I’m working on a prey-predator model. I’m using the following system of differential equations for it: \begin{align} x’&=-a_1x+a_2xy+a_3xz\\ y’&=b_1y-b_2xy\\ z’&=c_1z-c_2xz \end{align} Where $a_i, b_i, c_i >0$. One of the stationary points is $P=(\frac{b_1}{b_2},\frac{a_1}{a_2},0)$. Question: How can I determine the stability of this point $P$? Attempt: First I wrote the equation as: \begin{align} \frac{\mathrm d \underline{v}}{\mathrm […]

I have a question with respect to phase plots of repeated eigenvalue cases. For instance suppose that one is given a matrix with the following: $$\overrightarrow{y’} = \begin{pmatrix} 3 & -4 \\ 1 & -1 \end{pmatrix} \overrightarrow{y}$$ whereby the solution yields: $$\overrightarrow{y} = c_1e^t\binom{2}1 + c_2(te^t\binom{2}1 + e^t\binom{1}0)$$ I understand that it is asymptotically unstable […]

I was trying to solve the question of AeT. on the (local) Lyapunov stability of the origin (non-hyperbolic equilibrium) for the dynamical system $$\dot{x}=-4y+x^2\\\dot{y}=4x+y^2$$ The streamplot below indicates that this actually is true. Performing the change of variables to polar coordinates $x=r\cos\phi$, $y=r\sin\phi$ and after some trigonometric manipulations we result in $$\dot{r}=r^2(\cos^3\phi+\sin^3\phi)\\ \dot{\phi}=4+r^2\sin\phi(\sin\phi-\cos\phi)$$ From this […]

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