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