 # How to prove the asymptotic stability of the trivial equilibrium of this system? I was trying to pr Audrina Jackson 2022-07-12 Answered
How to prove the asymptotic stability of the trivial equilibrium of this system?
I was trying to prove the asymptotic stability of the trivial equilibrium $\left(0,0\right)$ of the two-dimensional non linear ODE system:
$\begin{array}{r}\frac{dH}{dt}=\mu \frac{\left(H+F{\right)}^{2}}{{K}^{2}+\left(H+F{\right)}^{2}}-{d}_{1}H-H\left({\sigma }_{1}-{\sigma }_{2}\frac{F}{H+F}\right)\\ \frac{dF}{dt}=H\left({\sigma }_{1}-{\sigma }_{2}\frac{F}{H+F}\right)-\left(p+{d}_{2}\right)F\end{array}$
where $H$ and $F$ are dependent variable and positive.
All other parameters are non negative with ${\sigma }_{2}>{\sigma }_{1}$.
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You need $K$ to be strictly positive. Because if $K=0$ and $\mu >0$, then in a neighborhood of the origin, $dH/dt\approx \mu >0$. So, suppose $K>0$.
In the first equation, the terms $\mu \frac{\left(H+F{\right)}^{2}}{{K}^{2}+\left(H+F{\right)}^{2}}$ and $H{\sigma }_{2}\frac{F}{H+F}$ are of second order of smallness near the origin. Thus, the sign of $dH/dt$ is determined by $-\left({d}_{1}+{\sigma }_{1}\right)H$ which of course suggests stability.
In the second equation, $-H{\sigma }_{2}\frac{F}{H+F}$ is of second order. Neglecting it, we are left with $H{\sigma }_{1}-\left(p+{d}_{2}\right)F$. This looks troublesome, but if $H$ goes to zero, $F$ will be forced to follow.
Let's summarize. For every $ϵ>0$ there is a neighborhood of the origin in which
$\begin{array}{rl}\frac{dH}{dt}& <-\left({d}_{1}+{\sigma }_{1}\right)H+ϵ\left(H+F\right)\\ \frac{dF}{dt}&
(I work in the positive quadrant $H,F>0$, which is what you are interested in). Hence,
$\frac{d\left(2H+F\right)}{dt}<-\left(2{d}_{1}+{\sigma }_{1}\right)H-\left(p+{d}_{2}\right)F+3ϵ\left(H+F\right)$
which is negative, provided $2{d}_{1}+{\sigma }_{1}>0$, $p+{d}_{2}>0$, and $ϵ$ is chosen sufficiently small.
I took $2H+F$ instead of $H+F$ so that the coefficient of $H$ on the right would have ${\sigma }_{1}$, increasing the chance of the coefficient being negative.