Problem 1
If a population is zero, its derivative contains that population as a factor and remains zero.
Faculty use
Competing Species and Invariant Regions
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
Nonnegative populations and parameter-dependent equilibria with interpretable stability.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind interpreting coexistence, exclusion, and boundaries in competition models. Use the formulas x′=x(a−bx−cy); y′=y(d−ex−fy).
Problem 4
A strong error analysis identifies the first step that violates classification, algebra, theorem hypotheses, units, initial/boundary data, or numerical assumptions; then repairs that step and re-verifies the final result.
Problem 5
Application responses vary. Require defined variables, a compatible differential-equation model connected to ecology, nonlinear mechanics, chemical oscillations, control, epidemiology, and coupled feedback systems, a complete computation, interpretation, and limitations.
Problem 6
Accept two independent checks appropriate to the task: differentiation/substitution, initial or boundary conditions, dimensional analysis, phase-line or stability evidence, residuals, step refinement, conservation/energy checks, data comparison, or a second method.
Scoring guidance
Six problems × 10 points: 5 mathematical accuracy, 2 structure/justification, 2 verification/interpretation, 1 notation and communication.