Math. Model. Nat. Phenom.
Volume 10, Number 6, 2015Nonlocal reaction-diffusion equations
|17 - 29
|02 October 2015
Patterns and Transitions to Instability in an Intraspecific Competition Model with Nonlocal Diffusion and Interaction
Department of Economics, Social Sciences University
of Ankara, 06100
⋆ Corresponding author. E-mail: firstname.lastname@example.org
We consider an intraspecific resource competition model with asymmetric nonlocal dispersal and interaction. Both interaction and dispersal are modeled using convolution integrals. We introduce two parameters for each convolution kernel describing the range of nonlocality and the extent of asymmetry. It is shown that the spatially homogenous equilibrium of this model becomes unstable for sufficiently small diffusion rates by performing a linear stability analysis. Our numerical observations indicate that traveling and stationary wave type patterns arise near the stability boundary. Away from the stability boundary, solutions to the model exhibit colony formation i.e. one can observe formation of a new colony island just ahead of a vanishing stationary wave island. We further analyze the behavior of solutions to the model near the stability boundary using the techniques of weakly nonlinear analysis. We obtain a Stuart-Landau type equation and give its parameters in terms of Fourier transforms of the kernels. This analysis allows us to study the change in amplitudes of the solutions with respect to ranges of nonlocalities and extents of asymmetries of two kernel functions. We show that both continuous and discontinuous transitions from disordered behavior to ordered one are possible. We also verify these results numerically.
Mathematics Subject Classification: 92B05 / 37N25 / 34K09 / 37G99
Key words: nonlocal diffusion / nonlocal competition / pattern formation / weakly nonlinear analysis.
© EDP Sciences, 2015
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