CritPt / Challenge_49_main / In a long-range dispersal model, a cluster of species can grow from a single…

# Problem setup:
In a long-range dispersal model, a cluster of species can grow from a single seed. The cluster expands into space with long-range power-law interactions. In one dimension, the asymptotic size of the cluster,  denoted by ℓ(t), satisfies a self-consistent equation:
\begin{equation}
    \frac{1}{\ell^{\mu}(t)} \int_0^t \ell(\tau) \ell(t - \tau) d\tau = K.
\end{equation}
Here, μ is the power-law index of the long-range interaction, and K is a constant independent of time.

# Main problem:

Consider the specific scenario where μ=2, and let us define φ=log₂ℓ and z=log₂t. Expand φ in terms of z  for large t and retain terms up to constant order in z. That is, you should fix any polylog(z) corrections such as log₂(z) and log₂²(z) if they are present; constant corrections can be ignored.

# Problem setup:
In a long-range dispersal model, a cluster of species can grow from a single seed. The cluster expands into space with long-range power-law interactions. In one dimension, the asymptotic size of the cluster,  denoted by $\ell(t)$, satisfies a self-consistent equation:
\begin{equation}
    \frac{1}{\ell^{\mu}(t)} \int_0^t \ell(\tau) \ell(t - \tau) d\tau = K.
\end{equation}
Here, $\mu$ is the power-law index of the long-range interaction, and $K$ is a constant independent of time.

# Main problem:

Consider the specific scenario where $\mu = 2$, and let us define $\varphi = \log_2 \ell$ and $z = \log_2 t$. Expand $\varphi$ in terms of $z$  for large $t$ and retain terms up to constant order in $z$. That is, you should fix any $\text{polylog}(z)$ corrections such as $\log_2(z)$ and $\log^2_2(z)$ if they are present; constant corrections can be ignored.