Understanding Lotka-Volterra Model in Mathematical Modelling
Explore the dynamics of predator-prey systems through the Lotka-Volterra model, including equilibrium points, behavior around equilibria, linearization, eigenvalue analysis, and classification of equilibria based on real and complex eigenvalues.
- Mathematical Modelling
- Lotka-Volterra Model
- Predator-Prey Systems
- Equilibrium Analysis
- Eigenvalue Analysis
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Presentation Transcript
Mathematical Modelling: The Lotka-Volterra Model Rouzhen Ma Stephanie Young(Mentor)
How many are there? Animal Capture Mark Recapture
Predator-Prey System P(t)=The number of predators at time t N(t)=The number of its prey at time t The rate of change of the prey population N (t) = aN(t) - bN(t)P(t)
Predator-Prey System The rate of change of the predator population P (t) = -dP(t) + cN(t)P(t)
The Lotka-Volterra Model First-order Nonlinear Autonomous Equilibrium: P=a/b, N=d/c; P=N=0
Rescaling the System =at u( )=(c/d)N(t) v( )=(b/a)P(t) =d/a Equilibrium: u=v=1
Behavior around the equilibrium: Linearization Linearized System
Behavior around the equilibrium: Linearization Linearized Lotka-Volterra Model at (1,1) Set U=u-1, V=v-1 Final result
Solving Linear system Eigenvalues |A- I|=0 solving the quadratic equation for Eigenvectors (A- I)v=0 Solving for the associated eigenvectors
Solving Linear system how to find eigenvalues/eigenvectors in our system?
Classification of the Equilibria Real Eigenvalues: 1, 2 <0 Stable node Real Eigenvalues: 1, 2 >0 Unstable node Real Eigenvalues: 1 < 0 < 2 Saddle point
Classification of the Equilibria 1= + i, 2= + i Complex Conjugate Eigenvalues: <0 Stable spiral Complex Conjugate Eigenvalues: >0 Unstable spiral Complex Conjugate Eigenvalues: =0 Centres
Limitations The equilibrium is a centre, which is structurally unstable General upper limit Improve: The Logistic Growth Law Improve: The Logistic Growth Law
