Beginning with Emlen (1966) and MacArthur and Pianka (1966) and extending through the last ten years, several authors have sought to predict the foraging behavior of animals by means of mathematical models. These models are very similar,in that they all assume that the fitness of a foraging animal is a function of the efficiency of foraging measured in terms of some "currency" (Schoener, 1971) -usually energy- and that natural selection has resulted in animals that forage so as to maximize this fitness. As a result of these similarities, the models have become known as "optimal foraging models"; and the theory that embodies them, "optimal foraging theory." The situations to which optimal foraging theory has been applied, with the exception of a few recent studies, can be divided into the following four categories: (1) choice by an animal of which food types to eat (i.e., optimal diet); (2) choice of which patch type to feed in (i.e., optimal patch choice); (3) optimal allocation of time to different patches; and (4) optimal patterns and speed of movements. In this review we discuss each of these categories separately, dealing with both the theoretical developments and the data that permit tests of the predictions. The review is selective in the sense that we emphasize studies that either develop testable predictions or that attempt to test predictions in a precise quantitative manner. We also discuss what we see to be some of the future developments in the area of optimal foraging theory and how this theory can be related to other areas of biology. Our general conclusion is that the simple models so far formulated are supported are supported reasonably well by available data and that we are optimistic about the value both now and in the future of optimal foraging theory. We argue, however, that these simple models will requre much modification, espicially to deal with situations that either cannot easily be put into one or another of the above four categories or entail currencies more complicated that just energy.

Citation classic - optimal foraging - a selective review of theory and tests

National Science Education Standards

미국의 “전국 수학 교사 협의회”(National Council of Teachers of Mathematics, NCTM)는 1989년부터 〈학교 수학의 교육과정과 평가 규준〉(1989), 〈수학 가르침(교수)의 전문성 규준〉(1991), 〈학교 수학의 평가(시험) 규준〉(NCTM, 1995), 〈학교 수학의 원리와 규준〉(2000)을 출판하여 미국의 수학 교육 의 전망(목표, 나아갈 길)과 규준(실행 지침)을 제시하였다. 수학 교사들로 구성된 미국의 NCTM은 학생, 학부모, 학교 행정가 등 많은 사람들과 힘을 합하여 모든 학생들에게 수준 높은 수학 교육을 받을 수 있는 여건(환경, 기회)을 조성하는 데 구심점의 역할을 하였다. 한편 많은 관련 단체들은 여러 배경과 능력을 가진 학생들이 전문성을 지닌 교사(특수 교사를 일컫는 밀이 아니다. 수학 교과를 이해하고 수학의 전문성과 특수성을 가르칠 수 있는 일반 교사를 일컫는 말이다.)로부터 미래를 대비해 평등하고, 진취적이며, 지원이 잘 이루어지고, 공학 도구(IT)가 잘 갖춰진 환경에서 중요한 수학적 아이디어를 이해하면서 학습할 수 있는 수학 교실(미국에서는 우리나라처럼 수학 교사가 수학 시간에 학생의 방(교실: Homeroom)에 찾아가지 않고 학생들이 선생의 방(수학 교실: Classroom)을 찾아온다. 전형적인 수학 교실의 사진은 2쪽에 나와 있다.)을 만들기 위해 함께 힘썼다. NCTM에서 출간한 여러 규준들은 우리나라의 제6차와 제7차 교육과정에도 큰 영향을 미쳤다. 이 글에서는 NCTM(2000)에서 제시한 학습 원리를 간단히 살펴본 다음 이를 중심으로 현재 미국 수학 교육의 교수ㆍ학습 이론의 동향을 살펴본다.

미국 NCTM의 Principles and Standards for School Mathematics에 나타난 수학과 교수,학습의 이론

Contents
 	Summary	30	
I.	Allocation in perspective	31	
II.	Topics of this review	32	
III.	Methodology	32	
IV.	Environmental effects	33	
V.	Ontogeny	36	
VI.	Differences between species	40	
VII.	Physiology and molecular regulation	41	
VIII.	Ecological aspects	42	
IX.	Perspectives	45	
 	Acknowledgements	45	
 	References	45	
 	Appendices A1–A4	49	


Summary
We quantified the biomass allocation patterns to leaves, stems and roots in vegetative plants, and how this is influenced by the growth environment, plant size, evolutionary history and competition. Dose–response curves of allocation were constructed by means of a meta-analysis from a wide array of experimental data. They show that the fraction of whole-plant mass represented by leaves (LMF) increases most strongly with nutrients and decreases most strongly with light. Correction for size-induced allocation patterns diminishes the LMF-response to light, but makes the effect of temperature on LMF more apparent. There is a clear phylogenetic effect on allocation, as eudicots invest relatively more than monocots in leaves, as do gymnosperms compared with woody angiosperms. Plants grown at high densities show a clear increase in the stem fraction. However, in most comparisons across species groups or environmental factors, the variation in LMF is smaller than the variation in one of the other components of the growth analysis equation: the leaf area : leaf mass ratio (SLA). In competitive situations, the stem mass fraction increases to a smaller extent than the specific stem length (stem length : stem mass). Thus, we conclude that plants generally are less able to adjust allocation than to alter organ morphology.

https://nph.onlinelibrary.wiley.com/doi/epdf/10.1111/j.1469-8137.2011.03952.x

Biomass allocation to leaves, stems and roots: meta-analyses of interspecific variation and environmental control

We e xplore the issues relevant to those types of ecosystems containing new combinations of species that arise through human action, environmental change, and the impacts of the deliberate and inadvertent introduction of species from other regions. Novel ecosystems (also termed ‘emerging ecosystems’) result when species occur in combinations and relative abundances that have not occurred previously within a given biome. Key characteristics are novelty, in the form of new species combinations and the potential for changes in ecosystem functioning, and human agency, in that these ecosystems are the result of deliberate or inadvertent human action. As more of the Earth becomes transformed by human actions, novel ecosystems increase in importance, but are relatively little studied. Either the degradation or invasion of native or ‘wild’ ecosystems or the abandonment of intensively managed systems can result in the formation of these novel systems. Important considerations are whether these new systems are persistent and what values they may have. It is likely that it may be very difficult or costly to return such systems to their previous state, and hence consideration needs to be given to developing appropriate management goals and approaches.

/pdf/novel-ecosystems-theoretical-and-management-aspects-of-the-20xgt41pzx.pdf

Novel ecosystems: theoretical and management aspects of the new ecological world order

Plant ecologists and evolutionary biologists frequently examine patterns of phenotypic variation across variable environments or genetic identities. Too often, we ignore the fact that most phenotypic traits change throughout growth and development of individual plants, and that rates of growth and development are highly variable. Plants growing in different environments are likely to grow at different rates, and will be of different sizes and stages of development at a particular age. When we compare plants as a function of plant size or developmental stage, as well as a function of age, we broaden our understanding of phenotypic variation between plants.

/pdf/interpreting-phenotypic-variation-in-plants-vx9j94i5s2.pdf

Interpreting phenotypic variation in plants.

We examined biomass allocation patterns throughout the entire vegetative growth phase for three species of annual plants along three separate gradients of resource availability to determine whether observed patterns of allocational plasticity are consistent with optimal partitioning theory. Individuals of the annual plant species Abutilon theophrasti, Chenopodium album, and Polygonum pensylvanicum were grown from locally field-gathered seed in controlled greenhouse conditions across gradients of light, nutrients, and water. Frequent harvests were used to determine the growth and allocation (root vs. shoot, and leaf area vs. biomass) responses of these plants over a 57-d period. Growth analysis revealed that each species displayed significant plasticity in growth rates and substantial amounts of ontogenetic drift in root:shoot biomass ratios and ratios of leaf area to biomass across each of the three resource gradients. Ontogenetically controlled comparisons of root:shoot and leaf area ratios across light ...

Biomass allocation in plants: ontogeny or optimality? a test along three resource gradients

1. We tested whether plants increase root :shoot ratios to compensate for limitations of below-ground resources in a manner consistent with optimal partitioning theory or whether the relative production of roots and shoots is controlled by species-specific developmental patterns. Individuals of two annual plant species, Abutilon theophrasti and Chenopodium album, were grown from seed in controlled greenhouse conditions under high- or low-nutrient regimes. Mid-way through the experiment, a sub-set of low-nutrient-grown plants were given high nutrient availability and a sub-set of high-nutrient-grown plants were transferred to a low nutrient environment. 2. Under continuous nutrient regimes : (1) high-nutrient-grown plants of both species grew faster and had a lower root :shoot ratio than low-nutrient-grown plants, consistent with optimal partitioning theory ; (2) both species exhibited a substantial amount of ontogenetic drift as root :shoot ratios decreased through ontogeny (subsequent to an initial increase in R/S shortly after germination) ; (3) allometric analyses revealed that increased allocation to roots occurred very early in ontogeny for both species, after which the relative growth of shoots exceeded that of roots in low-nutrient-grown plants compared to their high nutrient-grown counterparts - a result inconsistent with optimal partitioning theory. 3. Under temporally varying nutrient regimes : (1) growth substantially increased in low-nutrient-grown plants that were switched to a high-nutrient environment without a change in root :shoot partitioning ; (2) there was no change in growth or partitioning when plants were switched from a high- to a low-nutrient regime. 4. We conclude that, for these annual species, root/shoot partitioning is partially consistent with optimal partitioning theory but that is also highly ontogenetically constrained. This constraint is evident both in substantive ontogenetic drift in partitioning and in the period during development that plasticity in partitioning can be expressed.

Kelly D. McConnaughay

Papers

Interpreting phenotypic variation in plants.

Biomass allocation in plants: ontogeny or optimality? a test along three resource gradients

Plasticity in root/shoot partitioning: optimal, ontogenetic, or both?

Interpreting phenotypic plasticity : the importance of ontogeny

A non-functional interpretation of a classical optimal-partitioning example