Equilibria
Find the equilibria by solving for all
Linearization
Find the Linearization of the system.
where
where EP is each Equilibrium Point, where EP.x and EP.y are the x and y components
By definition at an equilibrium point, the function is equal to
Therefore the linearization becomes this:
The final expression for the linearization looks like this.
This is a Shifted Systems of Linear Differential Equations
Linearization in 1D with Complex Polynomials
The equilibrium points of this equation are
In short, if you have a polynomial of order 1, the linearization is just f, but you drop the EP term. If the order is greater than 1, then the linearization is 0.
Perterbation
Given
Which is the exact form of the homogenous part of our Linearization, except we have
Neutral Eigenvalues (
Approximating Dynamics
Given
We approximate the dynamics of the original system, by finding the dynamics of the Linearization
- TLDR: We find the Eigenstuff of the Jacobian, and use that to construct our Phase Portrait
- If the eigenvalues of the Jacobian lead to Unstable or Asymptotically Stable, then the approximation is sufficient to determine the dynamics of the original system
- If Stable, we consider the approximation “degenerate”, and conclude that the approximation is insufficient.
Lotka-Volterra Competition Model
where
where
where
where
- where
measures the negative effect tigers have on wolves - where
measures the native effect wolves have on tigers
Lotka-Volterra Predator-Prey Model
where🐰 is the prey population size
where 🐺 is the predator population size
where
where
where
where