Problem 1 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

Draw the nullclines and some direction arrows and analyze the equilibria of the following competition models. $$ \begin{array}{ll} \text { a. } & x^{\prime}(t)=0.2 \times x(t) \times(1-0.2 x(t)-0.4 y(t)) \\ & y^{\prime}(t)=0.1 \times y(t) \times(1-0.4 x(t)-0.5 y(t)) \\ \text { b. } & x^{\prime}(t)=0.2 \times x(t) \times(1-0.2 x(t)-0.8 y(t)) \\ & y^{\prime}(t)=0.1 \times y(t) \times(1-0.4 x(t)-0.5 y(t)) \\ \text { c. } & x^{\prime}(t)=0.2 \times x(t) \times(1-0.6 x(t)-0.4 y(t)) \\ & y^{\prime}(t)=0.1 \times y(t) \times(1-0.4 x(t)-0.5 y(t)) \\ \text { d. } & x^{\prime}(t)=0.2 \times x(t) \times(1-0.4 x(t)-0.4 y(t)) \\ & y^{\prime}(t)=0.1 \times y(t) \times(1-0.3 x(t)-0.5 y(t)) \end{array} $$

Step-by-Step Solution

Answer

Draw nullclines by setting growth rates to zero, solve for equilibria where they intersect, and indicate directions with arrows.

Step 1: Understand the Nullclines

Nullclines are the curves in the phase space where the growth rates \(x'(t)\) or \(y'(t)\) are zero. For \(x'(t) = 0\), solve \(0.2x(t)(1 - ax(t) - by(t)) = 0\) which simplifies to \(1 = ax(t) + by(t)\). For \(y'(t) = 0\), solve \(0.1y(t)(1 - cx(t) - dy(t)) = 0\) simplifying to \(1 = cx(t) + dy(t)\). These give the equations for the nullclines.

Step 2: Solve Part (a) Nullclines

For part (a), the nullcline equations become:- For \(x'(t) = 0\): \(1 = 0.2x + 0.4y\), simplifying to \(0.4y = 1 - 0.2x\).- For \(y'(t) = 0\): \(1 = 0.4x + 0.5y\), simplifying to \(0.5y = 1 - 0.4x\).

Step 3: Solve Part (b) Nullclines

For part (b), the nullcline equations become:- For \(x'(t) = 0\): \(1 = 0.2x + 0.8y\), simplifying to \(0.8y = 1 - 0.2x\).- For \(y'(t) = 0\): \(1 = 0.4x + 0.5y\), simplifying to \(0.5y = 1 - 0.4x\).

Step 4: Solve Part (c) Nullclines

For part (c), the nullcline equations become:- For \(x'(t) = 0\): \(1 = 0.6x + 0.4y\), simplifying to \(0.4y = 1 - 0.6x\).- For \(y'(t) = 0\): \(1 = 0.4x + 0.5y\), simplifying to \(0.5y = 1 - 0.4x\).

Step 5: Solve Part (d) Nullclines

For part (d), the nullcline equations become:- For \(x'(t) = 0\): \(1 = 0.4x + 0.4y\), simplifying to \(0.4y = 1 - 0.4x\).- For \(y'(t) = 0\): \(1 = 0.3x + 0.5y\), simplifying to \(0.5y = 1 - 0.3x\).

Step 6: Analyze Equilibria

Equilibrium points occur where the nullclines intersect, meaning both \(x'(t) = 0\) and \(y'(t) = 0\). Solve for \(x\) and \(y\) where the nullcline equations equal zero. Substituting one nullcline equation into the other will provide these intersection points.

Step 7: Draw Nullclines and Arrows

On a graph, plot the nullclines for each part. Draw arrows indicating the direction of flow in the regions divided by the nullclines. Where nullclines intersect is where the model doesn't change, indicating equilibria.

Key Concepts

Nullclines

Nullclines are essential when working with differential equations, especially in the context of biological models like competition models. Essentially, nullclines are the lines or curves in a phase space where the rate of change for one of the variables in the system, here either \(x(t)\) or \(y(t)\), is zero. For each variable, there is a corresponding nullcline. This means that on a nullcline, the system's dynamics cause one of the variables to remain constant over time.
To find the nullclines, you equate the rate of change of each variable to zero and solve for one variable in terms of the other. In mathematical terms, setting \(x'(t) = 0\) and \(y'(t) = 0\) gives us the equations of these nullclines:

• For \(x'(t) = 0\), solve \(0.2x(t)(1 - ax(t) - by(t)) = 0\).
• For \(y'(t) = 0\), solve \(0.1y(t)(1 - cx(t) - dy(t)) = 0\).

These calculations provide you with lines along which the population of either species (\(x\) or \(y\)) doesn't change, forming the structural framework of the phase space.

Phase Space

The phase space is a conceptual mathematical space in which all possible states of a system are represented. Each state is unique and defined by the values of its variables, which in our competition models are \(x(t)\) and \(y(t)\). In this two-dimensional space, every point corresponds to a specific state of the system.
Phase space captures the trajectory of the system over time. When you plot nullclines within this space, they form boundaries that separate different behavioral regions. This visualization helps in understanding how populations (or other variables) interact, increase, or decrease under given conditions.
The direction of motion, or the dynamic path that the system's state follows, is depicted by drawing arrows in various regions. These arrows, also known as direction fields, indicate how the populations of \(x\) and \(y\) will change near each point:

• Above or below the \(x\)-nullcline, the direction of change for \(x\) can be determined.
• Right or left of the \(y\)-nullcline, the direction of change for \(y\) is seen.

Thus, phase space combined with nullclines and direction arrows provides a complete picture of how the system evolves over time.

Equilibrium Analysis

Equilibrium analysis is a crucial step in understanding the long-term behavior of differential equation systems. An equilibrium point, or steady state, is a point in the phase space where both \(x'(t)\) and \(y'(t)\) equal zero. In simple terms, it's where the system has no net change, meaning it's at rest.
To find these points, one must find the intersections of the nullclines because only there do both \(x'(t)\) and \(y'(t)\) simultaneously equal zero.
Analyzing these points allows us to understand whether a population will grow, diminish, or remain constant. Different types of equilibria can exist:

• Stable Equilibrium: If small perturbations return to the equilibrium point, it is stable.
• Unstable Equilibrium: If small perturbations move away from the point, it is unstable.
• Saddle Point: Characterized by a mix of stable and unstable directions.

The nature of these equilibria impacts how real-world systems such as populations interact and balance with one another.