I am trying to understand how to analyse a system of coupled, non-linear ODEs taken from this paper. I want to perform a fixed point analysis and plot a bifurcation diagram to show how fixed points and their stability depend on the parameters.The setup is we have for example two cells, each characterized by its level of $n$ and $d$.
–$n_{i}$ represses the activity of its own $d_{i}$
–$d_{i}$ increases the activity of $d$ of its neighbour $d_{j}$
This is the non-dimensionalized form of the equations
$$\dot n_{i}=f(d_{j}) – n_{i}$$
$$\dot d_{i}=\nu (g(n_{i}) – d_{i})$$
where, $f=\frac{x^n}{a-x^n}$ and is monotonically increasing, $g=\frac{1}{1+bx^n}$ and is monotonically decreasing and $\nu$ represents a ratio of time constants $\frac{\gamma_d}{\gamma_{n}} $.
I honestly have no idea how to approach finding the fixed points, performing stability analysis and doing the bifurcation diagram, I care about dependence on $\nu,a,b$ and $n$.
If I take the case where $\nu >> 1$, I can simplify by assuming $d_{i}$ is in quasi steady state and hence $g(n_{i})=d_{i}$, therefore
$$\dot n_i = f(g(n_j)) – n_i$$
$$\dot n_j = f(g(n_i)) – n_j$$
which can be simplified further by substituting
$$\dot n_i = f(g(f(g(n_i))) – n_i$$
$$\dot n_j = f(g(f(g(n_j))) – n_j$$
and this is where i'm stuck. Usually in the 1d case you would solve for the fixed point and then check stability dependent on derivative and parameter values. I have no idea where to start from here.
Any pointers/help will be vastly appreciated. Also if this post is too long I will edit it down, I couldn't really decide what background is necessary so to understand things..
Best Answer
I ended up brute-forcing it by simplifying fgfg using WolframAlpha, formulating the equation fgfg(x) - x = 0 and sovling it with a newton-rhapson root finder.
Producing the bifurcation diagram from there was easy.
I guess I was looking for a closed form way, where I get formulas for the fixed points dependent on parameters but I don't think that is possible or atleast I couldn't do it.