# 臺灣博碩士論文加值系統

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 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.
 摘要 ivABSTRACT vAcknowledgement viList of Figures viii1. Introduction 12. Conditions of m,k 6Proof of (2.1) and (2.2) 7Proof of (2.3) 93. Analysis of boundary and spike layer solution 123.1 Boundary layered solution 123.2 Spike layer solutions 144. Decay behavior of the spike layer solution 174.1.2 Location of inflection point in spike layer solution 184.1.3 Integratability of (4.13) 195. Numerical result of boundary layer solution 216. Conclusion 23Reference 24
 1. Shibata, T. (2016). Inverse bifurcation problems for diffusive Holling–Tanner population model. Mathematische Nachrichten, 289(14-15), 1934-1945.2. Sáez, E., & González-Olivares, E. (1999). Dynamics of a predator-prey model. SIAM Journal on Applied Mathematics, 59(5), 1867-1878.3. Llibre, J., & Salhi, T. (2013). On the dynamics of a class of Kolmogorovsystems. Applied Mathematics and Computation, 225, 242-245.4. Gasull, A., & Giacomini, H. (2013). Some applications of the extended Bendixson-Dulac Theorem. In Progress and challenges in dynamical systems (pp. 233-252). Springer, Berlin, Heidelberg.5. Marin-Ramirez, A. M., de Indias, C., Rodriguez-Ceballos, J. A., & Ortiz-Ortiz, R. D. (2014). Quadratic systems without periodic orbits. International Journal of Mathematical Analysis, 8(44), 2177-2181.6. Marin, A. M., Ortiz, R. D., & Rodriguez, J. A. (2013). A generalization of a gradient system. In International Mathematical Forum (Vol. 8, No. 17, pp. 803-806).7. Marin-Ramirez, A. M., de Indias, C., Jaramillo-Camacho, V. P., & Ortiz-Ortiz, R. D. (2015). Solutions for the Combined sinh-cosh-Gordon Equation. International Journal of Mathematical Analysis, 9(24), 1159-1163.8. Baron-Pertuz, C. F., de Indias, C., Marin-Ramirez, A. M., & Ortiz-Ortiz, R. D. (2014). An Approximation to the Benjamin-Bona-Mahony Equation. International Journal of Mathematical Analysis, 8(56), 2757-2761.9. Osuna, O., & Vargas-De-León, C. (2015). Construction of Dulac functions for mathematical models in population biology. International Journal of Biomathematics, 8(03), 1550035.10. Maiti, A., Pal, A. K., & Samanta, G. P. (2008). Effect of time-delay on a food chain model. Applied Mathematics and Computation, 200(1), 189-203.11. Arditi, R., & Ginzburg, L. R. (1989). Coupling in predator-prey dynamics: ratio-dependence. Journal of theoretical biology, 139(3), 311-326.12. Hassell, M. P., & Varley, G. C. (1969). New inductive population model for insect parasites and its bearing on biological control. Nature, 223(5211), 1133.13. DeAngelis, D. L., Goldstein, R. A., & O'neill, R. V. (1975). A model for tropic interaction. Ecology, 56(4), 881-892.14. J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 45, pp. 331–340, 1975.15. Arditi, R., & Akçakaya, H. R. (1990). Underestimation of mutual interference of predators. Oecologia, 83(3), 358-361.16. Gutierrez, A. P. (1992). Physiological basis of ratio‐dependent predator‐prey theory: the metabolic pool model as a paradigm. Ecology, 73(5), 1552-1563.17. Blaine, T. W., & DeAngelis, D. L. (1997). The interaction of spatial scale and predator-prey functional response. Ecological Modelling, 95(2-3), 319-328.18. Poggiale, J. C., Michalski, J., & Arditi, R. (1998). Emergence of donor control in patchy predator—prey systems. Bulletin of Mathematical Biology, 60(6), 1149-1166.19. Bernstein, C., Auger, P., & Poggiale, J. C. (1999). Predator migration decisions, the ideal free distribution, and predator-prey dynamics. The American Naturalist, 153(3), 267-281.20. Cosner, C., DeAngelis, D. L., Ault, J. S., & Olson, D. B. (1999). Effects of spatial grouping on the functional response of predators. Theoretical population biology, 56(1), 65-75.21. Berezovskaya, F., Karev, G., & Arditi, R. (2001). Parametric analysis of the ratio-dependent predator–prey model. Journal of Mathematical Biology, 43(3), 221-246.22. May, R. M. (1975). Stability in ecosystems: some comments. In Unifying concepts in ecology (pp. 161-168). Springer, Dordrecht.23. Korman, P., Li, Y., & Ouyang, T. (2003). Perturbation of global solution curves for semilinear problems. Advanced Nonlinear Studies, 3(2), 289-299.
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