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.


c. Find the equilibrium points for the system.


x′(t) = −3x + 6xy, y′(t) = y − 4xy

Verified step by step guidance

1. Start by writing down the system of differential equations clearly: \[x\prime(t) = -3x + 6xy\] \[y\prime(t) = y - 4xy\]
2. To find the equilibrium points, set both derivatives equal to zero because equilibrium occurs where the populations do not change: \[-3x + 6xy = 0\] \[y - 4xy = 0\]
3. Factor each equation to isolate terms: From the first equation: \[x(-3 + 6y) = 0\] From the second equation: \[y(1 - 4x) = 0\]
4. Solve each factor equal to zero separately to find possible values of \(x\) and \(y\): - For the first equation: either \[x = 0\] or \[-3 + 6y = 0\] - For the second equation: either \[y = 0\] or \[1 - 4x = 0\]
5. Combine these results to find all pairs \((x, y)\) that satisfy both equations simultaneously. These pairs are the equilibrium points of the system.

Key Concepts

Equilibrium Points in Differential Equations

Equilibrium points occur where the rates of change of all variables are zero, meaning the system is in a steady state. For a system of differential equations, this means setting each derivative equal to zero and solving for the variables. These points help understand long-term behavior and stability of the system.

Predator-Prey Model Dynamics

Predator-prey models describe interactions between two species: one as prey (x) and the other as predator (y). The equations typically include growth and interaction terms, where prey growth is affected negatively by predators, and predator growth depends on prey availability. Understanding these dynamics is essential to interpret the system's behavior.

Solving Nonlinear Systems of Equations

Finding equilibrium points often requires solving nonlinear algebraic equations derived from setting derivatives to zero. Techniques include substitution or factoring to find all possible solutions. Mastery of solving such systems is crucial for analyzing complex models like predator-prey interactions.