Werner Jan, Sfakianakis Nikolaos, Rendall Alan D, Griebeler Eva Maria
Evolutionary Ecology, Institute of Organismic and Molecular Evolution, Johannes Gutenberg-University Mainz, P.O. Box 3980, D-55099 Mainz, Germany.
Applied Analysis, Institute of Mathematics, Johannes Gutenberg-University Mainz, Germany; Institute of Applied Mathematics, University of Heidelberg, Heidelberg, Germany.
J Theor Biol. 2018 May 7;444:83-92. doi: 10.1016/j.jtbi.2018.02.007. Epub 2018 Feb 13.
Ectothermic and endothermic vertebrates differ not only in their source of body temperature (environment vs. metabolism), but also in growth patterns, in timing of sexual maturation within life, and energy intake functions. Here, we present a mathematical model applicable to ectothermic and endothermic vertebrates. It is designed to test whether differences in the timing of sexual maturation within an animal's life (age at which sexual maturity is reached vs. longevity) together with its ontogenetic gain in body mass (growth curve) can predict the energy intake throughout the animal's life (food intake curve) and can explain differences in energy partitioning (between growth, reproduction, heat production and maintenance, with the latter subsuming any other additional task requiring energy) between ectothermic and endothermic vertebrates. With our model we calculated from the growth curves and ages at which species reached sexual maturity energy intake functions and energy partitioning for five ectothermic and seven endothermic vertebrate species. We show that our model produces energy intake patterns and distributions as observed in ectothermic and endothermic species. Our results comply consistently with some empirical studies that in endothermic species, like birds and mammals, energy is used for heat production instead of growth, and with a hypothesis on the evolution of endothermy in amniotes published by us before. Our model offers an explanation on known differences in absolute energy intake between ectothermic fish and reptiles and endothermic birds and mammals. From a mathematical perspective, the model comes in two equivalent formulations, a differential and an integral one. It is derived from a discrete level approach, and it is shown to be well-posed and to attain a unique solution for (almost) every parameter set. Numerically, the integral formulation of the model is considered as an inverse problem with unknown parameters that are estimated using a series of empirical data.
变温脊椎动物和恒温脊椎动物不仅在体温来源（环境与新陈代谢）上存在差异，在生长模式、生命中性成熟的时间以及能量摄入功能方面也有所不同。在此，我们提出一个适用于变温脊椎动物和恒温脊椎动物的数学模型。该模型旨在测试动物生命中性成熟时间的差异（达到性成熟的年龄与寿命）及其个体发育过程中的体重增加（生长曲线）是否能够预测动物一生的能量摄入（食物摄入曲线），并解释变温脊椎动物和恒温脊椎动物在能量分配（生长、繁殖、产热和维持之间，后者涵盖任何其他需要能量的额外任务）上的差异。利用我们的模型，我们根据五个变温脊椎动物物种和七个恒温脊椎动物物种的生长曲线以及达到性成熟的年龄，计算出了能量摄入功能和能量分配情况。我们表明，我们的模型产生了变温动物和恒温动物中观察到的能量摄入模式和分布。我们的结果与一些实证研究一致，即在鸟类和哺乳动物等恒温动物中，能量用于产热而非生长，并且与我们之前发表的关于羊膜动物恒温进化的假设相符。我们的模型为变温鱼类和爬行动物与恒温鸟类和哺乳动物在绝对能量摄入上的已知差异提供了解释。从数学角度来看，该模型有两种等效的形式，一种是微分形式，一种是积分形式。它源自离散水平方法，并且对于（几乎）每个参数集都被证明是适定的且具有唯一解。在数值上，该模型的积分形式被视为一个具有未知参数的反问题，这些参数使用一系列实证数据进行估计。
