We consider contact processes on locally compact separable metric spaces with birth and death rates that are heterogeneous in space. We formulate conditions on the rates that ensure the existence of invariant measures of contact processes. One of the crucial conditions is the so-called critical regime condition. To prove the existence of invariant measures, we use the approach proposed in our preceding paper. We discuss in detail the multi-species contact model with a compact space of marks (species) in which both birth and death rates depend on the marks.

中文翻译：

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具有依赖于国家的出生率和死亡率的接触过程的不变测度

我们考虑局部紧可分离度量空间上的接触过程，出生率和死亡率在空间上是异质的。我们制定速率条件，确保接触过程不变度量的存在。关键条件之一是所谓的临界状态条件。为了证明不变测度的存在，我们使用前一篇论文中提出的方法。我们详细讨论具有紧凑的标记（物种）空间的多物种接触模型，其中出生率和死亡率都取决于标记。