The current study uses Turing's reaction-diffusion instability model as an analytical framework to analyze the relationship between population density and information density.  It’s aim is to illustrate possible by using this mathematical approach to illuminate the complex bidirectional feedback loops underlying the interaction of population and information in a spatial context.

Introduction
Population growth is now seen as an open, evolving and self-organising system and growth is determined by internal processes. Mathematical modelling provides itself efficient when it comes to describing and predicting human development. Most of the concepts come from physics such as-collective interactions, chaos and self-organisation with causality expressed in probabilistic terms. Understanding the factors that drive population growth is essential for policymakers to plan and implement effective measures to manage population size. Individual-based models, matrix models, and delayed-differential equation models are some of the mathematical modelling methodologies utilised in population studies. Individual-based models are used to replicate the behaviour of individual agents within a population and investigate how their interactions affect population growth. In this dissertation, we explore mathematical models of human population growth and discuss how they can be used to understand the dynamics of population growth. We will analyse different mathematical modelling approaches used in population studies and examine their pros and cons. Turing's reaction-diffusion model, which was originally intended to describe morphogenesis patterns, has been widely altered to represent a wide range of systems, including ecology, chemistry, and social phenomena. Because of the inclusion of response and diffusion factors, which can be modified to represent intrinsic population growth and information dissemination, the model is particularly well-suited for examining the spatiotemporal dynamics of population and information density.

Past world population growth models
The Malthusian population growth model is a theory in demography that argues that population growth is limited by the availability of resources. In this regard, the model predicts that when resources become exhausted, population would eventually outpace growth. This idea was first proposed by Thomas Malthus, an English cleric and economist, in his 1798 book "An Essay on the Principle of Population." [1]
The Malthusian exponential model:

dN/dt=(b-d)N=rN (1a)
b=births
d=deaths
r=growth rate
N=current population
It assumes that the rate of population is proportional to the current population. This differential equation leads to the model of the form 
N=Noe^rt (1b)
No= initial population
r=growth rate
t=time
This model even if its accurate for specific time period, it fails to express larger time period. It was based on several key assumptions. Firstly, Malthus argued that population growth is exponential, while the growth of resources is only linear. This indicates that while the pace of population growth doubles over time, the rate of resource growth remains constant. In other words, resources expand at a slower pace, such as 10% or 20%, but the population doubles every generation. Second, according to Malthus, the population could only be kept in check by checks and balances like hunger, war, illness, and vice. Thirdly, Malthus contended that there is a threshold beyond which the population cannot be supported. He argued that resource competition intensifies, which lowers living standards and raises death rates.
Malthus underestimated the possibilities for societal and technological advancement. As societies develop, they might discover new methods for increasing food production and improving resource management.
Furthermore, he failed to consider the impact of demographic transition on population growth. Demographic transition refers to the shift from high fertility and mortality rates to low fertility and mortality rates as a society undergoes modernization. This shift typically occurs as a result of increased access to education, health care, and other basic services. People often have fewer children as child mortality rates decrease and as they become more aware of the advantages of family planning, which slows down population growth.

Logistic model (Verhulst model)
The logistic model was proposed to induce a saturation in the population dynamics in an attempt to address the problem of the Malthus model. Its foundation is the idea that a population's rate of expansion is influenced by two variables: the rate of reproduction and the accessibility of resources. A Belgian mathematician named Pierre Verhulst created the concept in the middle of the 19th century as a means of explaining how population growth is naturally constrained.[2]
The logistic model's basic principle is that all organisms compete with one another for scarce resources like food, water, and shelter. As a result, the number of people that a certain setting can support is limited. This concept is known as the carrying capacity, which is the maximum size a population can reach when given access to a finite amount of resources.
The logistic model is derived from the differential equation:
                                                                      dN/dt=rN(1-N/K)(2)
dN/dt= rate of change in population size over time
r=growth rate
N=population
K=carrying capacity
As N approaches K, dN/dt approaches zero and the population stops growing at N=K.
The general solution to (2) is
N=K/(1+1/c^''  e^(-rt) )(2a)
For t=0 and N=No:
No/(K-No)=c^''
Leading to:
N=K/(1+(K/No-1) )
The logistic model has been used to model a wide range of biological systems, from bacteria to fish to humans. When it comes to simulating the expansion of the human population, the logistic model has some drawbacks. The fact that human populations are influenced by a variety of elements, including cultural, social, and economic considerations, in addition to resource availability, is one of the key challenges. For instance, as a result of advancements in healthcare that prolong life and lower new-born mortality rates, human populations may expand quickly. Similar to how fertility rates can affect population increase, social and economic issues may also have an impact.
Hyperbolic equation 
Around 1951, it was discovered that the quasi-hyperbolic equation would fit the world population[3]:
P=a/(D-t)^M (3a)
Where a is a constant and M was calculated as M=0.99.
D is the point of singularity which was estimated by the authors as D=2026.87, the number correlates with the day 13th of November 2026. At that time, in relation to the equation, world population becomes infinite.

In differential form:
dP/dt=Ma/(D-t)^(M+1) =(M/a^(1/M) ) P^(1+1/M)      (3b)
 


Kapitsa’s model
His definition of ‘’information in the concept of population is that of knowledge. Knowledge essentially passing from one generation to the other. This separates the humankind from all the other population in nature. The ability to transfer that knowledge detach us from our solely biological nature and it manifests itself into culture and technology. Therefore, in information-based societies, the real limit to growth is not limitation of resources but a limitation of information- the limitation of culture, science, and education. He analyzed population growth with three differential equations corresponding to the three different stages of the demographic transition. [1]
	dN/dt=N^2/C+1/t
	dN/dt=  C/〖(T1-T)〗^2 
	dN/dt=C/(〖(T1-T)〗^2+τ^2 )
The addition of τ essentially works as taming of the hyperbolic function
 

Later refinements of the model suggest values of constants:
C=163 , T1=1995, τ= 45
Kapitsa’s model was examined and plotted against real population figures up to 2022;
 

Demographic transition theory
The demographic transition theory explains how economic progress and modernization alter the population growth rate. This shift often takes place across four or five stages, each of which is distinguished by a different birth and death rate that is driven by developments in infrastructure, technology, and public health.
1. Pre-Transition Stage: This stage is characterized by pre-industrial, agrarian societies with high birth and mortality rates and little population increase. High infant mortality, frequent illnesses, insufficient food supply, and lack of access to high-quality healthcare all contribute to heightened death rates. High birth rates can be ascribed to a number of things, such as lack of access to family planning, religious convictions, and the requirement that children contribute to agricultural labor or provide care for the family through illness and old age.
2. Early Transition Stage: As society begins to industrialize and modernise, advances in nutrition, sanitation, public health, and access to healthcare significantly lower mortality rates. Due to entrenched cultural norms and restricted access to family planning, birth rates continue to be high. As a result of the imbalance between the declining mortality rate and the continually high birth rate, this combination causes rapid population expansion.
3. Mid-Transition Stage: In this stage, overall population growth slows as birth rates steadily fall and mortality rates continue to fall. Increased access to education, especially for women, urbanization, and the availability of birth control technologies are factors resulting in reduced birth rates. Women frequently choose to have fewer children when they are given better educational and economic prospects, and urban living typically calls for smaller family sizes.
4. Late Transition Stage: Population growth is minimal during the late stage of demographic transition, which is characterised by low and steady birth and death rates. Reduced fertility rates are a result of increased economic growth, widespread education, and widespread access to family planning services. Additionally, when the significance of the traditional extended family declines, cultural values and attitudes may change in favour of smaller families and delayed childbirth.
 
Figure 1.Demographic Transition Model; Source: UNEP Grid-Sioux Falls
 

The Role of Information in the Demographic Transition
1. Education: Wider access to education, particularly for women, is made possible by the diffusion of information, which significantly lowers fertility rates. Women can make decisions that are in line with their personal preferences and family circumstances as they gain more understanding about reproductive options, childcare, and family planning.
2. Economic Development: Information exchange can stimulate economic growth by promoting innovation and knowledge transfer. Global communication and data access may boost markets and industries, generate employment possibilities, and improve living conditions. Population growth often reduces as fertility rates stabilise as countries advance economically.
3. Urbanisation: Due to financial and spatial constraints, relocating from rural to urban contexts often implies a change in family arrangements and a decrease in family size.
4. Cultural Changes: With more worldwide communication and information sharing, cultural norms and values may change, affecting views on marriage, parenthood, and gender roles. Increased acceptance of smaller families, dual-income households, and postponed childbearing might result from the introduction of new ideas and exposure to other cultural practises.
Information imperative / Information-population feedback loop
The distribution of information is one of the primary forces behind population self-organization. There are many different ways that information can be used to promote social and economic progress. People can share their knowledge and experiences with one another, and they can also change their behaviour in response to environmental cues. Information flow can be thought of as the adhesive that binds populations together and enables them to work as a collective.
The influence of population density on the rate at which information is distributed is one feedback loop. As a population grows in density, communication pathways become more efficient, allowing for the rapid and widespread diffusion of information. This effect is increased when people engage with one another in social and professional situations, which tend to grow in frequency as population density increases.
On the other hand, information transmission has the potential to influence population dynamics through altering reproductive patterns, migration rates, and total growth rates. Information about health and family planning, for example, can lead to reduced birth rates, whereas information about job possibilities or living conditions can motivate people to relocate, leading in population shifts.
Population density could be modelled as an evolving function of time and space, while the information density could be influenced by this population density structure through a mathematical relation


Experimental approach
Linear stability study 
In this report, linear stability study is used to analyze the behavior of the relation between population growth and information density, as well as computational modelling
Linear stability study is a mathematical analysis technique is used to look at a system of equations' stability characteristics. To ascertain the stability of a dynamic system or process, it is frequently utilized in several disciplines, including physics, engineering, and economics. The system is represented by a collection of differential equations or difference equations in a study of linear stability. The system is represented by a collection of differential equations or difference equations in a study of linear stability. As part of the study, the equations are linearized at a certain equilibrium point, which is frequently the system's steady state, and the behavior of minor perturbations around the equilibrium is then examined.
Determining whether or if small perturbations—that is, disturbances, fluctuations, or changes—in the system have a tendency to develop or degrade over time—is the main goal. The study specifically looks at the linearized system's eigenvalues, which reveal information on the direction and rate of change of perturbations
The system is considered to be stable if all of the eigenvalues have negative real components, in which case any perturbations near the equilibrium point will gradually decay to zero. The system is considered to be unstable if one or more of the eigenvalues have positive real components, which means that even minor perturbations might accumulate and eventually cause big changes.
Turing instability 
The mathematical model of Turing's diffusion-driven instability explains how patterns emerge in reaction-diffusion systems. The model was first proposed by Alan Turing in 1952 to explain how certain biological organisms such as zebrafish get their stripes, spots, and other distinctive patterns. Turing proposed that these patterns were created by the interaction of chemicals, which he called "morphogens," and that their creation could be explained by mathematical models.
He outlined a system of partial differential equations that represent the spatial distribution of morphogens in a developing organism. He suggested that the interaction between at least two morphogens, one activating and one inhibiting, could generate pattern differentiation. The substances diffuse through the tissue and react with one another such that small, random fluctuations in their concentrations are amplified, ultimately creating patterns. The foundation of Turing's theory is the idea of reaction-diffusion systems. While the diffusion component relates to how these chemicals are distributed throughout the organism, the reaction component describes how morphogens interact chemically.
When two or more chemicals interact and disperse through a medium, their concentration changes over time. This process is referred to as a reaction-diffusion system. A set of partial differential equations can be used to describe the dynamics of these systems. In a traditional reaction-diffusion system, two chemical species can interact and undergo a chemical transformation, which is represented by a set of differential equations.
This system is condensed in Turing's model to consist of two chemical species, X and Y, with distinct diffusivity values, D1 and D2. When X diffuses more quickly than Y, a feedback loop is produced because the concentration of X falls in the low-concentration zone more quickly than in the high-concentration region. As a result of this X depletion encouraging an increase in Y and vice versa, spatial patterns start to appear in the system. The following is a representation of the equations describing the system's dynamics:

∂X/∂t = F(X, Y) + D1∇2X

∂Y/∂t = G(X, Y) + D2∇2Y

Where F and G are functions that describe the rate at which X and Y react with each other, respectively. These functions can describe a wide range of chemical reactions, including autocatalytic and activator-inhibitor reactions. The Laplace operator, ∇2, describes the second-order partial derivative of the concentration gradient, and the diffusion constants, D1 and D2, determine how fast X and Y diffuse through the medium.
If a local perturbation causes the concentration of species X to fall while species Y grows in a particular area of the medium to see how patterns could form in this system. As a result, there is an increase in the gradient between the concentrations of X and Y, which encourages the diffusion of X from the highly concentrated area to the nearby low-concentration area. Since X diffuses more quickly than Y, its concentration will drop in the low-concentration zone more quickly than it will in the high-concentration region. As a result, a positive feedback loop is created in which the decrease of X encourages the growth of Y and vice versa.
To analyze the behavior of Turing’s model mathematically, it needs to compute the eigenvalues of the linearized system near the homogeneous steady state. These eigenvalues give insight into the stability of the homogeneous state and the possible formation of a spatial pattern.





For an information-based world population model it applies as;
I(t) represents the density of information in the environment, which can affect the rate of population growth.
P(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:

∂P/∂t = f(P, I) + D∇^2P

∂I/∂t = g(y, x) + C∇^2I

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:

f(P, I) = rP(x,t)^2 I(x,t)


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 modified logistic model:

g(P,I) = b*P(x,t) +a*(1- P(x,t)/K)


where a and b is a constant that determines the effect of information density on population growth
To linearize the system, we can assume small deviations from an equilibrium point where both P and I are constant. We can write P(x,t) = P0 + p(x,t) and I(x,t) = I0 + i(x,t), where P0 and I0 are the equilibrium values and p and i represent small perturbations from the equilibrium. Substituting these into the equations and neglecting terms of second order or higher, we obtain:
Substituting these into the original system and linearizing, we obtain:
∂p/∂t = r 2P p + r  P^2 i + D∇^2p
∂i/∂t = b Pi + a (1- P/K) p + C∇^2i
To find the eigenvalues, we can assume solutions of the form:
p(x,t) = P(x) exp(λt)
i(x,t) = I(x) exp(λt)



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