| 摘要: | 微分方程可以用來描述我們周圍的世界。 它們被應用於生物學，經濟學，物理學，化學和工程學等廣泛領域。
例如，微分方程模型可以顯示物種的種群增長。 在這裡，我們研究了雲杉芽蟲系統在鳥類捕食和殺蟲劑作用下的動力學。 在一定條件下，還表明了動力系統極限環的存在。 此外，我們找出了在特定情況下產生極限環唯一性的一些條件。 如果殺蟲劑曾經用於控制芽蟲的種群，即$n = 1$，
如果由於殺蟲劑的作用而導致的最大芽蟲死亡率低於某個特定值，我們將獲得極限環的唯一性。 同樣，在兩次噴灑殺蟲劑的情況下（即 $ n = 2 $）的情况下，我们也可以获得周期轨道的唯一性。
在後面的部分中，我們考慮 $\Delta u + V_{1}(\theta) u + V_{2}(\theta) |u|^{p-1}u=0$ 上 $\mathbb{S}^n$
並僅根據方位角研究正解的結構 $\theta$.
我們將給出在所有非線性情況下所有奇異解的存在
$p$ 和潛在功能 $V_{1}, V_{2}$.
在臨界情況下，顯示了非振盪奇異解的唯一性，圍繞該奇異解，存在無窮多個奇異解。 我們還展示了當非線性指數越過臨界值時奇異解的結構變化 $p_S$.;Differential equations can be used to describe the world around us. They are applied into a wide range of fields, from biology, economics, physics, chemistry and engineering.
For example, differential equations model can present the population growth of species. Here we study the dynamics of the spruce budworm system with the effect of bird predation and insecticide. The existence of limit cycle of dynamical system is also shown under certain conditions. Moreover, we figured out some conditions which yield the uniqueness of limit cycle in specific cases. In case insecticide is used once to control budworm population, i.e, $n=1$, if the maximum budworm mortality rate due to the action of insecticide is less than a particular value, we obtain the uniqueness of limit cycle. Similarly, we also achieve the uniqueness of periodic orbit in case insecticide is sprayed twice, i.e, $n=2$.
In the later part, we consider $\Delta u + V_{1}(\theta) u + V_{2}(\theta) |u|^{p-1}u=0$ on $\mathbb{S}^n$ and investigate the structures of positive solutions depending only on the azimuthal angle $\theta$. We shall give the existence of all %positive singular-
singular solutions in all nonlinear cases of $p$ and potential functions $V_{1}, V_{2}$. In critical case, the uniqueness of non-oscillatory singular solution is shown, around which infinitely many singular solutions are oscillating. We also show the change of structure of singular solutions when the exponent of the nonlinearity crosses the critical value $p_s$. |
