Mathematical Model on the Effects of Global Climate Change and Decreasing Forest Cover on Seasonal Rainfall

This study involves the study of the long-term behaviors of rainfall as it is affected by changes to forest area and the rise in global temperature. Global temperature and forest cover are considered annually while the amount of rainfall are considered seasonally to best capture the effects of severe weather hazards such as drought and puberty. A differential equation model was developed and verified using the mean global temperature annually, forest area, the daily amounts of rainfall. The rise in global temperature as well as the decline in forest area can be, as shown in the seminar, represented by logistic equations. Rainfall is, however, represented as a periodic function; hence, second order differential equation, of which the solution is periodic, is used to represent the rate of change in the amount of rainfall. In addition, by correlation analysis, the predator-prey terms of forest, global temperature and rainfall are presented in the models. DOI: 10.7176/MTM/9-1-03

Analysis of climate change and its extreme is becoming more important as it clearly affects the human society and are essential for exploration of ecological and societal changes. Climate change is an irreversible consequence of the global warming phenomenon. Global warming, or the greenhouse effect, has been brought about by an increase in greenhouse gases (GHG) in the atmosphere. Climate change will impact on bio-physical systems and ultimately will have consequences to human well being. Understanding climate change is fundamental to prepare and to cope with future risk. Climate change is not uniform over space and time and its impact on bio-physical system varies from place to place. Therefore, it is necessary to understand climate change at the local scale and aim to get site-specific information. Climate change, which is induced by global warming, has become a global concern because it has the potential to impact many systems and sectors which would threaten human well being. The El Nino phenomenon has played an important role on the behavior of rainfall in different areas of the world; it is also a major cause of extreme weathers in many regions of the world [?]. Changes in global temperature, therefore, lead to variations in the amount of rainfall in different areas of the world, an amount which depends on the level of forest transpiration. Deforestation is the single largest cause of forest area diminution, even compared to natural disasters such as wildlife. The amount of rainfall is an important indicator for severe weather conditions such as drought and flood. Many mathematical models in ecology have been developed to describe the relationship between carbon dioxide and global warming, forest and carbon dioxide or greenhouse gases, climate change and extreme weathers, species (or population) survival and pollution, i.e., industrialization, species (or population) survival and forestry or biomass resources and industrialization, population, and pollution. There is also a study to control the amount of pollution in the environment in order to restrain the global carrying capacity of population. In another study, the dynamic of the relationship between biomass (which can be viewed as forestry resources), industrialization, population, pollution, and pollution released by the biomass resources (which obviously includes carbon dioxide) was written in the form of a system of differential equations. Furthermore, there is a mathematical model related to the competition among rain forest species using the Lotka-Volterra predator-prey model [?]. However, a mathematical model that simultaneously represents the relationship between global temperature, amount of forest area and amount of rainfall has never been presented.

Statement of the Problem
Changes in global climate would significantly affect human health, natural aquatic and terrestrial ecosystems, and agricultural ecosystems. World-wide attention recently has turned to these issues and scientists from many disciplines and many countries are working to assess the potential magnitude and direction of the changes and the risks to the biota. This study is intended to answer the following basic questions: 1. Can we formulate a mathematical model that describes the relationship between global temperature dynamics, forest area and amounts of rainfall? 2. What are the impacts of the rise in global temperature and decline in forest area on stability of seasonal rainfall? 3. Is there the interaction between global temperature, forest area and rainfall? 1.3 Objective of the Study

General Objective:
The general objective of this study is to develop a mathematical model of the effects of global climate change and decreasing in forest cover on seasonal rainfall.

Specific Objectives:
The principal objectives of this study are: ➢ To develop a mathematical model that describe the relationship between global climate change and decreasing forest area on seasonal rainfall ➢ To study the effects of the rise in global temperature and decline of forest area on the stability of seasonal rainfall ➢ To know the interactions between global temperature, forest area and rainfall

Significance and Beneficiaries
This study provides reliable information on how we can use the mathematical modeling to know the effects of global temperature and decreasing forest cover on seasonal rainfall. The outcome of the study benefits building and running of a model is a process by which theory and observations are mathematically evaluated, codified and integrated. Also, initiate other researchers to identify and then assimilate observational measurements that are initially incomplete.

Chapter 2 Literature Review
The first global models (models of world dynamics) were developed by J.W. temperature T = T(t). The annual global mean insolation is known as the solar constant Q. The incoming energy absorbed by the Earth is then modeled by the term Q(1-), where is the average global insolation reflected back into space. The loss of energy is modeled by a linear approximation A + BT, with parameters A and B. and finally its model equation for the annual global mean temperature is then given by where the left-hand side of this equation represents the change in energy stored in the Earth's surface. The parameter R is the heat capacity of the Earth's surface. Bampfylde et al., (2005) introduces a mathematical model related to the competition among rain forest species using the Lotka-Volterra predator-prey model. Dubey and Narayanan, (2010) models a dynamic of the relationship between biomass, industrialization, population, pollution, and pollution released by the biomass resources was written in the form of a system of differential equations. Md. Hamidul Islam and Md. Abdus Salam et al.,(2011) uses a mathematical model to compute the crucial roles of water vapor in global warming by expressing the temperature T as a function of carbon dioxide C, that is T(t) = f(C). The developed differential equation is, = = The term ( = ) denotes the change of temperature per unit change in CO2 concentration in the atmosphere.
This model of global warming is without the effect of water vapor. However, a mathematical model that simultaneously represents the relationship between global temperature, amount of forest area and amount of rainfall has never been presented.

Chapter 2 Mathematical Preliminary
In this chapter, we state some theorems and give the definitions of terminologies which are most crucial for the seminar as an input.

Existence and Uniqueness Theorem
Before one spends much time attempting to solve a given differential equation, it is wise to know that solutions actually exist. We may also want to know whether the solution is uniquely satisfy the equation with the given initial condition. We consider here the autonomous system in R n. i.e., a collection of equations that do not explicitly contain the independent variable. More generally, autonomous systems have the form ′ = ( ) ( , ) = + ℎ 0 ( , ) is therefore always well defined in neighborhood of ( 0 , 0 ). Applying the Picard mapping, ( )( , ) = + ( ℎ 0 )( , ) = + ℎ 0 ( . ) = ( , ) Which proves that, is a solution of a differential equation which satisfies the initial condition ( 0 , ) = as long as is in a neighborhood of the point 0 defined by | − 0 | ≤ ′ and is any point such that | − 0 | ≤ ′ . about that same equilibrium, where ( 0 )is the × matrix of partial derivatives of . = ( ⁄ ) evaluated at 0 . A linear system of first order ordinary differential equations is defined by where the unknown is a map on an interval of R, say Γ, taking values in a normed vector space over a field , defined differentiable on a sub interval of Γ. The name linear for system the above equation is an abuse of language. The associated linear system of (2.1.4) is ′ ( ) = ( ) ( ) (2.1.5) Another way to distinguish systems (2.1.4) and (2.1.5) is to refer to the former as a non -homogeneous linear system and the latter as an homogeneous linear system. as long as ∅( , ) remains in since (∅( , )) is decreasing. Thus ∅( , )cannot intersect the boundary of ( 0 , ) for ≥ 0, so ∅( , ) remains in ( 0 , ) for ≥ 0, and 0 is stable. Now suppose is a strict Liapunov function, but 0 is not asymptotically stable. Then there is an ∈ ( 0 , ) so that ∅( , ) does not go to 0 as → ∞t. Since the orbit is bounded, there is an 1 ≠ 0 , and a sequence → ∞ so that∅( , ) → 1 → ∞. Note that by semi-group property for orbits ∅( + 1, ) = ∅(1, ∅( , ))

Chapter 3 Modeling the Effects of Global Climate Change
In this chapter, we present model description and formulation. The mathematical theory of climate is a branch of the theory of climate, which investigates the behavior of the climate models solutions on the arbitrarily large time scales by the use of a collection of mathematical methods. This seminar relies on the hypothesis that changes in global temperature, forest area and amount of rainfall are related.

The assumptions
The following assumptions are made in order to construct the model: (i) The behavior of global temperature is apparently increases exponentially. However, the global temperature cannot possibly increase forever. It should be bounded at some temperature level.
(ii) Today the forest area have been decreased exponentially from the past. Since the forest area cannot decrease below zero and cannot be larger than the maximum area of the studied region, be an example of a logistic decay. (iii) The amount of rainfall is apparently seasonal and therefore periodic. The second order differential equation with periodic solution should be considered to capture the seasonal rainfall. The amount of periodic rainfall will be represented by the following second order differential equation

Model Formulation and Analysis
Since the other two variables can be represented by first order differential equations, this second order equation representing the behavior of rainfall will be reduced to two first order differential equations. Introducing an intermediate variable S as the rate of change of rainfall, the amount of periodic rainfall will be represented by the following system of first order differential equations. = Is det( − ) = 2 + 2 = 0 Solving we obtain 1,2 = ± . Since all the Eigen values of A have zero real parts by theorem 3.3.1 the above system of equation has a stable solution.

Modified models with predator-prey terms
The amount of forest area, amount of rainfall and global temperature are used to obtain the correlations between the rates of change of each variable, , , , and the other terms including the cross products or the predator-prey terms. Note that under our preliminary hypothesis, the rates of change in forest area and rainfall are affected by the global temperature. On the other hand, there can be many other factors affecting the rate of global temperature change such as waste gases from industry or transportation, etc. The cross product terms between global temperature and others are not shown in the dynamics of the global temperature. The modified models with predator-prey terms are The new parameter, , represents the rate of change in forest area caused by the amount of water (rainfall) absorption. The terms in Eq. (3.3.1) represent the proportion of the amount of rainfall in the forest and FS is the proportion of the rainfall difference rate (or the rate of change in rainfall) in the forest. The coefficient , is the growth rate of the forest affected by the proportion of rainfall in the forest. The constant is the rate of decrease that indicates some amount of rainfall remaining in the forest. The parameters, R and S, are both the difference rates representing the changing behavior of the rain falling in the forest. The TF terms in Eq. (3.3.1) indicate the proportion of the forest cover at each temperature level. The parameters, and , are the difference rate of the forest cover and the rainfall difference rate, respectively. The TR terms in Eq.(3.3.1) represent the proportion of rainfall at each temperature level, TS is the proportion of the rainfall difference rate at each temperature level. The parameter is the decay rate of the forest caused by water absorption at each temperature level.
And are the decreasing and increasing rainfall difference rates, respectively, affecting by the proportion of rainfall at each temperature level. The parameters, and , are the difference rates with which the rainfall changes its behavior at each temperature level. The RS terms in Eq. (3.3.1) represent the relationship between the rainfall difference rate and the amount of rainfall. Consider the situation where the rainfall amount is near the saturation point; the rainfall difference rate could either be substantially reduced or slightly increased. Conversely, at the low level of rainfall, the rainfall difference rate could either be substantially increased or slightly reduced. The parameters are, respectively, the difference rate of the amount of rainfall and the rainfall difference rate affected by the relationship between the change in the behavior of the rainfall and the amount of rainfall.

Chapter 4 Conclusion
The relationship between global temperature dynamics, forest area and amounts of rainfall has been mathematically formulated. In this study, there are two models representing the long-term behaviors of rainfall as affected by the rise in global temperature and the decline in the forest cover. The hypotheses for this study are the logistic patterns of global temperature and the amount of forest cover, and the periodic behavior of the rainfall pattern. Since the behavior of rainfall is seasonal the increase in global temperature and decreasing in forest cover cause water management difficult and eventually affect to drought and flood. In the second part of the study, the model is refined to cover interactions between variables. The parameter indicates that the rainfall difference rate is affected by the proportion of rainfall at each temperature level. Therefore, global temperature influences the fluctuation of rainfall. The model on the rainfall pattern can further be improved to capture the fluctuations in the annual rainfall amount. To do so, the amplitude of the periodic functions representing the rainfall amount has to fluctuate. It would also be interesting to project the amount of rainfall to the future and to work on some scenario analysis using varying reforestation policies.