∂P/∂t = f(P,I) + D1(∂^2P/∂x^2)

∂I/∂t = g(P,I) + D2(∂^2I/∂x^2)

Where P is the population density, I is the information density, t is time, x is space, f(P,I) represents the reaction term for the population growth, g(P,I) represents the reaction term for information diffusion, D1 is the diffusion coefficient for population density, and D2 is the diffusion coefficient for information density.

To incorporate the idea that growth is determined by information, we can use a logistic growth function that is dependent on the amount of information available. This can be expressed as:

f(P,I) = rP(1 - P/K)I

Where r is the intrinsic growth rate, K is the carrying capacity, and (1 - P/K) represents the constraint on population growth as population density approaches carrying capacity.

For the diffusion term, we can assume that information spreads through social networks and communication channels, and that this diffusion is influenced by the population density. This can be expressed as:

g(P,I) = kP(I - I0)

Where k is the diffusion coefficient for information density, and I0 is the baseline information density.

Combining these equations, we get the following system of partial differential equations:

∂P/∂t = rP(1 - P/K)I + D1(∂^2P/∂x^2)

∂I/∂t = kP(I - I0) + D2(∂^2I/∂x^2)

This is a system of reaction-diffusion equations that describes the growth of a population based on the interaction between population density and information density. The logistic growth function represents the population’s self-regulating behavior as carrying capacity is approached, and the information diffusion term represents the spread of information through social networks. The diffusion coefficients capture the idea that both population and information can spread through space, but at different rates.
using Turing’s model of reaction-diffusion create a mathematical model for human population growth where growth is determined by information. The reaction term is population density and the diffusion term is information density. Don’t include the carrying capacity
To create a mathematical model for human population growth based on Turing’s model of reaction-diffusion where growth is determined by information and without including the carrying capacity, we can use the following equations:

∂P/∂t = f(P,I) + D1(∂^2P/∂x^2)

∂I/∂t = g(P,I) + D2(∂^2I/∂x^2)

Where P is the population density, I is the information density, t is time, x is space, f(P,I) represents the reaction term for the population growth, g(P,I) represents the reaction term for information diffusion, D1 is the diffusion coefficient for population density, and D2 is the diffusion coefficient for information density.

To incorporate the idea that growth is determined by information, we can use a simple linear growth function that is dependent on the amount of information available. This can be expressed as:

f(P,I) = rPI

Where r is the intrinsic growth rate and P and I are both positive quantities.

For the diffusion term, we can assume that information spreads through social networks and communication channels, and that this diffusion is influenced by the population density. This can be expressed as:

g(P,I) = kP(I - I0)

Where k is the diffusion coefficient for information density, and I0 is the baseline information density.

Combining these equations, we get the following system of partial differential equations:

∂P/∂t = rPI + D1(∂^2P/∂x^2)

∂I/∂t = kP(I - I0) + D2(∂^2I/∂x^2)      linearize this system, find the steady-states of the system and the eigenvalues