In the 1920s, Italian mathematician Volterra established the differential equation "predator model" of fishing according to the relationship between predator and prey, which proved that if the catch exceeds a certain amount, the big fish will decrease and the small fish will increase, and if the catch is properly reduced, it will be beneficial to the survival of the big fish. People catch fish according to the best catch, which is beneficial to the stable and high yield of fish, thus obtaining the best economic benefits.
The "connection" plan compiled by American econometrician Klein, winner of the Nobel Prize in Economics, is the largest econometric model in the world. It connects the economic information of many countries and can understand the world trade situation. The application of macroeconomic econometric model can predict the economic development trend and formulate economic policies, which fully shows the great power of mathematical model.
First, the definition of mathematical model
At present, there is no uniform and accurate definition of mathematical model, because different angles can have different definitions. But we can give the following definition. "A mathematical model is an abstract and simplified structure about a part of the real world, used for special purposes." Specifically, a mathematical model is an equation or inequality established by letters, mathematics and other mathematical symbols, and it is a mathematical structural expression that describes the characteristics of objective things and their internal relations, such as charts, images, block diagrams, etc.
2. Methods and steps of establishing mathematical model
First, model preparation.
First of all, we should understand the actual background of the problem, clarify the modeling purpose, collect all kinds of necessary information, and try our best to understand the characteristics of the object. Second, the model hypothesis.
According to the characteristics of the object and the purpose of modeling, it is a crucial step to simplify the problem reasonably and make assumptions with accurate language. If all the factors of the problem are taken into account, it is undoubtedly a courageous act and the method is very poor. Therefore, a superb modeler can give full play to his imagination, insight and judgment, be good at distinguishing priorities, and linearize and homogenize problems as much as possible in order to simplify the handling methods.
Third, the model composition.
According to the assumptions made, the causal relationship of the object is analyzed, and the equation relationship between various quantities or other mathematical structures are constructed by using the internal laws of the object and appropriate mathematical tools. At this time, we will enter a vast world of applied mathematics, where there are many lovely children at the knees of the elderly with high numbers and probabilities. They are graph theory, queuing theory, linear programming, game theory and many other theories. They are really a great country with unique views. But we should remember that the mathematical model is established for more people to understand and apply, so the simpler the tool, the more valuable it is.
Fourth, the model is solved.
We can use all kinds of traditional and modern mathematical methods, especially computer technology, such as solving equations, drawing pictures, proving theorems, logical operations, numerical operations and so on. Solving a practical problem often requires complicated calculation, and in many cases, the system operation has to be simulated by computer, so programming ability and familiarity with mathematical software packages are very important.
Fifth, model analysis.
Mathematically analyze the model solution. "Cross as a ridge side into a peak, far and near? Quot Whether you can analyze the model results carefully and accurately determines whether your model can reach a higher level. Also remember that in either case, error analysis and data stability analysis are needed.