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  Mathematical models

  It had long been understood that one of the most basic ecological processes is the struggle for survival: for herbivores to find food, predators to find prey, and prey to avoid being eaten. Predators do everything they can to hunt and eat prey, and the latter do all they can to avoid being eaten. In 1910, Alfred Lotka introduced one of the first mathematical models ever applied to ecology. Now known as the Lotka-Volterra model, its predator–prey equations help predict the population fluctuations of these two groups.

  In the early years of the 20th century, Joseph Grinnell conducted extensive research into animals’ habitat needs in the western United States. He observed that species had different “niches” within a habitat—and that if two species have approximately the same food requirements, one will “crowd out” the other. Darwin had observed this on his travels aboard HMS Beagle, but Grinnell’s axiom developed the idea further, as did subsequent research. In 1934, Georgy Gause demonstrated what he called the competitive exclusion principle in laboratory projects. As William E. Odum put it in 1959, “the ecological niche of an organism depends not only on where it lives, but also on what it does.”

  From field to lab

  Laboratory experiments and field observations are the main methods of providing data for the study of ecological processes, but field experiments—in which a local environment is manipulated to test a hypothesis—were not conducted with scientific rigor until Joe Connell’s work with barnacles in Scotland. His experiments—the results of which were published in 1961—were meticulously planned and observed, and were repeatable.

  Connell set the “gold standard” for fieldwork, but experiments in laboratories still have a vital role to play, too—as Earl Werner demonstrated 30 years later. His work revealed the nonconsumptive impact of predatory dragonfly larvae on the behavior and physical development of their tadpole prey.

  Since the mid-20th century, many new ideas on ecological processes have emerged. Work by Robert MacArthur and others on competition between species led to the development of optimal foraging theory, which seeks to explain why animals choose to eat some food items and not others. Mutualistic relationships became better understood through the research of biologists such as Daniel Janzen. Robert Paine’s work with starfish and mussels also highlighted the concept of keystone species—those that have a disproportionate influence on their ecosystems.

  New technology

  Technological advances—including sophisticated chemical sampling techniques, satellites with remote sensing equipment, and computers capable of rapidly processing huge quantities of data—have opened up new areas of study.

  Ecological stoichiometry, for example, studies the flow of energy and chemical elements throughout food webs and ecosystems, from the molecular level up. Like so many ideas in ecology, its origins can be traced back many years, but only took hold with Robert Sterner and James Elser’s 2003 book Ecological stoichiometry: The biology of elements from molecules to the biosphere. New techniques such as this will undoubtedly continue to deepen our understanding of processes in ecology.

  IN CONTEXT

  KEY FIGURES

  Alfred J. Lotka (1880–1949), Vito Volterra (1860–1940)

  BEFORE

  1798 British economist Thomas Malthus shows that the rate at which the population changes increases as the size of the population grows.

  1871 In Lewis Carroll’s novel Through the Looking Glass, the Red Queen tells Alice, “you have to run just to stay in the same place.”

  AFTER

  1973 American biologist Leigh Van Valen proposes the Red Queen effect, which describes the constant “arms race” between predators and prey.

  1989 The Arditi–Ginzburg equations offer another model of predator–prey dynamics by including the impact of the ratio between predator and prey.

  The predator–prey equations are an early example of the application of mathematics to biology. Formulated in the 1920s by American mathematician Alfred J. Lotka and Italian mathematician and physicist Vito Volterra, the two equations—also known as the Lotka–Volterra equations—describe the way in which the population of a predator species and that of its prey fluctuate in relation to each other.

  Lotka proposed the equations in 1910, as a way of understanding the rates of autocatalytic chemical reactions—chemical processes that regulate themselves. In the following decade, he applied the equations to the population dynamics of wild animals.

  In 1926, Vito Volterra arrived at the same conclusions. He had become interested in the subject after meeting Italian marine biologist Umberto D’Ancona. D’Ancona told Volterra how the percentage of predatory fish caught in nets in the Adriatic Sea had greatly increased during World War I. This change was clearly linked to the drastic reduction in fishing during the war years, but D’Ancona could not explain why less fishing did not produce more fish of all kinds in the nets. Using the same equations as Lotka, Volterra eventually explained the fluctuations in both the predator and the prey species.

  A cheetah pursues a Thomson’s gazelle. The predator–prey equations are able to model the way populations of both species will change in response to the activities of the other.

  “The food species cannot, therefore, be exterminated by the predatory species, under the conditions to which our equations refer.”

  Alfred J. Lotka

  Population principles

  At the time Lotka and Volterra made their calculations, the science of population dynamics was still in its infancy, having barely moved on since the population studies of British economist Thomas Malthus in the late 18th century. According to Malthus’s theory, a population grows or declines rapidly as long as the environmental factors for survival are constant, and the rate at which that population changes increases as the population grows. From this theory, Malthus predicted a catastrophic future for humanity. The number of humans was growing much more quickly than the amount of food that could be produced by the world’s farmlands. Eventually, Malthus argued, a point would be reached when the human population would succumb to global famine and decline.

  Malthus’s bleak vision did not happen, thanks to technological advances in agriculture and the development of artificial fertilizers, but his population model became applicable to species populations within ecosystems. Every habitat, and the niche occupied by a species within its community of organisms, has a carrying capacity—the maximum population that can be supported by the resources available, such as water, space, food, and light. Any rise in population above this level is likely to be reduced by naturally occurring factors. As a result, wild populations should in theory be more or less static, fluctuating only around the carrying capacity, assuming the random impacts of catastrophic events are ignored.

  However, this relative equilibrium did not always match up with observations—as in D’Ancona’s account of a sudden increase in the population of predatory sea fish. One theory to explain this discrepancy started from the premise that the population of predators is related to the size of the population of their food supply, such as prey species. The relationship suggests that when a lot of food is available, there will be a large predator population. The growing predator population should then begin to reduce the amount of prey, which will in turn lead to a drop in the number of predators. The size of both populations will rise and fall, but the ratio of predators to prey will remain stable.

  Such a balanced theory was still at odds with species observations. Through mathematical modeling, Volterra was able to show that the average sizes of predator and prey populations do indeed oscillate but the rate at which each population is growing or declining is always changing and almost never matches the changes experienced by the other population. To eliminate variables, Volterra made a series of assumptions: first, that the prey and predator species have no reproduction limits and the rate of change in a population is proportional to its size; second, that the prey population—presumed to be a herbivore—is always able to find enough f
ood to survive. Next, they assumed that the prey population is the predators’ only source of nourishment, and that the predators never become full and never stop hunting. Finally, they assumed that environmental conditions, such as weather or natural disasters, had no impact on the process. The effect of the genetic diversity of the predators and prey animals on their ability to survive was not taken into account.

  When plotted on a graph, the predator population trails the rise and fall of the prey population, and is still rising as the prey population starts to decline. This explained D’Ancona’s observation of the larger proportion of predators after the prey population had been allowed to boom by a reduction in fishing.

  The relative fluctuations of the populations depends on the relative reproductive rates of the two species and the predation rate. For example, oscillations in the size of an ant population and that of an anteater are barely noticeable because they reproduce at such different rates. The oscillations in the populations of species that breed at similar rates, such as the Iberian lynx and rabbit, are much more pronounced.

  The predator and prey populations rise and fall over time in regular cycles. Although the degree to which they change varies, the cycle follows a broadly similar pattern.

  “Mathematics without natural history is sterile, but natural history without mathematics is muddled.”

  John Maynard Smith

  British mathematician and evolutionist

  Nature’s arms race

  The predator–prey equations revealed that species are locked together in a never-ending struggle, swinging from near disaster and extinction to times of abundance and fertility. In this biological “arms race,” the evolutionary pressure on the prey species is to escape predation and survive, so as to have more offspring. Meanwhile, the predator is under pressure to have a higher predation rate in order to provide food for more offspring. However, neither species is superior, responding instead to the adaptations of the other. The predator–prey relationship between even-toed hoofed mammals—such as antelopes and deer—and mammalian carnivores, like the big cats and wolves, is an example of this evolutionary arms race. The hoofed animals have long legs, extended by walking on the very tips of thickened and fused toe bones. This adaptation allows them to outrun and outjump their predators. In response, big cats—such as lions and tigers—have evolved speed and strength to bring down large, fleet-footed prey in surprise attacks. Wolves have evolved the stamina to run for long distances without stopping. This allows them to work as a team to chase down their prey and kill them when the exhausted prey collapse.

  While the predator–prey equations offer an insight into the population dynamics of two species, the assumptions they rely on are rarely reflected in real life. Some predators do specialize in killing a single prey species, but other factors in the ecosystem also affect their populations.

  Other applications

  The Lotka–Volterra equations have been used to study the dynamics of food chains and food webs in which one species may be a predator of another species but also the prey species of a third. They have also been used to examine the relationship between host and parasite species, which bears some resemblance to that between prey and predator. Parasites often specialize in one host species—a relationship that should resemble the one described by the Lotka–Volterra equations. However, in practice the process of evolution is thought to interfere with this. A parasite does not usually kill its host (those that do are called parasitoids), but can reduce its fitness. The Red Queen evolutionary theory, proposed in the 1970s by Leigh Van Valen, describes how, thanks to beneficial genes, certain individuals in a host population are able to maintain their fitness despite the attacks from parasites. The parasites constantly evolve to exploit these seemingly immune individuals, and therefore the beneficial genes in the host population also change. In this way, evolution is happening all the time, as the parasite and host battle it out—although everything appears to stay the same.

  The parasitoid wasp lays its eggs in aphids (the smaller, yellow insects shown above). It is called a parasitoid because the wasp’s larvae later eat the aphids as they grow.

  “Volterra was interested in a mathematical theory of ‘the survival of the fittest.’”

  Alexander Weinstein

  Russian mathematician

  VITO VOLTERRA

  Born in 1860 in Ancona, Italy, the son of a Jewish cloth merchant, Vito Volterra grew up in poverty. Despite this, in 1883, aged just 23, he secured a position as professor of mechanics at the University of Pisa and began a career as a mathematician. Further professorships at the universities of Turin and Rome followed. In 1900, Volterra married, fathering six children, although only four survived to adulthood. He was made a senator of the Kingdom of Italy in 1905 and worked on the development of military airships during World War I. In 1931, Volterra refused to swear loyalty to Italy’s fascist dictator Benito Mussolini and was dismissed from the University of Rome. Forced to work abroad, he only returned to Italy for a short time before his death in 1940.

  Key works

  1926 “Fluctuations in the Abundance of a Species Considered Mathematically,” Nature

  1935 Les associations biologiques au point de vue mathématique

  See also: Evolution by natural selection • The selfish gene • Ecological niches • Competitive exclusion principle • Mutualisms • Keystone species • Optimal foraging theory

  IN CONTEXT

  KEY FIGURE

  Joseph Grinnell (1877–1939)

  BEFORE

  1910 In a paper about beetles, Roswell Hill Johnson, a US biologist, is the first person to use the word “niche” in a biological context.

  AFTER

  1927 British ecologist Charles Elton stresses the importance of an organism’s role as well as its “address” in his definition of an ecological niche in his book Animal Ecology.

  1957 In an academic paper called “Concluding Remarks,” British ecologist George Evelyn Hutchinson expands the theory of niches to embrace an organism’s entire environment.

  1968 A study by Australian D.R. Klein of the introduction, increase, and die-off of reindeer on St. Matthew Island, Alaska, identifies the destructive niche.

  An organism’s niche is a combination of its place and its role in the environment. It encompasses how the organism meets its needs for food and shelter, as well as how it avoids predators, competes with other species, and reproduces. All its interactions with other organisms and the nonliving environment are also part of what makes up its niche. A unique niche is an advantage for any animal or plant because this reduces competition with other species. For ecologists, a full knowledge of an organism’s niche is vital to inform interventions to compensate for the environmental changes caused by habitat destruction and climate change.

  The pioneer of the niche concept was Joseph Grinnell, a US biologist who studied a bird called the California Thrasher. In 1917, he published his observations, which showed how the bird fed and bred in the underbrush of a scrubby habitat known as chaparral, and how it escaped predators by running through the underbrush. The thrasher’s camouflage, short wings, and strong legs were perfectly adapted for life in this environment. Grinnell saw the chaparral habitat as the thrasher’s “niche.” His idea also allowed for “ecological equivalence” in plants and animals, whereby species distantly related and living far apart could show similar adaptations, such as feeding habits, in similar niches. In the Australian outback, for instance, babbler bird species forage in the scrubby vegetation in a similar way to the unrelated thrasher. Grinnell also identified “vacant” niches—habitats that a species could potentially occupy, but where it was not present.

  Widening the niche

  In the 1920s, ecologist Charles Elton looked beyond a simple habitat definition for “niche.” For him, what an animal ate and what it was eaten by were the primary factors. Thirty years later, George Evelyn Hutchinson expanded the definition yet further. He argued that a niche should take in
to account all of an organism’s interactions with other organisms and its nonliving environment, including geology, acidity of soil or water, nutrient flows, and climate. Hutchinson’s work encouraged others to explain the variety of resources used by a single organism (niche breadth), the ways in which competing species coexist (niche partitioning), and the overlap of resources by different animals and plants (niche overlap).

  “[A niche] is a highly abstract multi- dimensional hyperspace.”

  George Evelyn Hutchinson

  The importance of habitat

  Ecological niches depend on the existence of a stable habitat; small changes can eradicate niches that organisms once filled. For example, dragonfly larvae only develop within a certain range of water acidity, chemical composition, temperature, and prey, and with a limited number of predators. The right vegetation is needed by adult females for egg-laying, and by larvae for metamorphosis. The dragonfly also impacts its environment: its eggs are food for amphibians; its larvae, which are both predators and prey, add nutrients to the water; and the adults prey on insects. These requirements and impacts define its ecological niche. Hutchinson argued that for a species to persist, conditions had to be within the required ranges. If conditions moved outside the niche requirements, a species could face extinction.