关于浓缩悖论的一个解析
A Resolution of the Paradox of Enrichment
摘要：
M.一类捕食-食饵模型中的Rosenzweig。模型中的两个参数对悖论至关重要。这两个参数分别是被捕食者的载畜量和被捕食者的半饱和度。直觉上，由于猎物环境的丰富而增加的承载能力应该会导致更稳定的捕食者-猎物系统。分析表明，增加承载能力总是导致一个不稳定的捕食者-被捕食者系统，该系统容易因环境随机扰动而灭绝。这就是所谓的致富悖论。我们的解决方案在这里依赖于一个无量纲数$H$形成的承载能力和猎物的半饱和度更仔细的调查。通过将模型转换为无量纲形式，模型实际上由包括$H$在内的几个无量纲数控制。两个参数的影响：承载能力和半饱和度被纳入数$H$。事实上，增加承载能力相当于（即对$H$具有相同的影响）降低半饱和度，这意味着更积极的捕食。由于更积极的捕食和捕食者-食饵系统的不稳定性之间不存在悖论，富集悖论得以解决。我们还进一步探讨了空间相关的模型相空间是无限维的。空间独立的极限环是由一个不稳定的定态产生的，在无穷维相空间中是线性稳定的。数值模拟表明，极限环的吸引域是充满的。这表明，空间扰动有时（既不总是也不永远）可以消除极限环附近的富集悖论！
英文摘要：
The paradox of enrichment was observed by M. Rosenzweig in a class of predator-prey models. Two of the parameters in the models are crucial for the paradox. These two parameters are the prey's carrying capacity and prey's half-saturation for predation. Intuitively, increasing the carrying capacity due to enrichment of the prey's environment should lead to a more stable predator-prey system. Analytically, it turns out that increasing the carrying capacity always leads to an unstable predator-prey system that is susceptible to extinction from environmental random perturbations. This is the so-called paradox of enrichment. Our resolution here rests upon a closer investigation on a dimensionless number $H$ formed from the carrying capacity and the prey's half-saturation. By recasting the models into dimensionless forms, the models are in fact governed by a few dimensionless numbers including $H$. The effects of the two parameters: carrying capacity and half-saturation are incorporated into the number $H$. In fact, increasing the carrying capacity is equivalent (i.e. has the same effect on $H$) to decreasing the half-saturation which implies more aggressive predation. Since there is no paradox between more aggressive predation and instability of the predator-prey system, the paradox of enrichment is resolved. We also further explore spatially dependent models for which the phase space is infinite dimensional. The spatially independent limit cycle which is generated by a Hopf bifurcation from an unstable steady state, is linearly stable in the infinite dimensional phase space. Numerical simulations indicate that the basin of attraction of the limit cycle is riddled. This shows that spatial perturbations can sometimes (neither always nor never) remove the paradox of enrichment near the limit cycle!
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