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.
a. Identify which equation corresponds to the predator and which corresponds to the prey.


x′(t) = −3x + xy, y′(t) = 2y − xy

Verified step by step guidance

1. Examine the given system of differential equations: \(x'(t) = -3x + xy\) and \(y'(t) = 2y - xy\).
2. Recall that in predator-prey models, the prey population typically grows exponentially in the absence of predators, and the predator population declines without prey.
3. Look at the terms without interaction: For \(x'(t)\), the term \(-3x\) suggests that \(x\) decreases exponentially when \(y=0\), indicating \(x\) is the predator population.
4. For \(y'(t)\), the term \$2y\( suggests that \)y\( grows exponentially when \)x=0\(, indicating \)y$ is the prey population.
5. Confirm that the interaction terms \(xy\) have opposite signs in each equation, representing the predator-prey interaction where predators benefit from prey and prey are harmed by predators.

Key Concepts

Predator-Prey Model Dynamics

Predator-prey models describe interactions between two species: one as prey and the other as predator. The prey population typically grows in absence of predators, while the predator population depends on consuming prey. These models use differential equations to capture how populations change over time based on these interactions.

Interpreting Differential Equations in Population Models

Each differential equation represents the rate of change of a population. Positive terms indicate growth factors, while negative terms represent decline. Interaction terms, often products of both populations, model how one species affects the other's growth or decline, helping identify which equation corresponds to predator or prey.

Equilibrium and Stability in Predator-Prey Systems

Equilibrium points occur when population rates of change are zero, indicating steady states. Analyzing these points helps understand long-term behavior of the system. Stability analysis determines whether populations return to equilibrium after disturbances, crucial for predicting population survival or extinction.