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

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 caudataParamecium 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

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

10. L-V model: possible outcomes Case 1: β 12 K 2 /K 1 Species 1 always out-competes species 2

11. L-V model: possible outcomes Case 2: β 12 > K 1 /K 2 and β 21 < K 2 /K 1 Species 2 always out-competes species 1

12. L-V model: possible outcomes Case 3: β 12 > K 1 /K 2 and β 21 > K 2 /K 1 Outcome depends on initial values N 1 (0)=N 2 (0)N 1 (0)=4 x N 2 (0)

13. L-V model: possible outcomes Case 4: β 12 < K 1 /K 2 and β 21 < K 2 /K 1 Co-existence and populations reach a steady-state

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

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

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

17. Modified competition model: Long term behavior (steady-states) Using numerical method for finding steady state (Newton method ) Steady state analysis based on estimated parameters Paramete r value β 12 3.9 β 21 0.86 ε 1 0.65 ε 2 0.13

18. Modified competition model: Stability analysis r 1 and r 2 > 0 then, (0,0) unstable equilibrium λ 1 = -0.3667 λ 2 = -1.3316 Asymptotically stable

19. N1 N2 Modified competition model: Phase-portrait

20. Modified competition model: Numerical simulation

21. Modified competition model: Comparison with L-V Likelihood ratio test H 0 : Both models fit data equally well H 1 : One model fits the data better Chi square = 84.14, d.f.=2, p < 0.0001, thus, we reject H 0 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

22. Acknowledgement And the volunteer instructors