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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
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
Introduction
Paramecium caudata Paramecium aurelia
-Competition for the same food source (bacteria)
-Good system to investigate the dynamic of two competing species
Methods used by Gause
-Pure culture of both species in controlled medium
- Mixed culture
Daily estimation of population density for a period of25 days
Medium was changed daily to prevent depletion of resources
Objective
Revisiting Gause’s data using extension of the Lotka-Volterracompetition model
Model
Pure cultures: logistic growth
Mixed culture: Lotka-Volterra competition model
Logistic growth models: Parameter estimation
r’s and K’s
L-V model: Parameter estimationβ’s
We found β values minimizing sum of square deviation betweenpredicted and observed values
L-V model: possible outcomes
Case 1:β12 < K1/K2 and β21 > K2/K1
Species 1 always out-competes species 2
L-V model: possible outcomes
Case 2:β12 > K1/K2 and β21 < K2/K1
Species 2 always out-competes species 1
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)
L-V model: possible outcomes
Case 4:β12 < K1/K2 and β21 < K2/K1
Co-existence and populations reach a steady-state
L-V model: phase plane analysis
N1
N2
Coexistence at thestable steady-state
N1=450N2= 56
Does the Lotka-Volterra model fit our data?
Modified competition model
Where δ is a positive constant close to 0
Modified competition model:Long term behavior (steady-states)
Using numerical method for finding steady state (Newton method)
Steady state analysis based on estimated parameters
Parameter
value
β12 3.9
β21 0.86
ε 1 0.65
ε 2 0.13
Modified competition model:Stability analysis
r1 and r2 > 0 then, (0,0) unstable equilibrium
λ1= -0.3667
λ2= -1.3316
Asymptotically stable
N1
N2
Modified competition model:Phase-portrait
Modified competition model:Numerical simulation
Modified competition model:Comparison with L-V
Likelihood ratio test
H0 : Both models fit data equally wellH1 : 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 500RSS of L-V = 119 713
RSS of the new model is 6 times smaller than that of L-V
Acknowledgement
And the volunteer instructors
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