Theoretical Study of Diadrmous Migration Evolution in Fish Species.

在大约3万种鱼类中，大约有130种是在淡水和海洋之间迁徙的巨型鱼类。双穴居鱼类包括两个明显不同的类群；(i)例如鲑鱼（鲑科）和七鳃鳗（七鳃鳗科），它们在淡水中出生，迁徙到海洋，然后返回淡水产卵（溯海迁徙）；（ii）如某些鳗鲡科（Anguillidae）和鲻鱼科（Mugilidae），它们在海洋中出生，迁徙到淡水，然后返回海洋产卵（海地迁徙）。长期以来，这些截然不同的移民方向的存在一直令人困惑。当鱼类在不同类型的介质之间洄游时，最严重的生理障碍是维持体内平衡。离子和渗透调节是必不可少的，并且在进化过程中得到了很好的安排。我将生长（dB/dt）和生存（dL/dt）分别表示为与生境类型（河流或海洋）和体型(B)有关的函数。进一步假设dL/dt与分配给离子和渗透调节的能量比例（u (t)）成函数。利用Pontryagin极大值原理分析了鱼类如何选择离子报复和渗透报复的能量分配，以及鱼类如何选择栖息地以最大化B (T) L (T)，其中T为考虑的终端时间。最优生境选择时间表表现出最优的洄游时间表。数学模型得出的最有趣的结果是，大卵出生的个体比小卵出生的个体更早下海。一些实验和观察也显示了同样的模式。由此可以得出结论，该数学模型得到的结果具有启发式鲁棒性。

Of roughly thirty thousand fish species, about 130 species are diadrmous that migrate between freshwater and the ocean in their life history. Diadromous fishes include two distinctly different groups ; (i) such as salmon (Salmonidae) and lamprey (Petromyzonidae) , which are born in freshwater, migrate to the ocean, and return to freshwater to spawn (anadrmous migration) ; and (ii) such as some eels (Anguillidae) and mullets (Mugilidae) , which are born in the ocean, migrate to freshwater, and return to the ocean to spawn (catadrmous migration). The existence of these contrasting directions of migration has long been perplexing. The most serous physiological barrier of diadrmous migration is maintenance of homeostasis, when fish migrate between different types of mediums. Ionic and osmotic regulation is essential, and it is well scheduled in the course of evolutionary process.I represented growth (dB/dt) and survival (dL/dt) as functions with respective to habitat type (river or sea) and body size (B). Further dL/dt was assumed as function with proportion of energy allocated (u (t)) to ionic and osmotic regulation. How fish choose energy allocating to ionic and osmotic retaliation, and how it choose habitat so as to maximize B (T) L (T) were analyzed by the Pontryagin's maximum principle, where T is a terminal time considered. Optimal habitat choice schedule exhibits optimal sea-run schedule in anadrmous species.The most interesting result acquired by the mathematical model is that individual born from large egg goes down sea earlier than that born from small egg. Several experiments and observations exhibit the same pattern. Thus it should be concluded that the result acquired by the mathematical model is heuristically robust.