离散种群的演化与自适应动力学的正则扩散
Evolution of discrete populations and the canonical diffusion of adaptive dynamics
摘要：
适应动力学的生物学理论提出了一个结构化的无性种群的长期进化的描述。它基于大种群、罕见突变和小突变步骤的假设，这导致了描述主导类型进化的确定性常微分方程，称为适应动力学的正则方程。在这里，为了包括随机性（遗传漂变）的影响，我们考虑自我调节的随机波动的种群受到突变，使共存类型的数量可能波动。我们对这些群体应用罕见突变的限制，同时保持总体规模有限。这导致了一个跳跃过程，即所谓的“性状替代序列”，在这个过程中，进化通过突变类型的连续入侵和固定而进行。然后，我们对这个跳跃过程施加一个小突变步骤的限制（弱选择），这导致了一个扩散过程，我们称之为“自适应动态的典型扩散”，其中遗传漂变与固定概率梯度驱动的定向选择相结合，也被解释为入侵适应。最后，我们详细研究了特定情况下的多类型的逻辑分支种群，并寻求明确的公式的入侵健身的突变偏离居民类型略有。特别是，固定概率的二阶项是初始突变频率的函数的乘积，乘以初始总总体规模的函数，称为居民通过增加生育力、防御力、侵略性、隔离或生存的入侵系数。
英文摘要：
The biological theory of adaptive dynamics proposes a description of the long-term evolution of a structured asexual population. It is based on the assumptions of large population, rare mutations and small mutation steps, that lead to a deterministic ODE describing the evolution of the dominant type, called the ``canonical equation of adaptive dynamics.'' Here, in order to include the effect of stochasticity (genetic drift), we consider self-regulated randomly fluctuating populations subject to mutation, so that the number of coexisting types may fluctuate. We apply a limit of rare mutations to these populations, while keeping the population size finite. This leads to a jump process, the so-called ``trait substitution sequence,'' where evolution proceeds by successive invasions and fixations of mutant types. Then we apply a limit of small mutation steps (weak selection) to this jump process, that leads to a diffusion process that we call the ``canonical diffusion of adaptive dynamics,'' in which genetic drift is combined with directional selection driven by the gradient of the fixation probability, also interpreted as an invasion fitness. Finally, we study in detail the particular case of multitype logistic branching populations and seek explicit formulae for the invasion fitness of a mutant deviating slightly from the resident type. In particular, second-order terms of the fixation probability are products of functions of the initial mutant frequency, times functions of the initial total population size, called the invasibility coefficients of the resident by increased fertility, defence, aggressiveness, isolation or survival.
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