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

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