目录
摘要 II
关键词 II
Abstract III
引言
1 引言 1
1.1 课题背景及研究意义 1
1.2 国内外研究现状 2
1.3 本文主要研究方向及内容 2
2 无扩散情况模型方程的研究 5
2.1 模型方程的建立 5
2.2 计算平衡点 5
2.3 平衡点稳定性分析 7
2.4 小结 9
3 有限空间内扩散模型方程的研究 11
3.1 扩散模型方程的建立 11
3.2 扩散模型方程的稳定性分析 11
3.3 小结 12
4 数值模拟 13
5 总结与展望 15
致谢 16
参考文献 17
附录 18
有限空间内扩散型捕食者—猎物模型的研究
摘要
对捕食者—猎物模型的相关研究是生物数学这门科学中比较重要的一个分支，可以通过数学模型计算，分析两个种群在不同因素影响下如何保持生态平衡，以及是否具有稳定性，可以为现实中的种群生态问题提供理论依据以便更好地解决问题。
本文以基本的捕食者—猎物模型为基础，考虑到当今生态环境的状况，着重研究思考由于人类大规模侵占土地资源，当在有限的空间内时猎物与捕食者如何保持生态平衡的问题。将稀疏效应相关参数添加进方程，同时考虑捕食者种群内部的相互作用关系，即捕食者在搜寻猎物以及捕到猎物后进食时的相互干扰情况。研究计算在这两种影响因素下：（1）不考虑捕食者与猎物种群存在扩散情况，找到种群平衡点，分析稳态存在条件；（2）考虑捕食者与猎物种群存在扩散情况，将其限制在有限空间内，给定扩散率，分析平衡点稳态存在条件。
在经过研究计算分析之后我们可以得到存在扩散率以及不存在扩散率时，唯一正平衡点是局部渐近稳定的。同时将无扩散率情况的模型方程进行数值模拟，得出捕食者种群在存在内部竞争的时候总是会趋于灭绝的，而被捕食者种群在度过平衡点的时间节点后会趋于稳定，这符合现实情况，具有生态意义。
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ORPREY MODEL IN A FINITE SPACE
ABSTRACT
The research on the predatorprey model is an important branch in the science of biological mathematics. You can use the mathematical model to analyze how the two populations maintain ecological balance under the influence of different factors and whether they have stability. The real population ecological problem provides a theoretical basis to solve the problem better.
This article is based on the basic predatorprey model, taking into account the current state of the ecological environment, focusing on the question of how to maintain the ecological balance between prey and predator when in a limited space due to largescale human occupation of land resources. The parameters related to the sparsity effect are added to the equation, and the interaction between the predator populations is considered, that is, the mutual interference between the predators during the search for prey and the prey. The research calculations are based on these two influencing factors: (1) Do not consider the predatorhunting species population to spread, find the population balance point, and analyze the steadystate existence conditions; (2) Consider the predatorhunting species population to spread the situation. Restricted to a limited space, given the diffusivity, find the population equilibrium point, and analyze the existence conditions of steady state.
After research, calculation, and analysis, we can find that the only positive equilibrium point is locally asymptotically stable when diffusivity exists and does not exist. At the same time, the model equations without diffusion rate are numerically simulated, and it is concluded that the predator population will tend to become extinct when there is internal competition, and the predator population will tend to stabilize after passing the time node of the equilibrium point, Which is in line with reality and has ecological significance.

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