2D Nonlinear Systems of Differential Equations

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\vec{F}(x,y)=\left[\begin{array}{c}f(x,y)\\ g(x,y)\end{array}\right]$$

Equilibria

Find the equilibria by solving for all $(x,y)$ that satisfy $\vec{F}=\vec{0}$

Linearization

Find the Linearization of the system.

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{c}f(x,y)\\ g(x,y)\end{array}\right]⟹\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\vec{F}(\text{EP})+J_{\vec{F}(\text{EP})}\left[\begin{array}{c}x-\text{EP.x}\\ y-\text{EP.y}\end{array}\right]$$

where $J_{\vec{F}(EP)}$ is the Jacobian matrix at a particular equilibrium point

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 $\vec{0}$. See Equilibria.

Therefore the linearization becomes this:

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=J_{\vec{F}(\text{EP})}\left[\begin{array}{c}x-\text{EP.x}\\ y-\text{EP.y}\end{array}\right]$$ $$J_{\vec{F}(\text{EP})}=\left[\begin{array}{cc}{f}_x& f_y\\ g_x& g_y\end{array}\right]@\text{EP}$$

The final expression for the linearization looks like this.

$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left(\left[\begin{array}{cc}{f}_x& f_y\\ g_x& g_y\end{array}\right]@\text{EP}\right)\left[\begin{array}{c}x-\text{EP.x}\\ y-\text{EP.y}\end{array}\right]$$

This is a Shifted Systems of Linear Differential Equations

Perterbation

Given

$${\vec{v}}^{\prime }=A\vec{v}$$

Which is the exact form of the homogenous part of our Linearization, except we have

$$\vec{v}=J_{\vec{F}(\text{EP})}(\vec{v}-\text{EP})$$

Neutral Eigenvalues ($\mathrm{Re}\lambda =0$) lead to structural instability, e.g. a perterbation leads to signficant change.

Approximating Dynamics

Given

$$\vec{v}=J_{\vec{F}(\text{EP})}(\vec{v}-\text{EP})$$

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

Lotka-Volterra Competition Model

$$\frac{d🐺}{dt}=g_{🐺}🐺\left(1-\frac{🐺}{K_{🐺}}-{\alpha }_{🐺🐯}\frac{🐯}{K_{🐺}}\right)$$ $$\frac{d🐯}{dt}=r_{🐯}🐯\left(1-\frac{🐯}{K_{🐯}}-{\alpha }_{🐯🐺}\frac{🐺}{K_{🐯}}\right)$$

where $🐺$, $🐯$ are population sizes

where $g_{🐺}$, $r_{🐯}$ are intrinsic per capita growth rates

where $K_{🐺}$, $K_{🐯}$ are carrying capacities.

where ${\alpha }_{🐺🐯}$, ${\alpha }_{🐯🐺}$ are competition coefficients

• where ${\alpha }_{🐺🐯}$ measures the negative effect tigers have on wolves
• where ${\alpha }_{🐯🐺}$ measures the native effect wolves have on tigers

Lotka-Volterra Predator-Prey Model

$$\frac{d🐰}{dt}=\alpha 🐰-\beta 🐰🐺$$ $$\frac{d🐺}{dt}=\delta 🐰🐺-\gamma 🐺$$

where🐰 is the prey population size

where 🐺 is the predator population size

where $\alpha$ is the prey growth rate (birth rate)

where $\beta$ is the predation rate/interaction rate

where $\gamma$ is the predator death rate

where $\delta$ is the conversion efficiency of prey into new predators.