Mathematics in Nature: Animals

Summary: Principles of engineering, physics, and mathematics are demonstrated by the physiology, movement, and behavior of animals.

Animals, including human beings, are living organisms that belong to the domain Eukaryota (having complex cellular structures enclosed with membranes) and the kingdom Animalia. Within this taxonomy, the kingdom is defined by several characteristics, including internal digestion of food (called “heterotrophism”) and the ability to move using its own energy in at least some stages of life (called “motility”). Some say that what distinguishes humans from other animals is mathematical ability. However, researchers have studied a diverse range of mathematical concepts as they relate to animal behavior and have found evidence of abilities such as symbolic calculation, efficiency in locomotion, and synchrony. There are questions about whether these findings are biased perceptions of mathematical significance. Many mathematical patterns and symmetry can also be found in the structure of animals, ranging from their cellular tissue to their coat patterns. Some of the motivation behind the development of many statistical measures and methods, such as standard deviation and regression, was to characterize natural variability and associations in animal species.

Biological Systematics, Set Theory, and Logic

Biological systematics is the field that describes and names living organisms, provides their classifications and keys for identification, and situates classes of organisms within evolutionary history and modern adaptations. In particular, classification of organisms (called “taxonomy”) is an empirical science, where description of classes is the final step in the discovery and description of organisms. Existing biological classifications may include the ranks of domain, kingdom, phylum, class, order, family, genus, and species.

The definition of the kingdom Animalia is intensional definition—it specifies necessary and sufficient conditions for belonging to the set of animals. The particular subclass of definitions used in systematics to define animals is called definition by genus and differentia. Such definitions rely on a structure of sets, subsets, and supersets as well as their differentiating conditions. For example, defining negative numbers as the set of rational numbers that are less than zero, mathematicians use the superset of rational numbers (defined elsewhere) and the differentiating condition of being less than zero. Animalia is one of several kingdoms (subsets) of the domain Eukaryota, differentiated from other kingdoms by particular conditions.

Careful decisions are made in the organization of kingdoms and in defining differentiated conditions. For example, if only the conditions of internal digestion of food and motility were used, the Venus flytrap would be considered an animal rather than a plant. However, plants are also differentiated by the sufficient condition of having plastids, such as chloroplast, in their cells. Internal digestion of food and motility are necessary but not sufficient conditions for declaring an organism an animal. There are historical and modern systems defining anywhere from two to eight kingdoms of living organisms, depending on the necessary and sufficient conditions used for definitions.

Animal Tissue Structures

All animal cells have extracellular matrix, the boundary that can serve many functions, including exchanging substances between cells, segregating tissues, and anchoring cells. Animal cells typically form tissues; groups of cells carry out particular functions within animal bodies. There are four types of animal tissues, defined by their functions: muscle, nervous, epithelial, and connective. Cells within tissues and tissues within organs may be tessellated (filling space or surface infinitely, without gaps).

Tissue engineering is an interdisciplinary field combining biology, material science, chemistry, and engineering to re-create, change, or replace tissues. It pays special attention to the mechanical and structural properties of tissues, often modeled mathematically before being implemented in the lab.

Technological Metaphors and Models

Beginning in the Renaissance, it was common for people to conceptualize living organisms in terms of human-made machines. This phenomenon worked both ways, since human constructions were informed by new understandings and observations of nature. During the Renaissance, animal tissues and organs were seen as combinations of relatively simple mechanisms such as levers. Attempts were made to imitate some functions of animals in construction, such as making bird-like wings. This analytic approach informed the development of scientific methods in biology—in contrast with a holistic view of living things as having a completely different nature from human-made mechanisms. In the seventeenth century, this philosophical approach of modeling animals on machines was supported by such influential scientists as Galileo Galilei, René Descartes, and Isaac Newton.

Engineering and mathematics developed along with explanations in biology. Developments in steam technology introduced the ideas of energy and work, which, in turn, led to the analysis of gas and liquid pressures as explanations of the interaction of tissues and organs in animal bodies. The metaphors of heart or cellular structures as pumps—or kidneys and the liver as filters—persist to this day. When electricity and magnetism were first discovered, there were numerous attempts to apply them directly to explanations of animal bodies, but many of these early models were discarded later. In the twentieth century, animal processes are often conceptualized as computer entities, such as nervous system as a computational network. Likewise, animal brains are observed for the purpose of building the artificial intelligence. Mathematical models in biology developed from simple measurements of weight, length, and proportion to those incorporating calculus, differential equations, statistics, computational science, and other areas of modern mathematics.

Animal Motility, Field Perception, and Gradients

Animals can move under their own power. Animals movement in response to external stimuli or gradients of stimuli is called taxis. In calculus, the gradient is a vector field; its vectors point in the direction of the greatest rate of increase in a variable and have the magnitude equal to that rate. Depending on the nature of the variable in the gradient, animals or animal cells can exhibit different types of taxis, such as thermotaxis along temperature gradients or phototaxis along light gradients. Mathematical models of taxis are based on calculus, differential equations, and statistics.

Chemotaxis is the movement along the gradient in a chemical substance. Animal cells may have multiple chemical receptors around their boundaries, allowing the cell to determine the direction of chemical gradient vectors. Animal cells can move toward chemoattractors, such as immune cells arriving where they need to be, or away from chemorepellents. The development of animal embryos involves the movement of cells and is regulated by gradients in signal chemicals. Sperm movement occurs because of chemotaxis and thermotaxis.

Magnetoperception (the ability to detect magnetic fields) is observed in migrating birds, sharks, rays, honeybees, and other animals. It is an important factor in regulating animal movement and navigation—for example, during bird migrations. Experiments and applications in magnetoperceptions usually involve attaching magnetic substances to animals and observing effects. For example, cows and deer grazing under power lines orient themselves differently. The mechanisms of magnetoperception continue to be actively investigated.

Animal Locomotion

The way animals move, in addition to being a matter of biological interest, is a source of engineering ideas. Until the twentieth century, the main source of data on animal movement was observation and, sometimes, experiments with animals or their body parts. Photography and videography added details to the observation. Animals may be equipped with miniature devices that track their positions in space, as well as the electric activity within muscles, the contraction of muscles, or the forces exerted by muscles. These devices allow the development of detailed models of animal bodies during movement.

Every type of locomotion has been modeled in physics, with a variety of relevant equations. There are three major types of terrestrial locomotion (movement on solid surfaces): legged movement, slithering, and rolling. Legged animals may have from two to 750 legs, with the geometry of leg and joint position defining posture, and the pattern and pace of leg use defining gaits. Snakes move by undulating in several patterns, such as sidewinding, or by lifting parts of their belly slightly off the ground, moving them forward relative to their ribs, and then pulling the body to them (rectilinear motion). These movements on land are described by kinematic equations, in water by hydrodynamic equations. Rolling animals, such as pangolins, can briefly achieve great speed, usually by forming a wheel or a ball out of their whole body and using gravity to escape predators.

Swimming is accomplished by body movement propulsion in fish, jet propulsion in mollusks, undulation in several types of animals, and limb movement in some birds and mammals. Jet propulsion requires relatively high energy but can provide animals with an occasional burst of speed. Models of swimming include such measures as buoyancy and are modeled with fluid dynamics and mechanics.

Gliding, soaring, and flying are energy-efficient ways of locomotion, and attract much interest in biomechanics and aerodynamics. Scientists study concepts like lift and drag as well as ratios of wing measurements such as loading (weight to area). Animals use different types of motion through the air, which are defined by a combination of timing and geometry. For example, falling with increased drag forces that prolongs the fall can be either parachuting (when the angle to earth is more than 45 degrees) or gliding (when the angle is less than 45 degrees). Gliding animals such as fish and squirrels have aerodynamic adaptations including streamlining. The variable glide ratio is the ratio between the horizontal and the vertical speed components (lift to drag). A flying squirrel has a glide ratio of about two, and a human in a glider windsuit modeled after gliding animals has a glide ratio of about two and a half. Soaring birds glide during parts of their flight.

The properties of winged flight in birds and bats depend on proportions of the animal’s body. Wingspan is the distance between wingtips, and the mean wing chord is the average of the distances between the front and the back edge of the wing, found using calculus. Aspect ratio of a wing is the ratio of wingspan to mean chord. Fast birds such as falcons have pointy short wings with high aspect ratio (narrow wings). Long wings with high aspect ratios such as the wings of albatrosses, on the other hand, can produce slow soaring and gliding flight. Wide, rounded wings with medium aspect ratios can be used for a variety of flight types, for example, in storks or sparrows.

Biophysicists first attempted to explain insect flight using bird flight mechanics. They found that the resulting forces were several times less than what would be needed to lift and to propel an insect. Current theories of insect flight are still controversial. The theories use computational differential equations to model effects such as vortexes created in front of wings. When wings flap with high enough frequency, such a vortex can provide significant additional suction force.

Relatively rare types of animal locomotion depend on surface tension and capillary forces for walking on the water surface, or moving faster over released liquid (Marangoni effect). These forces are studied in fluid dynamics and thermodynamics.

Researchers debate why the wheel, which provides several mechanical advantages in terrestrial locomotion, has never evolved in any animal. The relevant mathematical model is a graph measuring fitness of organisms to the environment, called fitness landscape. Fitness peaks are stable states, with genetic modifications meaning worse fitness. While wheel locomotion may be a fitness peak, it is surrounded by fitness valleys too deep to be crossed by evolutionary means.

Migrating Animals

Many animals migrate—periodically travelling among habitats—sometimes over long distances. Models of migration take into account the time of each leg of the journey as well as the full period of migration. These times can be synchronized with seasonal milestones, developmental stages in the life of each animal, and other natural events. Because migrations can take place across international boundaries, they can help promote international efforts in research and conservation. The Convention on the Conservation of Migratory Species of Wild Animals, for example, covers several endangered species of birds and fish, as well as migratory bats and turtles.

About a fifth of all bird species in the world migrate. Typically, birds migrate closer to the equator in winters, and farther from the equator in summers. Mathematical models of bird migration include the overall patterns for particular populations such as migration corridors as well as random events such as irruptions (large numbers of birds) migrating farther after population explosions. Biophysics involved in bird migration includes theories of energy efficiency, and various mechanical effects, such as wear on feathers that necessitates periodic molting synchronized with the migration period.

During migration, birds navigate by using the landscape clues they learn while young, orienting by the sun, or using magnetoperception. In some bird species, navigating is mostly a learned behavior; in others, it is mostly coded genetically. Sometimes the coding goes wrong, reversing the migration direction 180 degrees, thus causing birds to reverse-migrate in the opposite direction from the majority of their flock. Bird species that learn their migration routes from their elders, such as cranes, can be taught to use safer routes by following light aircraft of animal preservation specialists.

Shorter migration routes also exist. For example, many fish species rise to the water surface to feed at night—a type of diel vertical migration. Many fish species high in the food chain migrate to follow their prey, with varying times and lengths of migration journeys.

Because many insects are relatively short-lived, their migrations may involve multiple generations being born along the route. In these cases, none of the individual insects travels the full migration route. Some migrating insects, such as locusts, swarm for the purpose of migration. A swarm can be modeled using a system of differential equations where pairs of individuals move closer if they are too far, move away if they are too close, and orient themselves toward the same direction. However, studies of insects, including locusts, show complex mechanisms that include chemoregulation, physiological change in response to overcrowding (measured in contacts per unit of time), emission, and responsiveness to sounds and other variables involved in swarming.

Herds of animals, schools of fish, and flocks of birds can be modeled as groups of particles, with interactions among individuals determined by differential equations with some fixed and some random parameters to account for individual behavior variations. Such mathematical models (called “interacting particle models”) can describe flock behavior or predict school migration routes. To observe animal migration, researchers use tracking devices, satellite observation, and echolocation for marine species.

Food Webs

Food webs and food chains map food relationships in ecosystems. The key measurement of the position within the food web is called “trophic level.” Autotrophs (producers) are at trophism level one. Autotrophs are organisms that do not consume other organisms or carbon produced by them, and therefore are not animals. Two mechanisms of autotrophism are photosynthesis in plants, and chemosynthesis in archae and bacteria. The first organisms to evolve on Earth used chemosynthesis. A third mechanism, radiotrophy, is being researched in fungi in high-radiation areas. All food chains within all food webs on Earth start with level one autotrophs. Predator species that no other species predate upon are called apex predators.

More specifically, classes of organisms are named according to the flow chart with three branchings. The first branching determines the source of energy, either light (photo-) or chemical (chemo-). The second branching determines the source of extra electrons in reduction-oxidation reactions, either organic (-organo-) or inorganic (-litho-). The third branching defines the source of carbon, either organic (-heterotroph) or carbon dioxide (-autotrophs). For example, fungi are chemoorganotrophic. All eight combinations resulting from these three branchings exist in nature. Heterotrophic organisms that break down other dead organisms into simpler organic or inorganic compounds are called decomposers. Consumer organisms use other living organisms as their source of energy. Simplistically, the second trophic level comprises primary consumers that eat plants (herbivores) or chemosynthesizing creatures. The third trophic level, secondary consumers or predators, consists of animals that eat primary consumers. Animals that eat those at the third trophic level are said to have the fourth trophic level, and so on. However, most existing animal species obtain energy from several sources. For example, foxes eat rabbits and berries; chickens eat grains and insects.

To address the complexity of food chains, the trophic level of an animal is determined by the formula of adding all products of levels of its food by the fraction of that food in the animal’s diet, and adding 1. For example, if a chicken’s diet consists of 30% worms (level 2) and 70% grain (level 1), its trophic level is equal to

0.3(2) + 0.7(1) + 1 = 2.3.

Statistical analysis is used to determine the mean trophic level of a species in a particular ecosystem.

Changes in any part of the food web affect all other parts. For example, the effect of introducing predators that reduce the numbers of the prey and cause abundance in the next trophic level down is an example of a “trophic cascade event.” The ability of an ecosystem to withstand disturbances is measured by an index called ascendency, and is derived by formulas from the information theory field of mathematics. Variables in ascendency formulas include both the amounts of energy and matter circulated within an ecosystem and the information shared among members of the system. Low ascendancy values make ecosystems internally unstable; high ascendancy values make ecosystems oversensitive to external disturbances. Ascendency values corresponding to stable systems are called “the window of vitality.”

Ascendency is an example of using multiple indices and metrics to model, evaluate, and predict changes in food webs. For example, consider energy or biomass transfer from one feeding level to the next feeding level. The efficiency of this transfer is a measure of an ecosystem called ecological efficiency. For example, in a food chain that consists of four levels, with mean ecological efficiency of 1/10, the apex predator has the ecological efficiency of converting sunlight into its biomass of

Ecological efficiency restricts the number of possible trophic levels.

Fantastic Animals, Hybrids, and Genetic Chimeras

A variety of cultures describe fantastic animals or humanoids with animal traits. These animals—especially those invented before the nineteenth century—are used in mathematics education to help students understand concepts related to combinatorics because they are made by combining parts of existing animals. For example, ancient Greeks invented a chimera that had the body of a lion, the heads of a goat and a lion, and a snake for the tail. In genetics, chimeras are animals that have genetic material from more than one zygote—from four or more parents. Chimeras of different animals of the same species happen naturally when several eggs in one female are fertilized by sperm from different males and then fused. They may also happen artificially, in which case different animals species can be used. For example, a goat-sheep chimera called “geep” was first produced in the 1970s.

Hybrid animals are different from chimeras in that they have two parents, but the parents are of different species. Hybridization has been recognized and used for millennia. For example, humans have produced large populations of mules since ancient times. The mathematics of hybrids involves tracking the amount of genetic material from each species through generations, and calculating the probabilities of achieving particular traits in offspring. For example, a single-cross hybrid has 50% genetic material from either line of parents. Crossing such hybrids with the line of one of the parents (called backcrossing) produces hybrids with roughly 75% genetic material from that parent’s species—averaged across a species, as individuals will have either pure or half-and-half genetic material.

Symmetry and Fractals

Most animal bodies exhibit either rotational (radial) or reflection symmetry. Animals with bilateral reflection symmetry (having a plane separating bodies into roughly reflected halves) form the taxon Bilateria. Observation of symmetry is a major tool of evolutionary theory. For example, it is hypothesized that all Bilateria animals evolved from a common ancestor species, Urbilaterian, that lived around six hundred million years ago. This makes Bilateria a clade (a group of animals that come from a common ancestor). Bilaterians have the front end with the mouth and the back end with the anus, defined by the plane of symmetry.

Rotationally symmetric animals such as sea anemones and sea stars usually have the mouth on the axis of the symmetry. When animals have a certain number of body regions positioned around the axis symmetrically, they are called by the number of regions. For example, five-armed stars exhibit pentamerism, and many coral polyps exhibit hexamerism, or six-part rotational symmetry.

Combinations of reflections, rotations, and translations can produce repeated geometric patterns called tessellations or wallpaper groups in plane and crystallographic groups in space. There are 17 types of wallpaper groups and 230 types of crystallographic groups described by the area of mathematics called group theory. Wallpaper and crystallographic groups can be found in colonies of animals such as corals or in arrangements of animal body parts such as fish scales.

Fractals are shapes that can be split into parts that are copies of the whole. Fractals frequently occur in the living nature. For example, feathers are fractal-like structures of the tree type, with three or four levels. Nervous systems and lungs of mammals are also tree-type fractals. Beyond the literal meaning as a geometric shape, the idea of a fractal as a self-repeating structure is applied to many areas related to animals to describe patterns within systems behavior, evolution, migration, and development.

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