How to predict which fixed point a mapping converges to for initial conditions?

Question

Say there is a population of mass 1 in which individuals can choose one of three traits (1,2,3). The population shares of traits 1, 2, and 3 at time $t+1$ is mapped from the shares at time $t$ via: $$p_1(t+1)=\frac{e^{\phi_1}}{e^{\phi_1}+e^{\phi_2}+e^{\phi_3}},$$ $$p_2(t+1)=\frac{e^{\phi_2}}{e^{\phi_1}+e^{\phi_2}+e^{\phi_3}},$$ $$p_3(t+1)=\frac{e^{\phi_3}}{e^{\phi_1}+e^{\phi_2}+e^{\phi_3}},$$ where $\phi_1$, $\phi_2$, and $\phi_3$ are linear functions of $p_1(t)$, $p_2(t)$, and $p_3(t)$ (specifically, $\phi_i=a_i-b_jp_j(t)-c_kp_k(t)$). Say that there are two absorbing states (fixed points?) in this system, with full population shares of either trait 1 or trait 2: $[p_1^*,p_2^*,p_3^*]\approx[1,0,0]$ and $\approx[0,1,0]$.

How can I work out to which point the system will converge for initial states in the neighbourhood of $[0,0,1]$? Or for any initial state $[p_1(0), p_2(0),p_3(0)]$ in the three-trait simplex? Ultimately what I’m interested in saying is how changes in the parameters of $\phi$ can result in an initial state converging to either $\approx[1,0,0]$ or $\approx[0,1,0]$. I understand from a previous question’s answer that this might involve Lyapunov exponents and the Jacobian, but I’m a little lost from there.

Answer 1

How can I work out to which point the system will converge for initial states in the neighbourhood of $[0,0,1]$? Or for any initial state $[p_1(0), p_2(0),p_3(0)]$ in the three-trait simplex?

What you are interested in are called basins of attraction. For non-linear maps such as yours, they can be very complex and impossible to describe analytically. Without intensively investigating the situation myself, I can only give you a road map here:

1. Start by visualising the basins for some typical sets of parameters (which should be easy if I understood correctly that your state space is two-dimensional). To do this, take a grid of points in your state space as initial conditions and colour them according to the fixed point they converge to.

2. If you see a simple structure, rejoice. See whether you can understand it and describe it analytically.

3. Otherwise look for symmetries and other structures. If your basins have simple borders, you can try to understand them.

4. Depending on your question, the relevant quantity for you may be the size of the basins. This can be approximated empirically (just count the pixels) and you can then analyse its dependence on the control parameters.

I understand from a previous question’s answer that this might involve Lyapunov exponents and the Jacobian

Probably not. (Local) Lyapunov exponents tell you something about how quickly the state converges to a given fixed point within its vicinity. They cannot tell you to which fixed point the state converges (unless there is only one stable fixed point, in which case you can indeed find out using Lyapunov exponents). If your initial condition is near the border of basins of attraction, it is not in the vicinity of a fixed point (in the sense of the above statement).