We consider a population distributed between two habitats, in each of which it experiences a growth rate that switches periodically between two values, $1-\varepsilon >0$ or $-(1+\varepsilon )<0$. We study the specific case where the growth rate is positive in one habitat and negative in the other one for the first half of the period, and conversely for the second half of the period, that we refer as the $(\pm 1)$ model. In the absence of migration, the population goes to 0 exponentially fast in each environment. In this paper, we show that, when the period is sufficiently large, a small dispersal between the two patches is able to produce a very high positive exponential growth rate for the whole population, a phenomena called inflation. We prove in particular that the threshold of the dispersal rate at which the inflation appears is exponentially small with the period. We show that inflation is robust to random perturbation, by considering a model where the values of the growth rate in each patch are switched at random times: we prove that inflation occurs for low switching rate and small dispersal. We also consider another stochastic model, where after each period of time $T$, the values of the growth rates in each patch is chosen randomly, independently from the other patch and from the past. Finally, we provide some extensions to more complicated models, especially epidemiological and density dependent models.

我们考虑分布在两个栖息地之间的种群，在每个栖息地中，其增长率在两个值之间定期切换，$1-\epsilon >0$或者$-（1+\epsilon ）<0$。我们研究特定情况，其中一个栖息地的增长率在前半段为正，而在另一个栖息地为负，而在后半段则相反，我们将其称为$（\pm 1）$模型。在没有迁移的情况下，每个环境中的人口数量都会以指数速度快速降至 0。在本文中，我们表明，当周期足够大时，两个斑块之间的微小分散能够为整个人口产生非常高的正指数增长率，这种现象称为通货膨胀。我们特别证明，通货膨胀出现的分散率阈值随着时间的推移呈指数级减小。通过考虑每个斑块中的增长率值在随机时间切换的模型，我们证明通货膨胀对于随机扰动是稳健的：我们证明通货膨胀发生在低切换率和小分散的情况下。我们还考虑另一个随机模型，在每个时间段之后$\mathrm{时间}$，每个补丁中的增长率值是随机选择的，独立于其他补丁和过去。最后，我们提供了对更复杂模型的一些扩展，特别是流行病学和密度依赖模型。