族群演化的隨機性

Abstract

生物族群使用不同的演化策略在變化的環境下繁衍，以細胞不同表現型構成的族群為例，在有限生活資源下，它的演化過程可用三階(表現型轉換，增殖及淘汰)離散映射來描述。此論文探討在此映射下系統的“隨機性”，包括確定性混沌的不可預測性，內噪聲造成的漲落，以及週期外噪聲造成的環面晃動。在數學上，如同Hénon映射跟Szilard映射，此族群演化映射具備豐富動力學議題，包括吸子結構、分岔方式、以及Perron-Frobenius算子下的不變密度。在物理上，混沌區裡的轉換映射如同耗散，增殖及淘汰映射如同造成漲落的隨機力，因此族群演化行為有如統計力學裡布朗運動的漲落耗散定理。

Biological populations use different evolutionary strategies to grow in changing environments. Taking the populations of phenotypes of cells as an example, under a limited resource, the evolution of these populations can be described by a three step discrete map, corresponding to phenotype switch, growth, and natural elimination. This thesis is devoted to explore the "stochasticity" generated by that map, including the unpredictability of deterministic chaos, the fluctuations induced by internal noises, and the toroidal oscillations caused by extrinsic periodic noises. Mathematically, as with the Hénon map and the Szilard map, this evolution map contains a variety of dynamical issues, including the structures of attractor, the types of bifurcation, and the invariant densities generated by the Perron-Frobenius operator. In physics, in the chaotic regime, the map for phenotype switch in the chaotic area is like dissipation, and the map for growth and elimination plays the role of random forces for generating fluctuations. Therefore, the behavior of the above evolution dynamics shares the same structure as the fluctuation-dissipation theorem of Brownian motion in statistical mechanics.