Many systems of biological interest can be modeled as pointlike oscillators whose coupling is mediated by the diffusion of some substance. This coupling occurs because the dynamics of each oscillator is influenced by the local concentration of a substance which diffuses through the spatial medium. The diffusion equation, on its hand, has a source term which depends on the oscillator dynamics. We derive a mathematical model for such a system and obtain an integro-differential equation. Its solution can be obtained by an approximation scheme for which the unperturbed solution is used to obtain a first-order solution to the coupled oscillators and so on. We present numerical results for the special case of a one-dimensional bounded domain in which the oscillators are randomly placed. Our results show the influence of the coupling parameters on some aspects of the dynamics of the coupled oscillators, like phase and frequency synchronization.

可以将许多具有生物学意义的系统建模为点状振荡器，其耦合是由某种物质的扩散介导的。之所以发生这种耦合，是因为每个振荡器的动力学受到扩散通过空间介质的物质的局部浓度的影响。另一方面，扩散方程的源项取决于振荡器的动力学特性。我们推导了此类系统的数学模型，并获得了积分微分方程。可以通过一种近似方案来获得其解，对于该近似方案，未扰动解可用于获得耦合振荡器的一阶解，依此类推。对于一维有界域的特殊情况，其中振荡器随机放置，我们给出了数值结果。