1 / 22

Mathematical model of inter-specific competition for two protozoan species

Mathematics of Biological Systems 4th Annual PIMS-MITACS . Mathematical model of inter-specific competition for two protozoan species. Hassan Khassehkhan, Ross Macdonald and David Drolet Supervisor: Rebecca Tyson. Outline. Description of the system

saniya
Télécharger la présentation

Mathematical model of inter-specific competition for two protozoan species

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Mathematics of Biological Systems4th Annual PIMS-MITACS  Mathematical model of inter-specific competition for two protozoan species Hassan Khassehkhan, Ross Macdonald and David Drolet Supervisor: Rebecca Tyson

  2. Outline • Description of the system • Logistic growth and competition models (Lotka-Volterra) • Modified model • Long term behavior • Comparison of modified model with L-V model

  3. Introduction Paramecium caudata Paramecium aurelia • Competition for the same food source (bacteria) • Good system to investigate the dynamic of two competing • species

  4. Methods used by Gause • Pure culture of both species in controlled medium • Mixed culture • Daily estimation of population density for a period of • 25 days • Medium was changed daily to prevent depletion of • resources Objective Revisiting Gause’s data using extension of the Lotka-Volterra competition model

  5. Model Pure cultures: logistic growth Mixed culture: Lotka-Volterra competition model

  6. Logistic growth models: Parameter estimation r’s and K’s

  7. L-V model: Parameter estimation β’s We found β values minimizing sum of square deviation between predicted and observed values

  8. L-V model: possible outcomes Case 1: β12 < K1/K2 and β21 > K2/K1 Species 1 always out-competes species 2

  9. L-V model: possible outcomes Case 2: β12 > K1/K2 and β21 < K2/K1 Species 2 always out-competes species 1

  10. L-V model: possible outcomes Case 3: β12 > K1/K2 and β21 > K2/K1 Outcome depends on initial values N1(0)=N2(0) N1(0)=4 x N2(0)

  11. L-V model: possible outcomes Case 4: β12 < K1/K2 and β21 < K2/K1 Co-existence and populations reach a steady-state

  12. N2 N1 L-V model: phase plane analysis Coexistence at the stable steady-state N1=450 N2= 56

  13. Does the Lotka-Volterra model fit our data?

  14. Modified competition model Where δ is a positive constant close to 0

  15. Modified competition model: Long term behavior (steady-states) Using numerical method for finding steady state(Newton method) Steady state analysis based on estimated parameters

  16. Modified competition model: Stability analysis r1 and r2 > 0 then, (0,0) unstable equilibrium λ1= -0.3667 λ2= -1.3316 Asymptotically stable

  17. N2 N1 Modified competition model: Phase-portrait

  18. Modified competition model: Numerical simulation

  19. Modified competition model: Comparison with L-V Likelihood ratio test H0 : Both models fit data equally well H1 : One model fits the data better Chi square = 84.14, d.f.=2, p < 0.0001, thus, we reject H0 Residual sum of squares of the new model is less than that of L-V RSS of new model = 21 500 RSS of L-V = 119 713 RSS of the new model is 6 times smaller than that of L-V

  20. Acknowledgement And the volunteer instructors

More Related