We present a simple mathematical model for the dynamics of a successional pioneer–climax system using difference equations. Each population is subject to inter- and intraspecific competition; population growth is dependent on the combined densities of both species. Nine different geometric cases, corresponding to different orientations of the zero-growth isoclines, are possible for this system. We fully characterize the long-term dynamics of the model for each of the nine cases, uncovering diverse sets of potential behaviors. Competitive exclusion of the pioneer species and of the climax species are both possible depending on the relative strength of competition. Stable coexistence of both species may also occur; in two cases, a coexistence state is destabilized through a Neimark–Sacker bifurcation, and an attracting invariant circle is born. The invariant circle eventually disappears into thin air in a heteroclinic or homoclinic bifurcation, leading to the sudden transition of the system to an exclusion state. Neither global bifurcation has been observed in a discrete-time pioneer–climax model before. The homoclinic bifurcation is novel to all pioneer–climax models. We conclude by discussing the ecological implications of our results.

我们使用差分方程为连续先驱-高潮系统的动力学提供了一个简单的数学模型。每个种群都面临种间和种内竞争；人口增长取决于两个物种的综合密度。对于该系统，对应于零增长等倾线的不同方向的九种不同几何情况是可能的。我们充分描述了九个案例中每个案例的模型的长期动态，揭示了不同的潜在行为。根据竞争的相对强度，对先锋物种和高潮物种的竞争排斥都是可能的。两种物种也可能稳定共存；在两种情况下，共存状态通过 Neimark-Sacker 分岔不稳定，并产生一个吸引不变的圈。不变圆最终在异斜或同斜分叉处消失在稀薄的空气中，导致系统突然过渡到排除状态。以前在离散时间先驱 - 高潮模型中都没有观察到全局分叉。同宿分岔对所有先锋-高潮模型都是新的。我们最后讨论了我们结果的生态影响。