臺灣博碩士論文加值系統

In this article we study a Lotka-Volterra reaction-di¤usion-advection model
arising from the evolution of conditional dispersal of two competing species.
In this model two species can choose their own preference of living environ-
ment based on di¤erent dispersal strategies. The dispersal behavior may
produce coexistence of two species under heterogeneous environment. Our
main purpose is to …nd out some coexisting periodic solutions by using dif-
ferent numerical methods.
We …rst introduce some basic numerical schemes, like forward Euler
method, backward Euler method and Runge-Kutta method to solve the
reaction-di¤usion-advection system. In addition, we also use Poincare section
method to examine the behavior of solutions. Although we have not found
any nontrivial coexisting periodic solutions, we have a better understanding
for the problem.
keywords: Evolution of conditional dispersal, reaction, advection, di¤u-
sion, Euler Method, Runge-Kutta Method .

Contents
1 Abstract 2
2 Introduction 2
3 The Main Question 3
3.1 Evolution of Dispersal . . . . . . . . . . . . . . . . . . . . . . 3
4 Some Failed Attempts 4
4.1 The Method of Iteration . . . . . . . . . . . . . . . . . . . . . 4
4.1.1 Forward Scheme . . . . . . . . . . . . . . . . . . . . . . 5
4.1.2 Forward Scheme with Central Di¤erence in Time . . . 5
4.1.3 Backward Scheme . . . . . . . . . . . . . . . . . . . . . 6
4.2 The Reasons about Why These Methods Do Not Work . . . . 6
4.2.1 Forward Scheme . . . . . . . . . . . . . . . . . . . . . . 6
4.2.2 Backward Scheme . . . . . . . . . . . . . . . . . . . . . 6
4.2.3 Sti¤ Problem . . . . . . . . . . . . . . . . . . . . . . . 6
5 Numerical Analysis 7
5.1 The Discrete ODE System . . . . . . . . . . . . . . . . . . . . 7
5.2 Some Properties of Discrete ODE System . . . . . . . . . . . . 8
5.2.1 Result 1: The property of Periodic Solution . . . . . . 9
5.2.2 Result 2: The property of Approaching Equilibrium . . 9
5.3 Some Results for Peaks and Solutions . . . . . . . . . . . . . . 9
5.3.1 Result 3: Estimation of Boundary Peaks . . . . . . . . 10
5.3.2 Result 4: The Application of Energy Function N(t) . 12
6 Numerical Results 12
6.1 Runge Kutta Fehlberg Method . . . . . . . . . . . . . . . . . 13
6.2 Poincare Section Method . . . . . . . . . . . . . . . . . . . . . 14
6.3 Peaks and Middle Lines . . . . . . . . . . . . . . . . . . . . . 15
6.3.1 Upper Peak . . . . . . . . . . . . . . . . . . . . . . . . 15
6.3.2 Lower Peak . . . . . . . . . . . . . . . . . . . . . . . . 17
6.3.3 Middle Line . . . . . . . . . . . . . . . . . . . . . . . . 18
7 Conclusion 19
8 Future Work 19
8.1 A Linear Eigenvalue Problem . . . . . . . . . . . . . . . . . 19
9 Acknowledgments 20

References
[1] J.C. Butcher. A history of runge-kutta methods. Applied Numerical
Mathematics, 20:247–260, 1996.
[2] A. C. LAZER C. Cosner. Stable coexistence states in the volterra-lotka
competition model with di¤usion. SIAM Journal on Applied Mathemat-
ics, 44(6), 1984.
[3] G. H. Golub and C. F. Van Loan. Matrix Computations. Johns Hopkins
Univ. Pr., 1996.
[4] P. Hess. Periodic-Parabolic Boundary Value Problems and Positivity.
Longman Scienti…c Technical, 1991.
[5] K. Mischaikow J. Dockery, V. Hutson and M. Pernarowski. The evolu-
tion of slow dispersal rates:a reaction-di¤usion model. Journal of Math-
ematical Biology, 37:61–83, 1998.
[6] C. V. Pao L. Zhou. Asymptotic behavior of a comptition-di¤usion sys-
tem in population dynamics. Nonlinear Analysis, Theory, Methods Ap-
plications, 6(11), 1982.
[7] R. J. LeVeque. Finite Di¤erence Methods for Ordinary and Partial Dif-
ferential Equations: Steady-State and Time-Dependent Problems. SIAM
Journals, 2007.
[8] J. H. Mathews and K. K. Fink. Numerical Methods Using Matlab. Pear-
son, 2004.
[9] C. V. Pao. Global asymptotic stability of lotkaâ¼A
¸Svolterra 3-species
reactionâ¼A
¸Sdi¤usion systems with time delays. Journal of Mathematical
Analysis and Applications, 281:186–204, 2003.
[10] Y. Lou R. Hambrock. The evolution of conditional dispersal strategies
in spatially heterogeneous habitats. Bulletin of Mathematical Biology,
71(8):1793–1817, 2009.
[11] Y. Saad. Iterative methods for sparse linear systems. PWS Publishing
Company, 1996.
[12] J.W. Thomas. Numerical Partial Di¤erential Equations: Finite Di¤er-
ence Methods. Springer, 1998.
[13] K. Mischaikow V. Hutson and P. Polacik. The evolution of dispersal
rates in a heterogeneous time-periodic environment. Journal of Mathe-
matical Biology, 43:501–533, 2001.
[14] Y. Lou X. Chen, R. Hambrock. Evolution of conditional dispersal: a
reaction-di¤usion-advection model. Journal of Mathematical Biology,
57(3):86–361, 2008.