The question of whether and how two competing consumers can coexist on a single limiting resource has a long tradition in ecological theory. We build on a recent seasonal (hybrid) model for one consumer and one resource, and we extend it by introducing a second consumer. Consumers reproduce only once per year, the resource reproduces throughout the"summer" season. When we use linear consumer reproduction between years, we find explicit expressions for the trivial and semi-trivial equilibria, and we prove that there is no positive equilibrium generically. When we use non-linear consumer reproduction, we determine conditions for which both semi-trivial equilibria are unstable. We prove that a unique positive equilibrium exists in this case, and we find an explicit analytical expression for it. By linear analysis and numerical simulation, we find bifurcations from the stable equilibrium to population cycles that may appear through period-doubling or Hopf bifurcations. We interpret our results in terms of climate change that changes the length of the"summer" season.

中文翻译：

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混合模型中单个资源上竞争消费者的共存

在生态学理论中，两个竞争的消费者是否以及如何在一个有限资源上共存这一问题具有悠久的历史。我们基于一个消费者和一种资源的最新季节性（混合）模型，并通过引入第二个消费者进行扩展。消费者每年仅繁殖一次，该资源在整个“夏季”繁殖。当我们使用几年之间的线性消费者再生产时，我们发现了平凡和半平凡的均衡的明确表达式，并且证明了通常没有正均衡。当我们使用非线性的消费者再生产时，我们确定了两个半平凡的均衡不稳定的条件。我们证明在这种情况下存在一个唯一的正平衡，并且我们找到了一个明确的解析表达式。通过线性分析和数值模拟，我们发现从稳定均衡到人口周期的分叉可能通过周期倍增或霍普夫分叉出现。我们根据改变“夏季”季节长度的气候变化来解释我们的结果。