x(t) represents the density of information in the environment, which can affect the rate of population growth.
y(t) represents the density of the population at time t.
Based on Turing’s equation for reaction and diffusion, we can write the following system of partial differential equations:

∂y/∂t = f(y, x) + D∇^2y

∂x/∂t = g(y, x) + C∇^2x

where f(y, x) and g(y, x) are the reaction terms describing the population growth and the information density, respectively, and D and C are the diffusion coefficients for the population and the information density. The Laplacian operator (∇^2) represents the diffusion of the respective variable in space.

For the reaction terms, we can use a logistic growth model for the population:

f(y, x) = ryx*(1 - y/K)

where r is the intrinsic growth rate, K is the carrying capacity (maximum population density), and y/K is the effect of density-dependent regulation on the population growth.

For the information density term, we can use a linear model:

g(y, x) = ay-bx-

where a and b are constants that determine the effect of information density on population growth
linearize the system and find the eigenvalues and the stability of the steady states of the system 