Differential Equations for Engineers · Laplace transforms, discontinuous forcing, and impulses
For the weak-competition system x′ = x(4 − x − y), y′ = y(3 − x/2 − y), find and…
Problem
For the weak-competition system x′ = x(4 − x − y), y′ = y(3 − x/2 − y), find and classify all first-quadrant equilibria, prove that positive solutions are bounded, rule out periodic orbits in the open first quadrant with a Bendixson–Dulac argument, and determine the limit of every trajectory with x(0), y(0) > 0.
Hint
Weak competition means each carrying capacity lies below the competitor’s intercept, so both species can invade the other’s axial state.
Check your work
Work the problem yourself first. Then open it in Training to check your answer and read the full worked solution.
The answer check and full solution for this problem come with ProofAnvil Practice membership ($19 USD monthly). See membership. Or start with the free Differential Equations for Engineers sample problem: Try the free sample problem.
More Differential Equations for Engineers practice problems
- A 400-liter tank initially contains 80 liters of brine with 12 kg of dissolved saltModeling and first-order differential equations
- Consider the initial-value problem (x^2 + 9) y'' + 5x y' − 4y = 0, y(0) = 2, y'(0) = −1Linear higher-order equations and oscillatory response
- A taut string of length 6 occupies 0 ≤ x ≤ 6Linear higher-order equations and oscillatory response
- A colony initially contains 800 bacteria and grows at a rate proportional to the square…Linear systems, eigenstructure, and matrix exponentials
- Two masses m_1=m_2=1 lie on a frictionless line and are coupled by three springs of…Laplace transforms, discontinuous forcing, and impulses
- At each of x_0=0,1,-2, classify the point as ordinary, regular singular, or irregular…Nonlinear systems, phase planes, and stability
- In the cube 0<x<π, 0<y<π, 0<z<1, solve u_(xx)+u_(yy)+u_(zz)=0 with zero Dirichlet data…Boundary-value problems and PDE separation methods
- Resolve the startup layer hidden by a quasi-steady sensor modelModeling and first-order differential equations
- Reduce a coupled fast–slow subsystem without losing its startup stateLinear systems, eigenstructure, and matrix exponentials