping these growth dynamics.

Ecosystem

An ecosystem is a complex network of interactions among living organisms and their environment. It includes both biotic (living) and abiotic (non-living) components that work in harmony to sustain life.
In a balanced ecosystem, species coexist by occupying different niches and competing for resources like food, water, and space.
The competition model focuses on the food resource aspect, where two species compete for survival. These models help visualize the impact of limited resources on species' populations over time.
In understanding and analyzing these ecosystems, ecologists often study models to predict how populations might interact and change. This insight is crucial for conservation efforts and managing biodiversity.

Species Interaction

Species interaction encompasses the various ways organisms can affect each other's survival and reproduction in an ecosystem. One significant type of interaction is competition, where different species vie for the same resource.
In the model we've discussed, interaction occurs through food resource competition. The presence and abundance of one species can directly impact the growth and sustainability of another.
Species interactions also include other dynamics like predation, mutualism, and parasitism, each shaping the community structure in unique ways.

• Competition is often modeled mathematically as it allows prediction of outcomes based on specific assumptions and parameters.
• Understanding these interactions helps in conserving threatened species and managing pest populations effectively.

Learning how species interact enables us to predict changes in biodiversity and ecosystem health.