我們在此項研究中探討了Holling-Tanner的人口模型。
u^'' (t)+λ(mu(t)-u^2 (t)-ku(t)/(1+u(t) ))=0,
u(t)>0,t∈I=:(-1,1),u(1)=u(-1)=0,
其中為一分歧的參數並且λ>0，m,k則為滿足一些特定條件的正常數。從Shibata先前的研究我們得知，在不同區域下的m,k, u在λ趨近於無窮大時會有峰型和邊界層兩類解。為了能完整了解當λ≫1時解的結構，我們對u和u^'進行了漸近行為的逐點分析。更準確的說，對於邊界層的解，我們精確的描述了薄邊界層的漸近行為，u在邊界會以指數型式遞減至零。另一方面，我們也對峰型解的反曲點附近進行了精確的估計。最後我們以數值方法來證明前述推導中λ的值與曲線結構的關係。

In this thesis, we investigate the Holling-Tanner population model.
u^'' (t)+λ(mu(t)-u^2 (t)-ku(t)/(1+u(t) ))=0,
u(t)>0,t∈I=:(-1,1),u(1)=u(-1)=0,
where λ>0 is a bifurcation parameter, and m and k are positive constants satisfying some certain conditions. Based on Shibata’s work, under different regions of m and k, u may have boundary layer and spike layer solutions as λ approaches infinity. To completely study the layer structure with λ≫1, we establish the pointwise asymptotic behavior of solutions u and u'. More precisely, for the boundary layered solution u, we describe the refined asymptotics of thin boundary layer and show that u decays to zero exponentially in interior points. On the other hand, delicate asymptotics near the spike is also described for the spike layer solution. And the numerical method was done to see the relationship between λ and curves of boundary layered solution.

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