27–30. Predator-prey models Consider the following pairs of differential equations that model a predator-prey system with populations x and y. In each case, carry out the following steps.
b. Find the lines along which x'(t) = 0. Find the lines along which y'(t) = 0.


x′(t) = 2x − 4xy, y′(t) = −y + 2xy

Verified step by step guidance

1. Identify the given system of differential equations: \(x'(t) = 2x - 4xy\) and \(y'(t) = -y + 2xy\).
2. To find the lines where \(x'(t) = 0\), set the right-hand side of the first equation equal to zero: \$2x - 4xy = 0$.
3. Factor the equation \$2x - 4xy = 0\( to get \)2x(1 - 2y) = 0\(. This implies either \)x = 0\( or \)1 - 2y = 0$.
4. Solve \$1 - 2y = 0\( to find \)y = \frac{1}{2}\(. So, the lines where \)x'(t) = 0\( are \)x = 0\( and \)y = \frac{1}{2}$.
5. Next, find the lines where \(y'(t) = 0\) by setting \(-y + 2xy = 0\). Factor to get \(y(-1 + 2x) = 0\), which implies \(y = 0\) or \(x = \frac{1}{2}\).

Key Concepts

Predator-Prey Model

A predator-prey model describes the interaction between two species: one as prey (x) and the other as predator (y). The populations change over time based on birth, death, and interaction rates, often modeled by coupled differential equations. Understanding these dynamics helps analyze population stability and oscillations.

Nullclines (Lines where derivatives are zero)

Nullclines are curves in the phase plane where the rate of change of one variable is zero (x' = 0 or y' = 0). They help identify equilibrium points and the system's behavior by showing where populations neither increase nor decrease. Finding nullclines is essential for analyzing system stability.

Solving for Equilibrium Conditions

Equilibrium points occur where both derivatives are zero simultaneously (x' = 0 and y' = 0). Solving these conditions involves setting the differential equations to zero and finding the corresponding population values. These points indicate steady states of the system and are key to understanding long-term behavior.