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Rasheed, Shaker M. (2013)
Languages: English
Types: Unknown
Subjects:
This thesis deals with a two component reaction-diffusion system (RDS) for competing and cooperating species. We have analyse in detail the stability and bifurcation structure of equilibrium solutions of this system, a natural extension of the Lotka-Volterra system. We find seven topologically different regions separated by bifurcation boundaries depending on the number and stability of equilibrium solutions, with four regions in which the solutions are similar to those in the Lotka-Volterra system. We study RDS in the small parameter of the range $0< \lambda \ll 1 $ (fast diffusion and slow reaction), and in a few cases we assume $\lambda=O(1)$. We consider three types of initial conditions, and we find three types of travelling wave solutions using numerical and asymptotic methods. However, neither numerical nor asymptotic methods were able to find a particular travelling wave solution which connects a coexistence state say, $(u_0,w_0)$ to an extinction state $(0,0)$ when $0< \lambda \ll 1 $. This type can be found when the reaction-diffusion system satisfy the symmetry property and $\lambda=1$.
  • The results below are discovered through our pilot algorithms. Let us know how we are doing!

    • 2 Equilibrium solutions 23 2.1 Equilibrium solutions and bifurcation . . . . . . . . . . . . . . . . . . . . . 23 2.1.1 Stability of the equilibrium points and the phase portrait in the regions R1 R7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2 The effect of the parameters a1,2 on the bifurcation boundaries . . . . . . . 37
    • 3 Travelling wave solutions for the initial value problem (1.5.3) 41 3.1 Travelling wave solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
    • 4 Travelling wave solutions for l 1 and l = O(1) 80 4.1 Asymptotic solutions for l 1 . . . . . . . . . . . . . . . . . . . . . . . . 80 4.2 Regular perturbation solutions . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.2.1 Asymptotic solutions for type (Ib) . . . . . . . . . . . . . . . . . . . 82 4.2.2 g2 < 1, R1 and R4: saddle-node connection . . . . . . . . . . . . . . 85 4.2.3 g2 > 1, R2 and R5: saddle-saddle connection . . . . . . . . . . . . . 88 4.3 Singular perturbation solutions . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.4 Asymptotic solutions for type (Ia) . . . . . . . . . . . . . . . . . . . . . . . 92 4.4.1 Inner solution for (Ia) . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.5 Solutions for type (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
    • 5 Stability of travelling wave solutions in two dimensions 107 5.1 Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.2 Perturbation of the planar wavefront . . . . . . . . . . . . . . . . . . . . . 113 5.2.1 Example test: Gray-Scott . . . . . . . . . . . . . . . . . . . . . . . . 114 5.2.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.3 Stability analysis of travelling waves of (5.1.1) . . . . . . . . . . . . . . . . 121 5.4 Linearisation of (5.1.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.5 Asymptotic solutions for (5.4.3) . . . . . . . . . . . . . . . . . . . . . . . . 123 5.5.1 Multiple scale method . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.5.2 Calculating M1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.5.3 Calculating the Evans function for (5.5.11) and the travelling wave of type (Ia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.5.4 Computing the Evans function for (5.5.11) with type (IIIr) . . . . . 133 5.5.5 Computing the Evans function for (5.5.11) for type (Ib) . . . . . . . 134 5.6 Calculating the Evans function for inner problem . . . . . . . . . . . . . . 136 [Bil04] J. Billingham. Dynamics of a Strongly Nonlocal Reaction-Diffusion Population Model. Nonlinearity, 17:313-346, 2004.
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