Problem 3 - Calculus: Early Transcendentals

Question

The system of differential equations \( \frac {dx}{dt} = 0.5x - 0.004x^2 - 0.001xy \) \( \frac {dy}{dt} = 0.4y - 0.001y^2 - 0.002xy \) is a model for the populations of two species. (a) Does the model describe cooperation, or competition, or a predator-prey relationship? (b) Find the equilibrium solutions and explain their significance.

Step-by-Step Solution

Answer

(a) The model describes competition. (b) Equilibrium: (0,0), (125,0), (0,400), (50,100); (50,100) shows coexistence.

Step 1: Analyze the Interaction Terms

To determine the type of relationship, look at the interaction terms in the equations. The term \(-0.001xy\) in the first equation and \(-0.002xy\) in the second equation both suggest negative effects from interactions between species, implying competition.

Step 2: Set Up Equations for Equilibrium

Equilibrium occurs when \( \frac{dx}{dt} = 0 \) and \( \frac{dy}{dt} = 0 \). We set the right-hand sides of both differential equations to zero:\[ 0.5x - 0.004x^2 - 0.001xy = 0 \]\[ 0.4y - 0.001y^2 - 0.002xy = 0 \].

Step 3: Solve for Equilibrium Points - Species Zero

First, consider if either species leads to zero by substituting \(x = 0\) and \(y = 0\). For \(x = 0\), the second equation becomes \(0.4y - 0.001y^2 = 0\), leading to solutions \(y = 0\) or \(y = 400\). Similarly, for \(y = 0\), solve \(0.5x - 0.004x^2 = 0\), leading to solutions \(x = 0\) or \(x = 125\).

Step 4: Solve for Coexistence Equilibrium

To find non-zero coexistence solutions, solve the system:\[ 0.5x - 0.004x^2 - 0.001xy = 0 \]\[ 0.4y - 0.001y^2 - 0.002xy = 0 \].By substituting \(y = \frac{0.5x - 0.004x^2}{0.001x}\) into the second equation and solving, we find the coexistence equilibrium: \((50, 100)\).

Step 5: Interpret Equilibrium Solutions

The equilibrium solutions are \((0, 0)\), \((125, 0)\), \((0, 400)\), and \((50, 100)\). The solutions \((0, 0)\), \((125, 0)\), and \((0, 400)\) represent cases where one or both species are extinct. The solution \((50, 100)\) represents a stable coexistence point where both species survive together.

Key Concepts

Equilibrium Solutions in Differential Equations

In differential equations involving population models, an equilibrium solution occurs when the rate of change for each species is zero. This implies stable population sizes, where neither species is increasing nor decreasing. For the given system of differential equations:

• \( \frac{dx}{dt} = 0.5x - 0.004x^2 - 0.001xy \)
• \( \frac{dy}{dt} = 0.4y - 0.001y^2 - 0.002xy \)

To find the equilibrium solutions, we set each differential equation to zero. Solving these yields the equilibrium solutions:

• \((0, 0)\): Both species are extinct.
• \((125, 0)\): The first species survives alone.
• \((0, 400)\): The second species survives alone.
• \((50, 100)\): Both species coexist in balance.

The coexistence solution \((50, 100)\) is significant because it represents a balanced state where both species can sustain themselves in the environment without one outcompeting the other. Understanding these solutions helps predict long-term outcomes in population dynamics.

Species Population Model

Species population models use mathematical equations to describe how populations of species change over time. These models incorporate factors such as growth rates, carrying capacities, and interactions between species. In the given model:

• The equation for species \(x\) is \( \frac{dx}{dt} = 0.5x - 0.004x^2 - 0.001xy \), which includes linear growth, competition within the species, and competition with species \(y\).
• The equation for species \(y\) is \( \frac{dy}{dt} = 0.4y - 0.001y^2 - 0.002xy \), which also includes linear growth, competition within its species, and competition with species \(x\).

These equations model population numbers based on:

• The intrinsic growth rate (e.g., \(0.5x\) and \(0.4y\)), showing the growth in ideal conditions without limitation.
• The carrying capacity, limiting population when numbers are high (e.g., \(-0.004x^2\), \(-0.001y^2\)).
• Inter-species effects (e.g., \(-0.001xy\), \(-0.002xy\)), which are negative indicating competition.

The goal of such a model is to forecast how species populations will evolve and interact, helping conservationists and ecologists develop strategies for species management.

Understanding Competition Interactions

In ecological models, interaction terms in the differential equations reveal the nature of the species relationships. For this system, both equations include terms that represent competition interactions.

• For species \(x\) the competition term is \(-0.001xy\).
• For species \(y\) the competition term is \(-0.002xy\).

These negative terms imply that the presence of one species negatively affects the growth of the other. This tells us that the species are competitors rather than mutual helpers or predator-prey pair:

• Competition limits the resources available, hampering each other's capacity to thrive.
• The interaction strengths (coefficients of \(-0.001\) and \(-0.002\)) quantify how strongly the competition affects each species' growth.

Understanding these interactions is crucial for predicting the model's outcomes and for making decisions on managing the species environments. It helps in evaluating how changes in population numbers of one species may affect the other over time.