The dynamics of ecological systems are often described by integrodifferential equations that incorporate nonlocal interactions associated with facilitative, competitive interactions between plants, and seed dispersion. In the weak-gradient limit, these models can be reduced to a simple partial-differential equation in the form of a nonvariational Swift–Hohenberg equation. In this contribution, we perform this reduction for any type of kernels provided that their Taylor series converge. Some parameters such as linear and nonlinear diffusion coefficients are affected by the spatial form of the kernel. In particular, Gaussian and exponential kernels are used to evaluate all coefficients of the reduced model. This weak gradient approximation is greatly useful for the investigation of periodic and localized vegetation patches, and gaps. Based on this simple model, we investigate the interaction between two-well separated patches and gaps. In the case of patches, the interaction is always repulsive. As a consequence, bounded states of patches are excluded. However, when two gaps are close to one another, they start to interact through their oscillatory tails. The interaction alternates between attractive and repulsive depending on the distance separating them. This allows for the stabilization of bounded gaps and clusters of them. The analytical formula of the interaction potential is derived for both patches and gaps interactions and checked by numerical investigation of the model equation.

This volume is dedicated to Professor Ehud Meron on the occasion of his sixtieth birthday. We take this opportunity to express our warmest and most sincere wishes to him.

通常用积分微分方程来描述生态系统的动力学，积分微分方程包含与植物之间的促进性竞争相互作用和种子分散相关的非局部相互作用。在弱梯度极限中，这些模型可以简化为简单的偏微分方程，形式为无变量的Swift–Hohenberg方程。在此贡献中，我们对任何类型的内核（只要它们的泰勒级数都收敛）都执行了这种减少。诸如线性和非线性扩散系数之类的某些参数受内核的空间形式影响。特别是，高斯和指数核用于评估简化模型的所有系数。这种弱梯度近似对于研究周期性和局部植被斑块和间隙非常有用。在此简单模型的基础上，我们研究了两孔分离的面片和间隙之间的相互作用。在贴剂的情况下，相互作用总是排斥的。结果，排除了补丁的有界状态。但是，当两个间隙彼此接近时，它们便开始通过其振荡的尾部相互作用。交互作用取决于吸引力和排斥力之间的距离而交替。这样可以稳定有限的间隙及其簇。推导了膜片和间隙相互作用的相互作用势的解析公式，并通过对模型方程的数值研究进行了检验。补丁的有界状态被排除。但是，当两个间隙彼此接近时，它们便开始通过其振荡的尾部相互作用。交互作用取决于吸引力和排斥力之间的距离而交替。这样可以稳定有限的间隙及其簇。推导了膜片和间隙相互作用的相互作用势的解析公式，并通过对模型方程的数值研究进行了检验。补丁的有界状态被排除。但是，当两个间隙彼此接近时，它们便开始通过其振荡的尾部相互作用。交互作用取决于吸引力和排斥力之间的距离而交替。这样可以稳定有限的间隙及其簇。推导了膜片和间隙相互作用的相互作用势的解析公式，并通过对模型方程的数值研究进行了检验。

这本书是献给埃胡德·梅隆教授（Ehud Meron）六十岁生日之际。我们借此机会向他表示最热烈和最真诚的祝愿。