[2023 Jiangxi Postgraduate Mathematical Contest in Modeling] The third question is the diversity of plants, 48 pages of papers and Python code
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2023 Jiangxi Provincial Postgraduate Mathematics Competition Topic 3
plant diversity
As producers in the food chain, plants absorb carbon dioxide through photosynthesis to produce oxygen, and at the same time provide food and habitat for other organisms to support their survival. Plants also play a role in preventing soil erosion and alleviating the greenhouse effect in the ecosystem. Therefore, the diversity of plants helps to maintain the stability of the food chain, ecological balance and biological diversity.
There are many kinds of plants growing in a forest, and there are different relationships between different kinds of plants. Please build a mathematical model to solve the following problems:
Question **** 1 If there is a single relationship between various plants in the forest, try to study the law of the change in the number of various plants, and analyze how to maintain the diversity of plants in the forest.
Question **** 2 If there are several different relationships between various plants in the forest, try to study the law of the change in the number of various plants, and analyze how to maintain the diversity of plants in the forest.
Question ****3 Now it is discovered that a foreign plant has invaded this forest, which has caused a sharp decrease in the number of certain plants in the forest and is in danger of extinction. In order to clean up this invasive plant, the forest management department intends to use some special measures (for example: use the method of burning). Please give the specific implementation plan of this special measure (for example: measure intensity, duration, etc.), so that within a certain period of time, the scale of the original plants in the forest will recover without the risk of extinction.
2 Introduction to the paper
*Summary*
Plants are producers in the food chain, absorb carbon dioxide through photosynthesis, produce oxygen, provide food and habitat for other organisms, and support their survival. In addition, plants also play an important role in preventing soil erosion and alleviating the greenhouse effect. The diversity of plants plays a key role in maintaining the stability of the food chain, ecological balance and biological diversity. Therefore, protecting and maintaining plant diversity is of paramount importance to ensure the health and continued prosperity of ecosystems. Plant diversity is crucial to maintaining food chain stability, ecological balance and biodiversity. It is necessary to pay attention to the maintenance of plant protection and species diversity to ensure the healthy and sustainable development of the ecosystem. By maintaining the living environment of plants, protecting natural resources and adopting sustainable agricultural and forestry management measures, we can promote the prosperity of plant diversity, maintain ecological balance, and benefit the prosperity and health of the entire earth's ecosystem.
Aiming at problem 1, this paper establishes the difference equation model of the number of plants in the forest, and uses the ordinary differential equation (ODE) solution method to describe the change of the number of different plants in the forest over time. In order to maintain the diversity of plants in the forest, this paper can adopt A more uniform cleanup strategy, avoiding the complete removal of certain plants. This means that during the cleanup, the population of each type of plant should decrease in roughly the same proportion to maintain a balanced population.
For the second question, this paper uses a multivariate ecosystem model based on difference equations to simulate the changes in the number of various plants in the forest. The fifth and fourth order Runge-Kutta methods are used to solve the model. From the analysis of the results, it can be seen that the number of different types of plants shows different growth and decline trends during the simulation process, and the number of some plants may show periodic fluctuations, while the number of other plants may show a stable trend. This change may be due to the interaction of factors such as intrinsic growth rate, competition intensity, and environmental carrying capacity, which verifies the validity and accuracy of the model.
For the third question, this paper implements a predation-competition model based on python simulation, which is used to simulate the change of the number of different plant populations in the plant community over time, and uses the fourth-order Runge-Kutta method (RK4) to solve the difference equation model, and analyze the simulation results It can be seen that the number of different types of plants presents different trends during the simulation process. The number of some types of plants increases gradually and reaches a certain equilibrium state, while the number of some types of plants shows periodic fluctuations or gradually decreases, which reflects the complexity of dynamic changes in plant populations.
Keywords: population dynamics, fourth-order Runge-Kutta, difference equation model, dynamic change
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