Generally speaking, there are at least two ways to establish an ecological mathematical model:
One is the compartment method, which is used to study the material and energy flow in each compartment in the ecosystem and give a quantitative expression.
One is the experimental component method, which is mainly used to analyze the ecological processes of complex ecosystems (such as predation and competition).
The following is a schematic diagram of the general process of ecological modeling:
Can be summarized as follows:
To prepare the model, we must first make clear the research problem, determine the modeling purpose, determine the system boundary, determine the components of the model (input and output variables, initial and driving variables, parameters, space-time scale), and establish the flow chart. Understand the actual background of the problem, clarify the purpose of modeling, and collect all kinds of information necessary for modeling, such as phenomena and data. And try to find out the characteristics of the object, so as to preliminarily determine which model to use. In a word, we should make preparations for modeling. This step cannot be ignored. When encountering problems, we should humbly consult comrades engaged in practical work and try our best to master first-hand information.
According to the characteristics of the object and the purpose of modeling, model hypothesis can be said to be a key step to simplify the problem and make assumptions in accurate language. Generally speaking, it is difficult to turn a practical problem into a mathematical problem without simplifying the hypothesis, and even if it is possible, it is difficult to solve it. Different simplified assumptions will lead to different models. Unreasonable or oversimplified assumptions will lead to model failure or partial failure, so the assumptions should be revised and supplemented. If the assumptions are too detailed, trying to take all the factors of complex objects into account may make it difficult or even impossible for you to continue your next work. Usually, the basis of making assumptions is based on the understanding of the inherent law of the problem, the analysis of data or phenomena, or the combination of the two. When making assumptions, we should not only use the knowledge of physics, chemistry, biology and economy related to the problem, but also give full play to our imagination, insight and judgment. We should be good at distinguishing the primary and secondary problems, firmly grasp the main factors, abandon the secondary factors, and linearize and homogenize the problems as much as possible. Experience often plays an important role here. When writing assumptions, the language should be accurate, just like writing known conditions when doing exercises.
Model construction is to analyze the causal relationship of the object according to the assumptions made, and construct the equation (or inequality) relationship between various quantities (constants and variables) or other mathematical structures by using the internal laws of the object and appropriate mathematical tools. In addition to the professional knowledge of some related disciplines, it is often necessary to apply mathematics knowledge more widely to open up new ideas. Of course, proficiency in mathematics is not required. It is to know what kind of problems these disciplines can solve and how to solve them generally. Similarity analogy, that is, according to some similarities of different objects, borrowing mathematical models in known fields is also a method of constructing models. One principle to follow in modeling is to use simple mathematical tools as much as possible, because the model you build always wants to be understood and used by more people, not just a few experts.
Establishment of quantitative model (or quantification of conceptual model): selection of model type, establishment of model (determination of functional relationship between variables), parameter estimation and calibration, preparation of computer program, model verification: careful examination of mathematical formulas and computer programs, preparation of model documents.
Various traditional and modern mathematical methods, especially computer technology, can be used to solve models, such as solving equations, drawing, proving theorems, logical operations and numerical calculations.
Model analysis is a mathematical analysis of model solutions. Sometimes it is necessary to analyze the correlation or stability of variables according to the nature of the problem, sometimes it is necessary to give a mathematical prediction according to the obtained results, and sometimes it may be necessary to give a mathematical optimal decision or control. In these two cases, it is often necessary to analyze the error, stability or sensitivity of the model to data.
Model checking converts the results of mathematical analysis back to practical problems, and compares them with actual phenomena and data to test the rationality and applicability of the model. This step is very important for the success or failure of modeling and should be taken seriously. Of course, some models, such as nuclear war models, cannot be required to be tested in practice. If the results of model verification are not in line with reality or partially, the problem usually lies in model assumptions, and some models need to be modified, supplemented and re-modeled.
Spatio-temporal expansion of the model: the established model is expanded in time and space scales.
Model application: the way of application naturally depends on the nature of the problem and the purpose of modeling.
Model operation and evaluation Levins( 1966) once put forward three criteria for establishing mathematical models:
(1) authenticity, and the mathematical description of the model should conform to the reality of the ecosystem;
⑵ Accuracy refers to the difference between the predicted value and the actual value of the model.
(3) Universality, that is, the scope and breadth of application of the model.
In practice, it is difficult for a model to meet these three criteria at the same time. Walters had an incisive exposition on this, and also introduced two indicators related to authenticity and universality, namely, resolution and integrity. These two concepts were put forward by Bledsoe and Jamieson( 1969) and Holling( 1966) respectively.
In short, not all modeling processes have to go through these steps, and sometimes the boundaries between steps are not so clear. Don't stick to formal step-by-step modeling, you can use it flexibly in the actual modeling process.