Calculating convergence rates in a deterministic, dynamic system

Question

Say there is a population of mass 1 in which individuals can choose one of two traits (1 and 2). The population share with trait 1 at time $t+1$, $$p_{t+1}=\frac{1}{1+e^{-\beta\left(u_1-u_2+J(2p_t-1)\right)}}$$ for constants $u_1,u_2,\beta,J$, with $u_1>u_2$. I can't calculate them analytically, but it can be shown that for $\beta J>1$, there are at most three stationary points (of which the smallest and largest are stable) and at least one stationary point (which will be in the neighbourhood of $p^*\approx 1$).

Assume $p_{t=0}=0$. Consider two cases defined by two sets of parameters $u_1,u_2,\beta,J$ such that in each, there is one unique stationary point $p^*\approx 1$. How can I work out which system will reach its stationary point faster? In general I would like to know how to calculate the convergence rates analytically, if it's possible, but if not, knowing how to compare two convergent systems and work out the faster one would suffice.

For example, my hunch is that higher $J$ slows the system. For given $\beta,u_1,u_2$, if we take two environments with $J_1$ and $J_2$, respectively, where there exists one stationary point, then the curve of the function with lower $J$ more closely "wraps" around the line $p=p$, which should lead to more transition steps. (I don't know how to explain that better, but hopefully you understand what I mean.)

Even just knowing what terms to search for would be a big help. I've tried searching for literature on convergence rates but given the implicit transition expression, I'm not sure whether my case is applicable to (e.g.) Markov chain theory, etc. As you can tell, this is not my area :) Any search terms would be appreciated.

(Postscript: Ultimately I'm investigating a three-trait system, but I thought this would a good start to get the tools to solve it. However, if your answer also generalises, all the better!)

Answer 1

Let $f$ be your map, i.e.: $$f(p) := \frac{1}{1+e^{-\beta\left(u_1-u_2+J(2p-1)\right)}}.$$

Suppose your state $p$ is close to the fixed point $p^*$, namely it is $p := p^*+ε$. Then you can estimate how quickly the distance to the fixed point shrinks via the following quantity:

$$ Λ := \left | \frac{f(p) - p^*}{p-p^*}\right | = \left | \frac{f(p^*+ε) - f(p^*)}{ε}\right | ≈ \left | f'(p^*) \right |. $$

With more iterations, the distance behaves like $ε Λ^t$ (where $t$ is the number of iterations). The quantity $λ := \ln(Λ)$ is also known as the local Lyapunov exponent at the point $p^*$. For more details on the exponential behaviour, see this question.

So to compare convergence behaviours close to fixed points, you only need to take a look at the derivative of your map. For multidimensional systems, things are a bit more complicated: There are multiple Lyapunov exponents and the one you need is the largest one, which in turn you can obtain from eigenvalues of the Jacobian of $f$.