Mathematical model of interspecific competition for two protozoan species

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Mathematical model of inter-specific competition
for two protozoan species
Mathematics of Biological Systems 4th Annual
PIMS-MITACS 

• Hassan Khassehkhan, Ross Macdonald and David
  Drolet
• Supervisor Rebecca Tyson

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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

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Introduction
Paramecium caudata
Paramecium aurelia

• Competition for the same food source (bacteria)
• Good system to investigate the dynamic of two
  competing
• species

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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 Gauses data using extension of the
Lotka-Volterra competition model
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Model
Pure cultures logistic growth
Mixed culture Lotka-Volterra competition model
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Logistic growth models Parameter estimation rs
and Ks
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L-V model Parameter estimation ßs
We found ß values minimizing sum of square
deviation between predicted and observed values
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L-V model possible outcomes
Case 1 ß12 lt K1/K2 and ß21 gt K2/K1 Species 1
always out-competes species 2
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L-V model possible outcomes
Case 2 ß12 gt K1/K2 and ß21 lt K2/K1 Species 2
always out-competes species 1
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L-V model possible outcomes
Case 3 ß12 gt K1/K2 and ß21 gt K2/K1 Outcome
depends on initial values
N1(0)N2(0)
N1(0)4 x N2(0)
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L-V model possible outcomes
Case 4 ß12 lt K1/K2 and ß21 lt
K2/K1 Co-existence and populations reach a
steady-state
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L-V model phase plane analysis
Coexistence at the stable steady-state N1450 N2
56
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Does the Lotka-Volterra model fit our data?
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Modified competition model
Where d is a positive constant close to 0
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Modified competition model Long term behavior
(steady-states)
Using numerical method for finding steady state
(Newton method)
Steady state analysis based on estimated
parameters
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Modified competition model Stability analysis
r1 and r2 gt 0 then, (0,0) unstable equilibrium
?1 -0.3667
?2 -1.3316
Asymptotically stable
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Modified competition model Phase-portrait
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Modified competition model Numerical simulation
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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 lt 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
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Acknowledgement
And the volunteer instructors