Lotka-Volterra Model in Mathematical Modelling
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
●
τ=at
●
u(τ)=(c/d)N(t)
●
v(τ)=(b/a)P(t)
●
α=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-
λI
|=0
solving the quadratic equation for
λ
●
Eigenvectors
(A-λI)v=0
Solving for the associated eigenvectors
Solving Linear system
how to find eigenvalues/eigenvectors in our system?
Classification of the Equilibria
●
Real Eigenvalues:
λ
1
, λ
2
<0
→ Stable node
●
Real Eigenvalues:
λ
1
, λ
2
>0
→ Unstable node
●
Real Eigenvalues:
λ
1
< 0 < λ
2
→ Saddle point
Classification of the Equilibria
λ1=α+βi, λ2=α+βi
●
Complex Conjugate Eigenvalues:
α<0
→ Stable spiral
●
Complex Conjugate Eigenvalues:
α>0
→ Unstable spiral
●
Complex Conjugate Eigenvalues:
α=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
A
ny questions?
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.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. Download presentation by click this link. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
E N D
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
