Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation
Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation
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DOI：
10.1080/14689360116914
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发表时间：
2001-06
期刊：
影响因子：
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通讯作者：
S. A. Gourley;M. Chaplain;F. A. Davidson
中科院分区：
文献类型：
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作者：
S. A. Gourley;M. Chaplain;F. A. Davidson
我们研究了标量反应扩散方程，其中包含一个非局部项的形式在空间变量的积分卷积和证明，使用渐近，分析和数值技术，这个标量方程是能够产生时空模式。费雪方程是这个方程的一个特例。一个渐近展开得到的行波波前连接的两个均匀的稳定状态和定性差异，相应的解决方案的Fisher方程的注意。结合数值积分方程的稳定性分析表明，在某些情况下，非均匀的解决方案，形成在这方面的后续工作。利用全局分歧理论，我们证明了在很宽的参数范围内存在这样的非一致稳态解。数值分岔研究的行为的稳态解作为一个特定的参数是变化的，也提出。
We study a scalar reaction-diffusion equation which contains a nonlocal term in the form of an integral convolution in the spatial variable and demonstrate, using asymptotic, analytical and numerical techniques, that this scalar equation is capable of producing spatio-temporal patterns. Fisher's equation is a particular case of this equation. An asymptotic expansion is obtained for a travelling wavefront connecting the two uniform steady states and qualitative differences to the corresponding solution of Fisher's equation are noted. A stability analysis combined with numerical integration of the equation show that under certain circumstances nonuniform solutions are formed in the wake of this front. Using global bifurcation theory, we prove the existence of such non-uniform steady state solutions for a wide range of parameter values. Numerical bifurcation studies of the behaviour of steady state solutions as a certain parameter is varied, are also presented.