Problem 1
The logistic growth peaks at rK/4=25 units/time.
Faculty use
Population Harvesting and Threshold Management
Reward correct mathematics, explicit units, conditions, and hypotheses, interpretation, and verification. Equivalent valid reasoning should receive credit.
Problem 2
No positive equilibrium remains; the model predicts eventual collapse.
Problem 3
Definitions should identify the equation type, dependent and independent variables, parameters, conditions, units or domain, and the hypotheses behind analyzing logistic growth modified by constant or proportional removal. Use the formulas P′=rP(1−P/K)−H; equilibria solve rP(1−P/K)=H.
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 thermal systems, tanks and reactors, mechanics, ecology, pharmacology, and electrical circuits, 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.