Introduction A model is a representation or an abstraction of a real or abstract system or a process. Thus, mathematical modeling is the process of representing real or abstract phenomenon using mathematical tools, which includes mathematical notations, formulas, relations, and symbols. Models are necessary since they enable scientists define a problems, understand data, organize thoughts/ideas, make predictions and communicate with the world (Bender, 2000, p. 23). Most real problems or phenomenons are complex and, therefore mathematical models enable us to capture certain key aspects of the phenomenon. Usually, it is impossible or unnecessary to represent some facts about a phenomenon since they are either too complex or unnecessary. Models can take many forms, including statistical models, dynamical systems, or game theoretic models, differential equations (Korotayev, Malkov & Khaltourina, 2006). Other forms of mathematical models use logic and they are mainly applied in humanities and philosophy. Although models represents real phenomenon they do not necessarily agree related experimental measurement. This has been a basis for advanced theories to define the related phenomenon. The role that mathematical models Figure 1 Brief History There are two basic types of mathematical models namely deterministic and empirical models. ...

Through these models, scientists were able to explain basic physical properties such as motion. Newton’s models have been fundamental in Newtonian mechanics, which is still a fundamental aspect of modern mathematical physics. Initial models were deterministic and, they enabled scientist to write explicit mathematical formulas connecting two or more variables. Some of the well-known Newtonian models include The models occur naturally for a point mass m moving in space. such models could not be applied on multivariate problems. This necessitated the development of empirical models that are based on experimental observations. Key Developments Key developments in mathematical modeling include application of computer software to solve numerical equations resulting from mathematical models. Other developments involve the application of abstract theories in the development of models. Early models relied on experimental theories that resulted in deterministic models. However, advances in the field have proved that other branches of mathematics such as topology, algebra, and analysis are of equal significance in formulating mathematical models. Indeed, mathematicians are exploring the possibility of abstract approach in formulating models. Mathematical Content Fundamental theory in mathematical model is dependent on outcomes of natural laws and phenomenon. For instance, models of population studies have to incorporate basic theories on the subject. For example, the models have to account for population growth rate, implication of diminishing resources, carrying capacities of the environment, predation, death, and decay. Consequently, modelers have to have concrete knowledge in formulating natural problems using mathematical
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