﻿ 三种龙的生长：模型、数值模拟和可视化 Growth of Three Dragons: Model, Numerical Simulation and Visualization

Pure Mathematics
Vol. 09  No. 06 ( 2019 ), Article ID: 31700 , 10 pages
10.12677/PM.2019.96094

Growth of Three Dragons: Model, Numerical Simulation and Visualization

Lizhi Yang, Junyi Zeng, Yixuan Yang

School of Science, Southwest University of Science and Technology, Mianyang Sichuan

Received: Jul. 19th, 2019; accepted: Jul. 29th, 2019; published: Aug. 14th, 2019

ABSTRACT

In this paper, the three fictional dragons Drogon, Viserion and Rhaegal in The Game of Rights are used as prototypes. Under the condition of limited growth resources, the mathematical relationship between the growth of the three dragons and the environmental factors such as climate are considered. The growth of the three dragons was simulated to calculate the various indicators needed for their growth. Finally, sensitivity analysis is performed on it. The model predicts and regulates the competitive growth of multiple species. This keeps the ecological structure of the entire region relatively stable.

Keywords:Competition, Numerical Simulation, Growth Indicator, Sensitivity Analysis

1. 引言

2. 问题分析

3. 多因素生长模型

3.1. 模型分析

Figure 1. Overview of the three dragons’ distribution

3.2. 模型建立

Step1：环境资源充足时，确定临界面积

$\frac{\text{d}x}{\text{d}t}=\left(1-\frac{x}{{x}_{m}}\right)\left[ax+u\left(s,r,t\right)\right]$ (1)

$r\left(t\right)={r}_{0}\left[{\mathrm{sin}}^{2}\left(\text{π}t\right)+b\right]$ (2)

Figure 2. Diagram of the relationship between resources and seasons

$u\left(s,r,t\right)=s\cdot r\left(t\right)=s\cdot {r}_{0}\cdot \left[{\mathrm{sin}}^{2}\left(\text{π}t\right)+b\right]$ (3)

(4)

$s={s}_{1}+{s}_{2}+{s}_{3}$ (5)

Step2：综合环境和竞争因素模拟龙的生长情况

$\frac{\text{d}{x}_{1}}{\text{d}t}=\left(1-\frac{x}{{x}_{m}}\right)\left\{{a}_{1}{x}_{1}-{a}_{2}{x}_{2}-{a}_{3}{x}_{3}+s{r}_{0}\left[\mathrm{sin}\left(\text{π}t\right)+b\right]\right\}$ (6)

$\frac{\text{d}{x}_{1}}{\text{d}t}=\left(1-\frac{x}{{x}_{m}}\right)\left\{{a}_{1}{x}_{1}-{a}_{2}{x}_{2}-{a}_{3}{x}_{3}+s{r}_{0}\left[\mathrm{sin}\left(\text{π}t\right)+b\right]\right\}+\alpha {\epsilon }_{i}$ (7)

$\left\{\begin{array}{l}\frac{\text{d}{x}_{1}}{\text{d}t}=\left(1-\frac{{x}_{1}}{{x}_{m}}\right)\left\{{a}_{11}{x}_{1}-{a}_{12}{x}_{2}-{a}_{13}{x}_{3}+{s}_{1}{r}_{1}\left[{\left(\mathrm{sin}\text{π}t\right)}^{2}+b\right]\right\}+\alpha {\epsilon }_{1}\\ \frac{\text{d}{x}_{2}}{\text{d}t}=\left(1-\frac{{x}_{2}}{{x}_{m}}\right)\left\{-{a}_{21}{x}_{1}+{a}_{22}{x}_{2}-{a}_{23}{x}_{3}+{s}_{2}{r}_{2}\left[{\left(\mathrm{sin}\text{π}t\right)}^{2}+b\right]\right\}+\alpha {\epsilon }_{2}\\ \frac{\text{d}{x}_{3}}{\text{d}t}=\left(1-\frac{{x}_{3}}{{x}_{m}}\right)\left\{-{a}_{31}{x}_{1}-{a}_{32}{x}_{2}+{a}_{33}{x}_{3}+{s}_{3}{r}_{3}\left[{\left(\mathrm{sin}\text{π}t\right)}^{2}+b\right]\right\}+\alpha {\epsilon }_{3}\end{array}$ (8)

3.3. 模型的求解与分析

Figure 3. The weight gain of a dragon growing

Table 1. Corresponding parameter table

$s\ge 8000\text{\hspace{0.17em}}{\text{km}}^{2}$ 时，三条龙之间不存在竞争关系，当 $s<8000\text{\hspace{0.17em}}{\text{km}}^{2}$ 时，龙之间会产生资源的竞争，此时综合考虑季节气候因素，竞争因素和随机影响因素对三条龙的生长情况进行数值模拟，结果如图4所示。

4. 基于外界援助的多因素生长模型

Figure 4. Growth conditions of three dragons under multi-factor conditions

$\left\{\begin{array}{l}\frac{\text{d}{x}_{1}}{\text{d}t}=\left(1-\frac{{x}_{1}}{{x}_{m}}\right)\left\{\left({a}_{11}+{\eta }_{1}\right){x}_{1}-\left({a}_{12}-{\beta }_{1}\right){x}_{2}-\left({a}_{13}-{\beta }_{2}\right){x}_{3}+{s}_{1}{r}_{1}\left[{\left(\mathrm{sin}\text{π}t\right)}^{2}+b\right]\right\}+\alpha {\epsilon }_{1}\\ \frac{\text{d}{x}_{2}}{\text{d}t}=\left(1-\frac{{x}_{2}}{{x}_{m}}\right)\left\{-\left({a}_{21}-{\beta }_{1}\right){x}_{1}+{a}_{22}{x}_{2}-\left({a}_{23}-{\beta }_{3}\right){x}_{3}+{s}_{2}{r}_{2}\left[{\left(\mathrm{sin}\text{π}t\right)}^{2}+b\right]\right\}+\alpha {\epsilon }_{2}\\ \frac{\text{d}{x}_{3}}{\text{d}t}=\left(1-\frac{{x}_{3}}{{x}_{m}}\right)\left\{-\left({a}_{31}-{\beta }_{2}\right){x}_{1}-\left({a}_{32}-{\beta }_{3}\right){x}_{2}+\left({a}_{33}+{\eta }_{2}\right){x}_{3}+{s}_{3}{r}_{3}\left[{\left(\mathrm{sin}\text{π}t\right)}^{2}+b\right]\right\}+\alpha {\epsilon }_{3}\end{array}$ (9)

$\left\{\begin{array}{l}{\eta }_{1}=0.11\\ {\eta }_{2}=0.08\\ {\beta }_{1}=0.02\\ {\beta }_{2}=0.01\\ {\beta }_{3}=0.03\end{array}$ (10)

Figure 5. Growth status diagram of external factors

$\left\{\begin{array}{l}{\stackrel{¯}{Q}}_{s_1}=\frac{1}{Y}{\int }_{0}^{Y}\left({a}_{11}+{\eta }_{1}\right){x}_{1}\left(t\right)\text{d}t\\ {\stackrel{¯}{Q}}_{s_2}=\frac{1}{Y}{\int }_{0}^{Y}{a}_{22}{x}_{2}\left(t\right)\text{d}t\\ {\stackrel{¯}{Q}}_{s_3}=\frac{1}{Y}{\int }_{0}^{Y}\left({a}_{33}+{\eta }_{1}\right){x}_{1}\left(t\right)\text{d}t\end{array}$ (11)

$\left\{\begin{array}{l}{\stackrel{¯}{Q}}_{x_1}=\frac{1}{Y}{\int }_{0}^{Y}\left[\left({a}_{21}-{\beta }_{1}\right){x}_{1}\left(t\right)+\left({a}_{31}-{\beta }_{2}\right){x}_{1}\left(t\right)\right]\text{d}t\\ {\stackrel{¯}{Q}}_{x_2}=\frac{1}{Y}{\int }_{0}^{Y}\left[\left({a}_{12}-{\beta }_{1}\right){x}_{2}\left(t\right)+\left({a}_{32}-{\beta }_{3}\right){x}_{2}\left(t\right)\right]\text{d}t\\ {\stackrel{¯}{Q}}_{x_3}=\frac{1}{Y}{\int }_{0}^{Y}\left[\left({a}_{13}-{\beta }_{2}\right){x}_{3}\left(t\right)+\left({a}_{23}-{\beta }_{3}\right){x}_{3}\left(t\right)\right]\text{d}t\end{array}$ (12)

$\left\{\begin{array}{l}{\stackrel{¯}{Q}}_{r_1}=\frac{1}{Y}{\int }_{0}^{Y}{\eta }_{1}{x}_{1}\left(t\right)\text{d}t\\ {\stackrel{¯}{Q}}_{r_3}=\frac{1}{Y}{\int }_{0}^{Y}{\eta }_{2}{x}_{3}\left(t\right)\text{d}t\end{array}$ (13)

5. 灵敏度分析

1) 考虑在竞争时，Drogon和Rhaegal竞争比较激烈。在模型三的基础上，设置Drogon和Rhaegal自身能量消耗率分别为0.18和0.19。数值仿真结果如图6所示，可见Drogon不敌Rhaegal，在6.5年左右时死亡。

2) 考虑在竞争时，三条龙竞争都比较激烈。在模型三的基础上，在自身能量消耗率方面，设置Drogon为0.18和0.19；Rhaegal为0.19和0.18；Viserion为0.2和0.18。数值仿真结果如图7所示，可见Drogon和Viserion在竞争中处于劣势，且Drogon在6年左右时死亡，只有Rhaegal和Viserion存活下来。

Figure 6. Competitive survival simulation diagram a

Figure 7. Competitive survival simulation diagram b

3) 考虑在竞争时，三条龙所在领地不同。在模型三的基础上，设置三条龙Drogon, Rhaegal, Viserion自身所在领地面积分别为6000，2000和500。数值仿真结果如图8所示，可见Viserion在竞争中处于劣势，在6年左右时死亡，Drogon和Rhaegal存活下来。

4) 考虑在竞争时，三条龙所在领地的单位面积资源给予率不同。在模型三的基础上，设置三条龙Drogon，Rhaegal，Viserion自身所在领地的单位面积资源给予率分别为0.2，0.2和0.6。数值仿真结果如图9所示，可见Viserion在竞争中存活一段时间后，在16年左右时死亡。Drogon和Rhaegal存活下来。

5) 考虑在竞争时，三条龙领地面积较小，生存资源有限，竞争较为激烈。在模型三的基础上，设置三条龙Drogon，Rhaegal，Viserion自身所在领地面积均为1000；单位面积资源给予率分别为0.05，0.04和0.06；在自身能量消耗率方面，设置Drogon为0.28和0.29；Rhaegal为0.29和0.28；Viserion为0.3和0.28。数值仿真结果如图10所示，可见三条龙都无法正常生长，并且Rhaegal在第4年就死亡。

Figure 8. Competitive survival simulation diagram c

Figure 9. Competitive survival simulation diagram d

Figure 10. Competitive survival simulation diagram d

6. 结论

Growth of Three Dragons: Model, Numerical Simulation and Visualization[J]. 理论数学, 2019, 09(06): 702-711. https://doi.org/10.12677/PM.2019.96094

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