Math 2.0

Biology and other natural sciences Example

Lotka–Volterra equations

Builds on Equilibrium pointFirst integralPeriodic orbitLinearisation at an equilibrium

Example

Let α,β,γ,δ>0\alpha, \beta, \gamma, \delta > 0 be constants. Let x(t)>0x(t) > 0 be the size of a prey population and y(t)>0y(t) > 0 the size of a predator population at time tt. The Lotka–Volterra equations are the autonomous system x′=αx−βxy,y′=δxy−γy.x' = \alpha x - \beta x y, \qquad y' = \delta x y - \gamma y. The equilibrium points in the closed quadrant x,y≥0x, y \ge 0 are (0,0)(0, 0) and (γ/δ,α/β)(\gamma / \delta, \alpha / \beta). The function V(x,y)=δx−γln⁡x+βy−αln⁡yV(x, y) = \delta x - \gamma \ln x + \beta y - \alpha \ln y is a first integral on the open quadrant x,y>0x, y > 0.

Remarks

Along a solution, ddtV=(δx−γ)(α−βy)+(βy−α)(δx−γ)=0\frac{d}{dt} V = (\delta x - \gamma)(\alpha - \beta y) + (\beta y - \alpha)(\delta x - \gamma) = 0. The linearisation at (0,0)(0, 0) has the eigenvalues α\alpha and −γ-\gamma, so the origin is a saddle. At (γ/δ,α/β)(\gamma/\delta, \alpha/\beta) the eigenvalues are ±iαγ\pm i \sqrt{\alpha \gamma}. The level curves of VV in the open quadrant are closed curves around (γ/δ,α/β)(\gamma/\delta, \alpha/\beta). So each solution other than this equilibrium is periodic: the two populations oscillate.

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Differential equation, concept, no evidence levelDifferential equationconceptFunction, concept, no evidence levelFunctionconceptInterval, concept, no evidence levelIntervalconceptOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionLipschitz condition, definition, no evidence levelLipschitz conditiondefinitionGrönwall's inequality, statement, no evidence levelGrönwall's inequalitystatementOrder of a differential equation, definition, no evidence levelOrder of a differenti…definitionSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionAutonomous system, definition, no evidence levelAutonomous systemdefinitionInitial value problem, definition, no evidence levelInitial value problemdefinitionLinear differential equation of order n, definition, no evidence levelLinear differential e…definitionEquilibrium point, definition, no evidence levelEquilibrium pointdefinitionSystem of linear differential equations, definition, no evidence levelSystem of linear diff…definitionUniqueness of solutions for Lipschitz vector fields, statement, no evidence levelUniqueness of solutio…statementHomogeneous linear system with constant coefficients, definition, no evidence levelHomogeneous linear sy…definitionMaximal solution, definition, no evidence levelMaximal solutiondefinitionMatrix exponential as the solution of a linear system, statement, no evidence levelMatrix exponential as…statementFlow of a vector field, definition, no evidence levelFlow of a vector fielddefinitionLyapunov stability, definition, no evidence levelLyapunov stabilitydefinitionTrajectory, definition, no evidence levelTrajectorydefinitionAsymptotic stability, definition, no evidence levelAsymptotic stabilitydefinitionFirst integral, definition, no evidence levelFirst integraldefinitionPeriodic orbit, definition, no evidence levelPeriodic orbitdefinitionStability of linear systems via eigenvalues, statement, no evidence levelStability of linear s…statementLinearisation at an equilibrium, concept, no evidence levelLinearisation at an e…conceptLotka–Volterra equations, example, no evidence levelLotka–Volterra equati…example
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Differential equation concept
  2. Function concept
  3. Interval concept
  4. Ordinary differential equation definition
  5. Lipschitz condition definition
  6. Grönwall's inequality statement
  7. Order of a differential equation definition
  8. Solution of an ordinary differential equation definition
  9. Autonomous system definition
  10. Initial value problem definition
  11. Linear differential equation of order n definition
  12. Equilibrium point definition
  13. System of linear differential equations definition
  14. Uniqueness of solutions for Lipschitz vector fields statement
  15. Homogeneous linear system with constant coefficients definition
  16. Maximal solution definition
  17. Matrix exponential as the solution of a linear system statement
  18. Flow of a vector field definition
  19. Lyapunov stability definition
  20. Trajectory definition
  21. Asymptotic stability definition
  22. First integral definition
  23. Periodic orbit definition
  24. Stability of linear systems via eigenvalues statement
  25. Linearisation at an equilibrium concept
  26. Lotka–Volterra equations example

Further reading

  1. Web https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations, Population equilibrium
  2. Web https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations, Hamiltonian structure of the system
About this page
Tag00Y5QZ82AA
URIm2://00Y5QZ82AA
Aliasespredator-prey-equations, lotka-volterra-predator-prey-model
Kindexample
WritingW2 usable
MSC92D25, 34C25
Instance ofAutonomous system
UsesEquilibrium point, First integral, Periodic orbit, Linearisation at an equilibrium
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