Parsons Todd L, Quince Christopher
Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ont., Canada M5S 2E4.
Theor Popul Biol. 2007 Dec;72(4):468-79. doi: 10.1016/j.tpb.2007.04.002. Epub 2007 Apr 27.
We determine fixation probabilities in a model of two competing types with density dependence. The model is defined as a two-dimensional birth-and-death process with density-independent death rates, and birth rates that are a linearly decreasing function of total population density. We treat the 'quasi-neutral case' where both types have the same equilibrium population densities. This condition results in birth rates that are proportional to death rates. This can be viewed as a life history trade-off. The deterministic dynamics possesses a stable manifold of mixtures of the two types. We show that the fixation probability is asymptotically equal to the fixation probability at the point where the deterministic flow intersects this manifold. The deterministic dynamics predicts an increase in the proportion of the type with higher birth rate in growing populations (and a decrease in shrinking populations). Growing (shrinking) populations therefore intersect the manifold at a higher (lower) than initial proportion of this type. On the center manifold, the fixation probability is a quadratic function of initial proportion, with a disadvantage to the type with higher birth rate. This disadvantage arises from the larger fluctuations in population density for this type. These results are asymptotically exact and have relevance for allele fixation, models of species abundance, and epidemiological models.
我们在一个具有密度依赖性的两种竞争类型的模型中确定固定概率。该模型被定义为一个二维生死过程，其死亡率与密度无关，出生率是总人口密度的线性递减函数。我们处理两种类型具有相同平衡种群密度的“准中性情况”。这种情况导致出生率与死亡率成正比。这可以被视为一种生活史权衡。确定性动力学具有两种类型混合的稳定流形。我们表明，固定概率渐近等于确定性流与该流形相交处的固定概率。确定性动力学预测，在增长的种群中出生率较高的类型的比例会增加（而在收缩的种群中会减少）。因此，增长（收缩）的种群与流形相交时，该类型的比例高于（低于）初始比例。在中心流形上，固定概率是初始比例的二次函数，对出生率较高的类型不利。这种不利源于该类型种群密度的较大波动。这些结果在渐近意义上是精确的，并且与等位基因固定、物种丰度模型和流行病学模型相关。
