Mathematical Modelling of India’s Population and Rice Production with Entropy Maximisation
Abstract — Rice being the staple food in India occupies an important role in the country’s agrarian economy. India also stands second in human population in the world. So it is of at most importance to study the relationship between the rice production and human population. In this work, a Lotka-Volterra model is proposed for analyzing Indian population and India’s rice production from 1951 to 2013. The parameters of the model are estimated using genetic algorithm. The Maximum Entropy Principle (MEP) is used to get the best estimates of the sampled data by maximising the entropy of the model. The entropy for the actual and developed model is calculated using Shannon’s entropy. Further, the international standards of performance indices are computed along with the mean square error to evaluate the overall performance of the model. Results demonstrate that by maximizing the entropy of the Lotka-Volterra model, better estimates of the sample data are obtained and the error is found to be minimal. The performance too has been validated by the values of performance indices.
Keywords- Lotka-Volterra model, Maximum Entropy Principle (MEP), Human Population, Rice Production
*Department of Instrumentation Engineering,
Madras Institute of Technology Campus, Anna
University, Chennai – 600044, India.
Rice or Oryza Sativa as it is called by botanists has its origin 10,000 years ago. Rice is the main crop of India particularly in the eastern and southern parts. India, being the world’s second largest producer of rice after China accounts for 20% of the world’s total rice production and serves as the main food for more than half of the Indian population. Rice is a very important crop in terms of economy and food security. Since Indian economy is largely an agrarian economy, rice production inevitably plays a vital role in not only feeding the masses but also in terms of contribution towards the GDP of the country. Rice is a tropical kharif crop and requires sufficient amount of annual rainfall for its growth. This makes it easier for Indians to cultivate rice as India is a country with tropical climate. But to enhance the rice production, there are several challenges that are to be encountered like unpredictable climatic conditions, insufficient monsoon rainfall, lack of labour, decreasing area of agricultural land due to urbanization, excessive use of fertilizers and pesticides and so on. Also there are other natural factors like drought, flood, earthquakes which will affect rice production. So to increase the production all these factors must be taken into account. Hence the analysis of rice production in India is of high relevance in terms of studying its growth and predicting its trend so as to meet the growing demands.
Human Population is defined as the total aggregate of humans living in a certain geographical location under consideration with reproductive capabilities. The human population of the world has been increasing slowly over the years and has crossed the seven billion mark too. India is the second largest populated country in the world after China and the Indian population too has been tremendously increasing after the 1950’s. India’s poverty rate, illiteracy, rapid decline in mortality rates, high fertility rate and immigration from other foreign nations are some of the main reasons attributed to this rapid growth of its population. The current Indian population stands at about 1.34 billion and is expected to increase at a steady rate of 1.2% annually. Although many efforts have been initiated by the government in spreading awareness and implementing policies to curb its rapid increase, human population is found to be steadily increasing year by year. This increase in population is of concern as it stagnates the economy and reduces the impact of developmental progress. So presently it is of prime importance to study its growth rate and predict its nature.
Mathematical Model is termed as the formulation of mathematical equations and concepts describing and characterising real world phenomena and systems under certain constraints. The process of developing such a description is called mathematical modelling. It establishes a relationship with the physical variable whose dynamics are represented mathematically through variables and equations. Such a model actually helps in better understanding of the real world process and aids in prediction, control, estimation, optimization, decision making and determination of its causes and effects. Recently, mathematical models have been used for describing complex systems [1-3]. It finds its application in natural sciences for formulation of laws and theorems, business for maximising output, economics and finance for effective analysis of strategies and decision making, engineering discipline for design and study of numerous processes, social sciences, psychology and so on.
Entropy is a measure of randomness or microscopic disorderliness of a system. It represents the lack of predictability in the process. It is a state function and an extensive property which depends upon the mass of the substance in that system. The second law of thermodynamics states that the universe evolves in such a way that its total entropy always stays the same or increases which implies that entropy forms the basis of it. The spontaneity of a process is highly related to entropy. Always, spontaneous processes occur with increase in entropy. This indirectly means that the entropy of a system is dependent on the thermodynamic probability of the state of the system. The statistical information obtained from partial knowledge of process has entropy aspect incorporated in it. Maximisation Entropy Principle is used in statistical modelling of these processes while characterizing few unknown events. It is one of the least biased estimates possible when there is little information. This can be applied to many to Spatial Physics, Computer Vision and many other fields.
Edelstein-Keshet (2009)  has discussed about mathematical models like logistic model, Lotka-Volterra model for describing the dynamics of interaction among population in the same environment. Wang (1990)  has proposed a logistic model for a single population and Wake, Watt (1996)  have analyzed the noise fluctuation in carrying capacity of the population. Zhen et al. (2006)  have proposed an improved population model in which the growth rate was taken to be based on power law exponent. Regional sustainability was extensively studied by Ding (2008)  and Geng (2011)  which emphasised the importance of sustainable growth particularly in agriculture. Yu (2005)  has dealt with a time series model to determine the recurrent relationship in rice production and Vinodini et al. (2014)  have developed a grey model for prediction of rice productivity and consumption in India. The concept of entropy for nonlinear dynamical systems was looked upon by Balestrino, Caiti, and Crisostomi (2009)  and Balestrino et al. (2008)  based on Ordinary Differential equations (ODE).
The objective of this paper is to fit a Lotka-Volterra model for modelling of India’s rice production and population and to compute the estimates of the sampled data using Maximum Entropy Principle (MEP).
A. The Lotka-Volterra Model
The Lotka-Volterra model as discussed by Edelstein-Keshet (2009)  is a pair of nonlinear differential equations with deterministic and continuous solution, used to describe the dynamics of ecological prey-predator model. The first equation expresses the rate of change of the prey population which is the first state variable and the second equation expresses the rate of change of predator population which is the second state variable. In this work, rice production is taken as prey and Indian human population is considered as predator. Both the equations are represented as a function of rice production-human population and this model is supposedly built on certain assumptions. Some important assumptions made are that, during the process of evolution of both rice and humans, the environment that sustains them does not change significantly in favour of either of them. Further, the genetic adaptation and mutation of the species are considered inconsequential. It is also assumed that the food supply of Indian population depends predominantly on rice production and the rate of change of either of them depends on their respective sizes.
Mathematically the Lotka-Volterra model  is represented as,
where x is the size of the rice production, y is the size of the Indian human population, dx/dt and dy/dt represent the growth rates of the two state variables respectively. Here a, b, c and d are fixed real parameters representing the growth rate of rice production, the rate at which the humans consume the rice, the rate at which Indian human population increases by consuming rice and the mortality rate of humans respectively. These parameters describe the interaction among the two state variables.
This means that the rice production increases at a rate proportional to its size denoted by ax and simultaneously decreases at a rate proportional to product of both the state variables indicated by — bxy. The human population decreases at a rate proportional to its size indicated by —dy and increases at a rate proportional to product of both the state variables which is represented by cxy.
Maximum Entropy Principle (MEP) states that the best estimate of a missing data is the one which maximises the information entropy under specified constraints. There are several types of entropy measures that are used in information theory. Shannon entropy is one important measure used to measure the average missing information on a random source. The Shannon (1948)  defined shannon’s entropy mathematically as:
where X is the random variable and p is the probability distribution.
C. Estimation of Parameters using GA
Genetic algorithm (GA) as discussed by Sivanandam, Deepa (2007)  and Li Min-qiang et al. (2002)  is an adaptive heuristic search algorithm used for solving complex optimisation problems under constrained and unconstrained conditions. Basically, it draws its inspiration from optimisation problems based on process of natural selection and processes that have their base in biological and ecological evolution. This method could be applied to solve a variety of optimization problems in which the objective function is non differentiable, stochastic, discontinuous or nonlinear in nature. In this algorithm, a base population with individuals is first considered and a fitness or objective function is defined. It then randomly selects the individuals from the current population and uses them to produce the offspring of the newly generated population. Each generation consists of a population of character strings and each individual represents a point in a search space with a possible solution. So this step is done repeatedly until an optimal solution is found that satisfies the criterion of minimization of the fitness function. It uses two terminologies namely crossover and mutation to execute the algorithm. Crossover refers to the production of offspring from the parents of the previous generation. Mutation refers to the alteration of a particular chromosome so as to maintain diversity in the population. The objective function J is given by
where J is the fitness function to be minimised,
eH is the error in human population at time instant t
eR is the error in rice production at time instant t
E is the Shannon’s Entropy.
The errors are defined as:
where PR(t) is the amount of rice production at time instant t and PH(t) is the human population size at time instant t
D. Data Acquisition
The data for Indian population and rice production is considered for a wide range of years from 1951 to 2013 [18-20].
III. RESULTS AND DISCUSSION
The simulations have been carried out using MATLAB software. A Lotka-Volterra model for both Indian population as well as India’s rice production is developed and their corresponding results are shown in Figure 1 and Figure 2 respectively. The normalized values of both the data have been considered in this work for modelling. In Figure 1, the actual population increases almost linearly and it is found that the developed model is closer to the former. Also, the model error is very insignificant after 1985. This shows that the developed model captures the trend of the Indian population very closely. Although India’s rice production has been changing intermittently at irregular intervals, the overall production rate has been on the rise since the 1950’s as in Figure 2. The rice production in the country saw a considerable growth from the 1980’s and this was predominantly due to the green revolution brought in by the fourth five-year plan. During this period, a larger share of revenue was allocated towards agricultural development. In 2003, it was seen that India’s rice production declined sharply. The proposed model captures the entire dynamics of rice production over the years almost accurately.
The international standards of performance indices for evaluating the efficiency of a model are Integral Square Error (ISE), Integral Absolute value of magnitude of error (IAE), and Integral Time multiplexed Absolute value of Error (ITAE). These standards are calculated and tabulated in Table I. These indices depend on the absolute value of error thereby preventing the error cancellation due to negative error. Calculation of ISE and IAE reveal that they are both low for the human population and rice production where as the values of ITAE for both of them are quite large because of the larger time period considered. On the other hand, the values of mean square errors are extremely small for the two. Furthermore, the Maximum Entropy Principle is applied to the model by first sampling the population in discrete periods and it is found that the Shannon’s entropy for the developed model is relatively the same as that of the actual system. The entropy estimates are listed in Table II.
The modelling of rice production and Indian population finds its relevance due to the dependence of the Indian economy and its population on agriculture. In this work, a model for rice production and Indian population is proposed using Lotka-Volterra equations. The model is found be very accurate capturing the entire dynamics of the system. Also, the error present in the model is found to be very low. The genetic algorithm is employed to ensure wide range of search in estimating the parameters of the model. When the Maximum Entropy Principle was incorporated, the estimates of the missing data in the sample lie very closer to the actual ones. Furthermore, the calculated Shannon’s entropy value of the estimated model is found to be nearly the same as that of the actual data. The calculated performance indices validated the efficiency of the developed model.
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