Stimulation Modelling of the Effect of Internal Migration on Urbanization in Rivers State, Nigeria

Internal migration is defined as “a movement of people from one area of a country (especially one local government area) to another area of the same country for the purpose or with the effect of establishing a new residence” either temporarily or permanently, is a complex and multidimensional process. In this study we are expected to determine the impact of internal migration in Port Harcourt city and Bori town respectively and to show the effect of the variation of some parameters that affect the growth of Port Harcourt city, through Simplified Assumptions and Mathematical Formulations. The internal migration of effective working population from Port Harcourt city to Bori town is lesser than that of Bori town to Port Harcourt city due to lesser jobs and social infrastructural facilities in Bori than Port Harcourt. Method of Analysis used is a non-linear ordinary differential equation of order 45 which otherwise is called O.D.E 45(Matlab ODE 45). It is evident from the results of the study that the variation of the rate of internal migration can produce either a positive or a negative effect on the population growth patterns of the two interacting population predictions.


Introduction
Internal migration is defined as "a movement of people from one area of a country (especially one local government area) to another area of the same country for the purpose or with the effect of establishing a new residence" (IOM, 2011 as recommended in ACP Observatory Research Guide, 2011a) either temporarily or permanently, is a complex and multidimensional process.
Growth in population requires corresponding growth in urbanization and industrialization for economic and social development, which will in turn result to the viability of the people. The growth of a city is influenced by the infrastructures on ground such as: modern facilities, industries/companies, hospitals, schools and good polices that will make for a business friendly environment. These infrastructures attract people who lack them in their cities or rural areas to look for greener pasture and modern lifestyle elsewhere that is where they are available. This brings to the front burner the issue of migration to bear and in turn increase or decrease in population, which if not checked has its negative impact on the living and entering countries/communities. Migration of human population is a common phenomenon in Africa especially in the presence of political, environmental and economic crises (Myers, 2002;Hunter, 2005;Raleight, 2011). In the developing world, significant population continue to move within and between rural and frontier areas (Henry et al., 2004;Lopezicarr, 2012), as well as to cities. It is therefore necessary to put in check the rate of migration in order to avoid shortage of food, housing or high cost of commodities, high rate of unemployment due to shortage of jobs and unpleasant social and environmental situations.
Internal migration and its impact on regional development in Macedonia was investigated by Temova (2017) with the aim to analyse the causes of imbalanced growth among the region and in Macedonia and to identify the conditions for their future growth. Statistical data were used to determine the trend of internal migration and the causes. It was discovered that with implementation of decentralization policy, less developed areas can be further developed according to their geographical climate and natural characteristic.
Internal migration and its impact in the growth of Port Harcourt city of Nigeria is an interesting and challenging sustainable development problem that seeks solution. Internal migration has been an issue in the city of Port Harcourt, following the fact that it is the only well developed and industrialized city in Rivers state. This has brought about continuous movement of people to the city to seek for better jobs, schools and modern social amenities that are not available in their localities. This movement can enhance the growth of the city if the rate of migration is very low. But being the only developed and industrialized city, the rate might not be known. It is therefore important to determine the impact of this migration on the city and the way forward to resolving the issue of internal migration in order to reduce its effect in Port Harcourt and Bori town.
The earlier analysis of population growth and how demand for housing has affected land use in the city of Port Harcourt has involved the development of a mathematical model that defines the extent of the population growth and its implication. It is important to further determine the effect of this growth due to internal migration in Port Harcourt city of Nigeria.
Port Harcourt is an industrialized city in Rivers state compared to other urban cities in the state. This industrialization attracts people from the urban and rural areas to Port Harcourt to seek for jobs, education and medical care. The issue of internal migration comes to bear as people migrate to Port Harcourt for greener pastures. Migration has its impact on the population of the city, infrastructures, social facilities, the economy and the environment.
Ayotamuno and Ekaka-a (2017) investigated the extend on how the variation of population growth and demand for housing can affect land use in Port Harcourt and urban areas in Nigeria. Two main methods were utilized to answer the research questions. These methods are called Geographic Information System (GIS), remote sensing, mathematical modelling and numerical simulation. One of the key results of this important study is stated as follows: development of built up areas need to be controlled so that all the vegetation is not used up because the city needs some vegetation to allow the city breath.
Similarly, the proposed mathematical model has compliment the relationship of population growth, housing demand and land use in the city of Port Harcourt in Nigeria.
Port Harcourt city is growing faster in population due to migrants in search for urbanization and industrialization which is on the increase. When there is increase in the rate of migration, the demand for land increases to meet the increasing demand for food production, housing and modern infrastructures. Also there is increasing demand for jobs and social amenities, thus overstretching available facilities.
It is this impacting factors that have prompted this research work, but for this work we are considering the impact of internal migration on the growth of Port Harcourt city and how sensitive the population is to the parameter values of population that changes over time.
In this study, we are expected to determine the impact of internal migration in Port Harcourt city and Bori town respectively and to show the effect of the variation of some parameters that affect the growth of Port Harcourt city. It is against this background we have considered the following expected outcome.
1) To determine how internal migration affects the growth of Port Harcourt city and Bori town.
2) To determine how the rate of internal migration can be minimized to enhance the growth of Port Harcourt City and Bori town.

Method
The purpose of this chapter is to define the core method with which the proposed research problem is solved. The research aims at determining the impact of internal migration on the growth of Port Harcourt city in Nigeria. For the purpose of this work, the following are considered: i. The internal migrants are effective working population.
ii. The migration is from Bori town to Port Harcourt city.
iii. The migration will enhance the population of Port Harcourt city.

Simplifying Assumptions
The mutualistic interaction between a bigger city called Port Harcourt city council and a relatively smaller city called Bori town can be understood by considering the following simplifying assumptions: i The internal migration of the effective working population of Bori town tends to enhance the growth of Port Harcourt city, where as the internal migration of the Port Harcourt city working population also tends to enhance the growth of Bori town. ii The interaction between effective working population within the Port Harcourt city tends to inhibit the growth of Port Harcourt city.
iii The growth of Port Harcourt city is enhanced proportionately which is mathematically tractable based on the method of separation of variables in differential equation modelling but do not offer meaning in the study of human population growth because, the population of Port Harcourt city cannot continue to grow indefinitely due to constraints of job facilities and social infrastructural facilities.
iv The internal migration of effective working population from Port Harcourt city to Bori town is lesser than that of Bori town to Port Harcourt city due to lesser jobs and social infrastructural facilities in Bori than Port Harcourt.
v The initial population sizes of both Port Harcourt and Bori are assumed using the available data.
We will apply these assumptions to formulate the mathematical model next.

Mathematical Formulations
In this study a first order differential equation will be utilized to determine the effect of internal migration on the growth of Port Harcourt city and Bori town. The first order differential equations are having the following mathematical structure: These model equations describe the interaction between Port Harcourt city and Bori town population at time t. The parameters in the models are defined as follows:  P(t) represent the population size of Port Harcourt city at time t.
 B(t) represent the population size of Bori town at time t.
 α 1 denotes the growth rate of Port Harcourt city.
 α 2 denotes the growth rate of Bori town.
 β 1 represents intra-competition coefficient due to internal migration between the city of Port Harcourt.
 β 2 represents intra-competition coefficient due to internal migration between the city of Bori.
 r 1 denotes the rate of internal migration of Bori town to enhance the growth of Port Harcourt city.
 r 2 represents the rate of internal migration of Port Harcourt city to enhance the growth of Bori  r 1 = 0.000 000 12  r 2 = 0.000 000 8 Initial condition P(0) = 635 000, B(0) = 150 000 with a duration growth of fifteen years.

Method of Analysis: Illustration
A non-linear ordinary differential equation has been proposed for analysing the research work. Since the proposed interaction models are non-linear, it does not have a closed formed solution. Hence, we have proposed to implement the numerical simulation method that is based on the ordinary differential equation of order 45 which otherwise is called O.D.E 45. The matlab ODE 45 numerical scheme will be used to run the calculation with the parameter values given above.
In determining the effect (percentage) of the internal migration on the populations of Port Harcourt and Bori, the following equations were used with the results as shown in the tables.

Discussion
Tables 1 and 2 evaluate the effect of the rate of change on the population of Port Harcourt city and Bori town in the face of changing internal migration rate.
From Table 1 in the first row, column 1 and 2 the population of Port Harcourt denoted by P(t) old and P(t) new are the same (635 000) being the initial populations. Therefore, on the basis of either the theory of depletion or increase, the effect 1 in column 3 can be calculated using percentage depletion or increase formula to arrive at the value 0 for effect 1. The second row shows a slight increase in P (t) old column (647 233) and a sharp increase in P(t) new column (9 178 488). This has resulted in a percentage increase of 1318.o% which is very high. The likely implication of this result is that it will impact on Port Harcourt city negatively since the demand for jobs will be too far more than the jobs available.
From row 3 down there is a steady increase on P(t) old while that of P(t) new which was extremely highly reduces steadily. This has resulted in steady decrease in the effect 1 column (1318.0% to 116.4%) which is still not good because that is higher than the old population of Port Harcourt. The migration of the effective working population of Bori to Port Harcourt has its impact on the population of Bori town.
A close observation at row 1 column 4 and 5 of Bori population denoted by B(t) old and B(t) new, the population of the old and new are the same (150 000) that is initial populations. Hence, on the basis of either the theory of increase or depletion the value for effect 2 that is column 6 can be calculated using percentage increase or depletion formula to arrive at the value 0 as shown. In Table 3 the scenario is different as P(t) new shows a dramatic decrease due to a lower internal migration rate (0.000012). The values of P(t) new has dropped from 9 241500 as at second year of down the year (t).

Conclusion
The findings from the results showed that if growth rate and intra-competition coefficient due to internal migration of Port Harcourt city are fixed and the rate of internal migration varies, it impacts on the population of Port Harcourt. From the results the population of Port Harcourt grew along sides the migration rate. As the rate increased the P(t) new values increased and as the rate dropped the population dropped too. This shows that the population is sensitive to the rate of internal migration.
Also the population did not grow without bounds as the interaction between the other parameters tends to reduce the growth. But this reduction as a result of negative feedbacks did not give the required population growth that will enhance the population of Port Harcourt city.
Bori town is not left out in the share of the impact of internal migration, the population of Bori continued to deplete alongside with the increasing migration into Port Harcourt city. The decrease in the growth rate resulted in depletion which shows less sensitivity of the population. This is a better indicator for future prediction and it implies that if the trend is not checked it will cripple the economy of Bori town. The work clearly shows that internal migration drives population growth of a city and if not well managed the interaction within the city becomes unhealthy since too many persons will be competing for scarce resources. The infrastructures and the personnel will be over stretched since the population growth outweighs the facilities on ground. This brings to bear the issue of development of other cities and decentralizing industries and facilities in order to reduce the rate in which people in the urban and rural areas migrates to the cities.
This study has unveiled how the search for good jobs and a better life has led to migration into the city from the rural and urban areas which if not controlled may have a negative impact on the city and the rural areas. Therefore, the numerical ideals of this work can be extended to answer the following questions which we did not address in this work. Recommendations are; to model