新手进阶建模(6)平衡原理和机理分析应用
简单的说,平衡原理就是原来的平衡的因为一边发生了变化使平衡破坏 然后反应朝着能够削弱这次变化的方向发展。现实中绝大多数问题都满足平衡原理,交通红绿灯的设置与路宽的关系,收费站的卡口数与车流量的关系等...建模中我们拿到的题目并不会直接说明其中哪些因素满足平衡关系,所以这需要我们首先要查阅资料找到该问题中隐含的平衡关系因素。然后建立微分方程求解。
通过几道例题来说明一下:
1.建模描述一个地区内人口的自然增殖的过程。即考虑由于人口的生育和死亡所引起的人群数量变化的过程。
第一:找平衡关系。题目只说了出生与死亡,给了我们一对平衡因素(人口的增长还与该地区的人口迁入迁出、资源分配有关,出
于简便,只考虑出生率与死亡率)。
第二:找平衡变量,与出生死亡有关的我们可以联想到出生率、死亡率。
第三:约束条件,建立方程
1. 人群个体同质。 2. 群体规模大。 3. 群体封闭,只考虑生育和死亡对人口的 影响。 4. 从大群体的平均效应考虑生育和死亡对人口的影响。(生育率和死亡率) 5. 群体增长恒定。 6. 个体增长独立。
人口数在区间△t ]内的改变量N等于这段时间内出生的个体数与死亡的个体数之差。 令B(t, △t, N), D(t, △t , N) 分别表示在时间区间△t ]内生育数和死亡数, 则有:
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则出生率:
data:image/gif;base64,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
死亡率:
data:image/gif;base64,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
自然增长率 :
data:image/gif;base64,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
又因为初始条件:
data:image/gif;base64,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
继续推演可得:
data:image/gif;base64,R0lGODlh7AAuAMQAAP///wAAANzc3FRUVMzMzERERBAQEKqqqmZmZpiYmCIiIoiIiDIyMrq6uu7u7nZ2dj09PQAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACH5BAEAAAAALAAAAADsAC4AAAX+ICCOZGmeaKqubLs2bizPdG3ftpPg/Jw4KMGgRywajyqhYIZAOkmO5kn4rFqvKULBQHooAogHYoAAjhaEkiDtciiSbJY3EHcMAoWDqLGYDrGAgU8HfyMFUiJeJAwmBTttBQ8pjjELCoUiiCKMJlSCn6A4CI8jAXojB3QiCaSlSy1ABAFmJgGvLQkLsyMOp2e+I56hw8QsCrcADbsjA1wiBcgABM4tbIcn0zENAg4BfSIHtCIEmCLCxejFCwsHB9SJnJkGce8EDwwKD61BI7LI9vj0tfiGgNq3Eu/MlUvHMBAlAAPKMRhwIMEABgdFJATwkAUMQws7svgm62PGUp3+FjZc+SRBvAH7VGVqJeBNCVstGoiTFUcETlikGBQAIOBjCQXiiKpkybSIAVIGkCkjkSAAiZolGmxEAezZ0BFatd1KJWDfCKRqljZda0PAz2xd4gFAYJVXwgdqSYQzUXIE3hj7DDw4OWKjAAhsE/eQdWao0XyLbBYuQcmB0RNmn8WrfBnFvgcGCGtMqbj0DQNABDCQ9KjbZaFEP0IjoSDNNyFJ2TXYzXu3rlO1AdwekHTcPtfYFp4zzTzng3ZCFgC5F4AiWAa5RrDSq+/VtMABwosfH8Dmge7jnpooqGCfJhLsSDdXocPKD5YOvqroyoM/DbnB5DXfe0hEwdQDPZ3+4N8NC2qT2XLzwZfgGi7MUccdeewhmghuwHEEgfAVd4N0OBjoR4QoAMhRZihYUs57KvISySQs5lBjgzXgiIuISqFownY3RaNCLssA0EsJ8ZkQS5FB+igIhCjORgJcOXHjDSpJkYPNMyBS6SQWUK51wAIJlEXmaP3ck0+N8MlzxgmG9fPTOGoK9OUVYTJlmVs7DCAJABuJtAJJAZh0Ql0kXFZASI+Q5+ijkEYq6aSUVmqpeMEgBsoYEXXq6Z+JAnCAZOaQ6pOQKNS3yVBFnYAWWDvJ5MqdgOTZVBhqmBqWC9ugYguLr4LTiH7JbEUrErYyZUBnDtwlIFUIDYbCO3v+ldBXIs8ey0OyTnDq6adTIEpCQpyx8Flo00J7QgGb7WCZtk9w25BL6yIT3HAiEnBcocn90ltvv4lwbyhKmCYvQ3aWACQ46EmjXgnsudeiLwZMal7DgmixUYcpUPhktorlx4KOLcRYGiFKzoiCoHiCrBiCK5D8AptsjVLCkjzOWavLioF4Bo9t+FzaMdZyiY2xVRxcmqoK8nBfc+u0k9At/kxZJ81GKA3v1io8FFFWJCzaCNZHHMb12TbQKwJMicaaoM47oy23DA8Digx/BRC76ydaz821W95RW9y1APwFSt9+w8uYCAs4tsrK7RrZmRVKEwB0P2RumDhDqBG1GgCTO+j2r2+mCGwb3ws5ECMDqJbAycKbs9TAcwdEB0TFkl5cJuqdrGMC0mD9oUXsxAeyHBvBNoAARpcv8IcAwBcv/bblsGERfN9E8W1EAiwghVvTh48sJpcF90yCJzifnvjsEyHMu6jEE/0eXzVAbPv4zyCMWQrooTroRtoe9wDAiTPl74BtgZQzRoE+vugDVAgkQQgAADs=
data:image/gif;base64,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
给定初值 N(0)=N0,可得人口增长的指数模型(Maithus 模型)。但是实际情况下资源必定会限制人口的增长,即人口数量达到阈值时候会减速增长,模型变为 logistic 模型(s曲线)。
以上例题仅仅作为说明平衡原理的理解例题。实际应运中平衡关系从物理角度、经济学角度好找一些(只是本人感觉)。
附本次例题来源的word文件供大家阅读,更深入理解平衡原理。
怎么找不到4了???
页:
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