“The forest is beautiful, dense, high,” wrote Robert Frost. I remember this iambic every time I start walking along a forest trail located not far from my house. The trail is named after Frost, who spent several years in this part of Massachusetts teaching boys in blazers with brass buttons from Amherst College. Did the poet walk among these particular trees? This is possible, although then, a century ago, they were still young. Be that as it may, if he stopped near this forest, it would not be for long, because "they are expecting me today on time, and the way is long before bed."
As I walked the Frost Trail, it led me to an unremarkable lawn in the wooded area of the Northwest, sandwiched between highways, houses, and a city dump. She was neither dense nor tall, and the feeling of closeness of people never disappeared. It was not a virgin forest, but it was wooded enough not only to recall the rhymes of popular poets, but also to ask difficult questions about trees and forests - questions that have worried me for many years. Why are the trees so tall? Why don't they get taller? Why do their leaves have such a variety of shapes and sizes? Why are trees (in the sense used in graph theory) and not some other structure? And there was one more question that I would like to discuss today:
Today's question: why in a mixed forest the tree species remain mixed?
Moving along the Frost trail, I conducted a short census, fixing the Canadian hemlock, sugar maple and at least three types of oak (red, white and marsh), beech and birch, hazel ovate, Weymouth pine and two more trees that I could not identify even with with help from Peterson's guide and iNaturalist . The forest closest to my house is dominated by the Canadian hemlock, but broadleaf species are more common on the hillsides a few miles further down the trail. The photo below shows a pass (called Notch by locals) between two peaks of the Holyoke Range south of Amherst. I took this image on October 15th last year, at a time of year when autumn colors make it easier to identify species diversity.
Such forests cover most of the eastern half of the United States. Tree species vary with latitude and height, but anywhere in the forest canopy there are usually eight to ten species. Individual isolated areas are even richer; in some valleys of the southern Appalachians, called cove forests, there are up to 25 canopy species. And tropical rainforests are inhabited by a hundred or even two hundred species of tall trees.
From the point of view of ecological theory, this diversity is confusing. One could assume that in any environment one species will be slightly better adapted, and therefore will displace all the others, coming to dominate the landscape.
This principle was first described by Garrett Hardin. Its mathematical formulation was probably created by Vito Volterra.
Environmentalists call this idea the principle of competitive crowding out. It states that two or more species competing for the same resources cannot coexist permanently in the same habitat. Over time, all but one competitor will come to a local extinction. It seems that the trees of the forest canopy are competing for the same resources - sunlight, CO 2 , water, different mineral nutrients, so the persistence of mixed forests needs to be explained.
Here is a small demo of competitive preemption. Two types of trees - let's call them olives and oranges - share one piece of forest, a square area that can hold 625 trees.
The source code for six computer simulations from this article is available on GitHub.
Initially, each point is randomly assigned a tree of one type or another. When you click on the Start button (or touch an array of trees), you start a cycle of deaths and updates. At each stage of time, one tree is chosen, completely random and regardless of its type, and falls under the ax. Then another tree is selected as the parent of the replacement, thus defining its appearance. However, this choice is not completely random, it has a bias. One of the species is better adapted to the environment, uses available resources more efficiently, and therefore has an increased chance of reproduction and placement of its offspring at a free point. There is a “fitness bias” slider in the control panel under the array of trees; when shifting to the left, he prefers oranges, to the right, olives.
The result of this experiment shouldn't surprise you. Two species play a zero-sum game: whichever territory the olives capture, the oranges must disappear from it, and vice versa. Point by point, the fitter view captures everything. If the advantage is very small, then the process can take a long time, but in the end the less efficient organism is always destroyed. (What if the two species are completely equal? I'll come back to this question a bit later, but for now let's just pretend it never happens. And I've cleverly modified the simulation so you can't set the offset to zero.)
Competitive preemption doesn't prohibit anyone.coexistence. Let's say olive and orange trees use two mineral nutrients from the soil - say, iron and calcium. Suppose both of these elements are in short supply, and their availability limits the growth of tree populations. If olives are better at absorbing iron and oranges are more efficient at absorbing calcium, then the two can create a system in which they both survive.
In this model, none of the species goes extinct. With the standard sliders settings, when iron and calcium are in the same ratio in the medium, olive and orange trees also retain approximately equal numbers of species on average. Random fluctuations push them away from this equilibrium point, but not very far and not for very long. Populations are stabilized due to negative feedback. If an accidental variation increases the proportion of olive trees, then each of these trees receives a smaller proportion of the available iron, thus reducing the potential for further growth in the population of the species. Iron deficiency is less harmful to orange trees, and therefore their population returns to its original values. But if then the orange trees exceed them,they will be limited by overuse of a limited calcium supply.
By moving the slider to the left or right, we change the balance of iron and calcium in the medium. A 60:40 ratio in favor of iron will shift the balance between the two tree species, allowing olives to take up more land. But as long as the ratio of resources does not become too high, the outnumbered species is not threatened with extinction. Two types of trees adhere to the principle of "live and let the others live."
In ecology terms, olive and orange tree species have escaped the rules of competitive displacement because they occupy separate niches (roles) in the ecosystem. They are professionals who prefer different resources. Niches don't have to be completely separate. In the simulation shown above, they partially overlap: olives, along with iron, also need calcium, but only 25%; orange trees have opposite requirements.
Will this gap in the law of competitive repression allow more than two species to exist? Yes: N competing species can coexist if there are at least N independent resources or natural aspects limiting their growth, and if each species has its own limiting factor. Everyone must have a specialization.
The principle of dividing an ecosystem into many niches is an established practice in the field of biology. This is how Darwin explained the diversity of finches in the Galapagos Islands, where a dozen species differ in their habitat (soil, shrubs, trees) or food (insects, grains and nuts of different sizes). Forest trees can be organized in a similar way, with different microenvironments to suit different species. The process of creating such a diverse community is called ecological niche formation.
A certain niche differentiation is clearly visible among forest trees. For example, gum trees and willows prefer moist soil. However, in my nearest forest, I could not find any systematic differences in areas colonized by maples, oaks, hazel trees and other trees. They are often close neighbors on parcels of land with the same steepness and height, growing on soil that seems to me the same. Perhaps I just haven't yet learned to recognize these subtleties of tree growth.
The formation of ecological niches is especially staggering in the tropics, where hundreds or more of individual limiting resources are required. Each tree species is in charge of its own tiny monopoly, claiming rights to some environmental factor that is completely irrelevant to everyone else. At the same time, all trees compete fiercely for the most important resources, namely sunlight and water. Each tree seeks to occupy a void in the forest canopy with direct access to the sky, where it can spread its leaves and absorb photons daily. Given the existential importance of winning this competition for light, it seems strange to associate the diversity of forest communities with the struggle for other, less important resources.
In the creation of ecological niches, each species wins its own little competition; another theory removes the question of competition by assuming that trees are not trying to bypass other species. They just move randomly. According to this concept, called ecological neutral drift, all trees are equally well adapted to their natural environment, and the multitude of species appearing in any particular place and at any time is a matter of probability. The site can be occupied by oak, but maple or birch can flourish there. Natural selection has nothing to take away. When a tree dies and another grows in its place, nature does not care about the appearance of the replacement tree.
This idea brings us back to the question I skirted above: What happens if two competing species have exactly the same fitness? The answer will be the same for both two and ten species, so for the sake of visual variety, we'll look at the broader community.
If you run the simulation and wait patiently for it to complete, you will see a monochrome array of trees. I can't guess what the only color will be left on your screen, or in other words, what kind will take over the whole forest, but I know that there will be only one kind. The other nine will die out. In this case, such a result can be considered at least rather unexpected. Earlier we found out that if a species has even the slightest advantage over its neighbors, then it will capture the entire system. Now we see that even an advantage is not required. Even when all players are absolutely equal, one of them will become the king of the mountain, and all the rest will be destroyed. Cruel, right?
Here is a graph of one run of the program showing the number of representatives of each species as a function of time:
At the beginning, all 10 species are represented by approximately equal numbers, grouped next to an average number of 625/ten... After starting the program, the grid begins to quickly and repeatedly change colors. However, in the first 70,000 time steps, all but three species disappear. Three surviving species take turns taking the lead in spreading contrasting colors across the array. Then, after about 250,000 steps, the species represented by the bright green line declines to zero population, or extinction. The last stage of the one-on-one struggle is very uneven - the orange species is close to complete domination, and the burgundy one flounders near the extinction line, and yet this tug of war lasts another 100,000 steps. (Once the system reaches the mono-view state, nothing else can change, so the program stops execution.)
This skewed result cannot be explained by any hidden bias in the algorithm. At any time, for all species, the probability of obtaining a new representative of the species is fully equal to the probability of losing a representative. Let's pause for a while to be convinced of this fact. Let's say the formX has a population x, which should be in the interval 0≤x≤625... A randomly selected tree will be of the speciesX with probability x/625; therefore, the probability that the tree is of some other species must be(625-x)/625... X receives one representative, if it is a substitute species, but not a prey species, such an event has a joint probability x(625-x)/625... Xloses one representative if he is a victim, but not a replacement, which has the same probability.
This is a fair game, there are no cheating dice. Nevertheless, someone breaks the jackpot, and the rest of the players lose everything every time.
The spontaneous extinction of species diversity in this simulated area is entirely due to random fluctuations. Let's take a look at the populationx as a label randomly wandering along a line segment, where from one end 0and from the other 625... At each time step, the mark is shifted one unit to the right with equal probability(+1) or left (-1); upon reaching either end of the segment, the game ends. The most fundamental fact about this wandering is that it always ends. Wandering infinitely oscillating between two boundaries is not impossible, but it has a probability0; collision with one or the other wall has a chance1...
What is the expected duration of such a random walk? In the simplest case, with one label, the expected number of steps starting at positionx, equals x(625-x)... This expression has a maximum when the walk starts in the middle of a line segment; the maximum duration is slightly shorter100,000steps. In a forest simulation with ten species, the situation is more difficult because several wanderings are correlated, or rather anticorrelated: when one tag steps to the right, the other must move to the left. Computational experiments let us know that the median time required for ten species to shrink to one is approximately320,000steps.
From these computational models, it is difficult to see how a neutral ecological shift can be the savior of forest diversity. On the contrary, it seems to ensure that we end up with a monoculture - one species wipes out all the others. But this is not the end of the story.
Here one should not forget about the time scale of the process. In the simulation, time is measured by counting the cycles of death and replacement of trees in the forest. I don't know exactly how to convert this to calendar years, but I can assume that 320,000 death and replacement events for 625 trees could take 50,000 or more years. This is a very long time for forest life in New England. The entire existing landscape was cleared by the Laurentide Ice Sheet just 20 thousand years ago. If the local forests are losing sight at the rate of random drift, then we have not yet had time to reach the end of the game.
The problem is that this statement implies that early forests are diverse and gradually evolve towards monoculture, which is not supported by observations. On the contrary, it seems that diversity increases over time .... The Cove forests of Tennessee, which are much older than New England forests, contain more, not fewer species. And the super-diverse tropical rainforest ecosystem is believed to have existed for millions of years.
Despite these conceptual impediments, many ecologists actively advocate the idea of environmental neutral drift. The most notable example is Stephen Hubbell's 2001 book The Unified Neutral Theory of Biodiversity and Biogeography . The main aspect of Hubbell's defense of this idea (if I understood him correctly) is that 625 trees do not constitute a forest, and certainly do not form the entire ecosystem of the planet.
Hubbell's theory of neutral drift was inspired by previous studies of island biogeography, in particular the collaboration of Robert MacArthur and Edward Wilson in the 1960s. Suppose that our small section from625trees grows on an island separated from the continent. For the most part, the island evolves in isolation, but from time to time the bird brings seed from a much larger forest on the mainland. We can simulate these rare events by adding immigration to the neutral drift model. In the demo below, the slider controls the immigration level. With the default value1/100 every hundredth tree that comes for replacement is taken not from the local forest, but from a stable reserve, in which everything ten species are equally likely to be selected.
During the first several thousand cycles, the evolution of the forest proceeds in much the same way as in the model with normal drift. First, there is a brief period of complete chaos, then waves of flowers spread through the forest, it turns pink, burgundy, or bright green. The difference is that none of these expansive species can ever capture the entire massif. As shown in the graph below, they never greatly exceed 50% of the total population, and then disappear into the shadow of other species. Later, another tree color tries to create an empire, but meets the same fate. (Since this process does not have a clear end point, the simulation stops after 500,000 cycles.)
Immigration, even at low levels, provides a qualitative change in the behavior of the model and the fate of the forest. The important difference is that we can no longer say that extinction is permanent. The species may disappear from the array with 625 trees, but sooner or later it will be re-imported from the permanent reserve. Therefore, the question is not whether the species lives or is extinct, but whether it exists or not at the current time. At an immigration rate1/100 the average number of existing species is approximately equal nine,6, that is, none of them disappears for a long time.
At higher levels of immigration, 10 species remain thoroughly mixed, and none of them makes any progress towards world domination. On the other hand, they have a small risk of extinction, albeit temporary. Drag the slider all the way to the left, setting the immigration rate to1/tenand the forest will turn into randomly twinkling lights. There are no extinctions in the graph below.
If you drag the slider all the way to the other side, then less frequent immigration will allow the distribution of species to move much further from the state of equality. On the graph where the immigrant appears on each1000-th cycle, one or two species dominate in the population most of the time; other species are often on the verge of extinction or beyond , but sooner or later return. The average number of living species is approximately equal to4,3, and sometimes only two species remain.
With a coefficient 1/10000the impact of immigration is barely noticeable. As in the non-immigration models, one species covers the entire site; in the example below, it took about400,000steps. Thereafter, the infrequent immigration events create small spikes on the curve, but a different species will be able to replace the winner for a very long time.
The insular structure of this model makes it possible to understand how sporadic, weak ties between communities can have a huge impact on their development. But islands are optional in this model. Trees, known for their immobility, rarely have long-distance relationships, even if they are not separated by bodies of water. Hubbell formulated an ecological drift model in which many small areas of forest are ordered into a hierarchical metacommunity. Each site is both an island and part of a larger reservoir of species diversity. By choosing the appropriate sizes of sites and the coefficients of migration between them, it is possible to maintain a balance of many species. Hubbell also takes into account the emergence of completely new species, which is also seen as an accidental or selection-neutral process.
The theories of ecological niche formation and neutral ecological drift raise mirror questions among skeptics. In the case of the formation of ecological niches, we look at tens or hundreds of coexisting tree species, and ask the question: "Can each of them have a unique limiting resource?" In the case of neutral drift, we ask: "Could all of these species have exactly the same fitness?"
Hubbell answers the last question by turning it upside down. The very fact that we observe coexistence implies equality:
« , , - , . , .. - , , . - , ».
Herbert Spencer proclaimed survival of the fittest. We have a corollary to this motto: if they all survived, then all should be equally adapted.
And now something completely different.
Another theory of forest diversity was developed specifically as an answer to the most difficult question - the riddle of the extreme diversity of trees in tropical ecosystems. In the early 1970s, Joseph Heard Connell and Daniel Hunt Jensen, field biologists who worked independently in different parts of the world, came up with the same idea almost simultaneously. They hypothesized that in tropical rainforests, trees employ social distancing as a protective measure against infestations, and this stimulates diversity.
(The phrase "social distancing" does not appear in Connell and Jensen's works written fifty years ago, but today I cannot resist the temptation to describe their theory in this way.)
Surviving in a tropical rainforest is not easy. Trees are under pressure from frequent attacks by gangs of marauders: predators, parasites and pathogens. (Connell summarized all of these villains under the heading “enemies.”) Many of the enemies are specialists, targeting only one tree species. Specialization can be explained by competitive displacement: each tree species becomes a unique resource supporting one enemy type.
Suppose a tree is attacked by a large population of host-specific enemies. A swarm of villains attacks not only an adult tree, but also all the descendants of the owner, who have taken root next to the parent. Since young trees are more vulnerable than adults, the entire group will be destroyed.
Sprouts farther from their parent are more likely to remain undetected until they are mature and resilient enough to resist attacks. In other words, evolution probably favors the apple that fell farther from the apple tree. Jensen illustrates this idea with a graphical model similar to the one shown above. With increasing distance from the parent, the likelihood of a seed emerging and rooting becomes less (red curve) , but the likelihood that such a sprout will survive to maturity becomes more (blue curve) . The overall probability of successful reproduction is the product of these two factors (purple curve) ; its top is at the intersection of the red and blue curves.
Connell-Jensen's theory predicts that trees of one species will be widely distributed throughout the forest, leaving a lot of space between them for trees of other species, which will also have an equally dispersed distribution. This process leads to anticlustering: trees of the same species are, on average, farther apart from each other than they would be in a completely random structure. This pattern was noticed by Alfred Russell Wallace in 1878 based on his long experience in the tropics:
, . , , , - . , , . , .
My artificial model of the social distancing process implements a simple rule. When a tree dies, it cannot be replaced by another tree of the same species, and this replacement cannot match in appearance with any of the eight nearest neighbors surrounding the vacated space. Therefore, there must be at least one other tree between trees of the same species. In other words, each tree has an exclusion zone around it, in which other trees of the same species cannot grow.
It turns out that social distancing is a pretty effective way to preserve diversity. When you press Start, the model comes to life and begins to feverishly flicker like the front panel of a computer from a Hollywood movie of the 50s. Then it just keeps on flickering, nothing else happens. There is no spreading waves of flowers when successful species take over territory, nor extinction. Fluctuations in population sizes are even smaller than they would be in a completely random and uniform distribution of species in the array. This stability is evident from the graph below, which shows ten species holding close to an average of 62.5:
When I finished writing this program and pressed the button for the first time, I expected to see the very short-term survival of all ten species.
Pencil sketches influenced my expectations. I made sure that only four colors are enough to create a pattern in which there are no two trees of the same color, adjacent to each other horizontally, vertically, or on both diagonals. One of these patterns is shown above. I suspected that the social distancing protocol could cause the model to condense into such a crystalline state, with the loss of species that are not lined up in a repeating pattern. I was wrong. While four is indeed the minimum number of colors for a social distancing 2D grid, there is nothing in the algorithm to stimulate the system to find the minimum.
By examining the program in action, I was able to figure out what makes all species survive. There is an active feedback process that prioritizes rarity. Suppose, at the current moment, oak trees have the lowest frequency in the general population. As a result, oak trees are the least likely to be present in the exclusion zone surrounding any vacant space in the forest, which in turn means that they are most likely to be chosen as a replacement. As long as oak trees remain more rare than average, their population tends to grow. Symmetrically, it is more difficult for representatives of redundant species to find free space for their descendants. All deviations from the average population level are self-regulating.
The initial configuration in this model is completely random and ignores the constraints that prevent trees of the same species from being adjacent. Usually there are about 200 exclusion zone violations in the initial pattern, but all of them are eliminated in the first few thousand time steps. After that, the rules are strictly followed. Note that with ten species and a nine-seat exclusion zone, there is always at least one species that can fill the empty space. If you try to experiment with nine or fewer species, then some empty spaces should remain empty spaces in the forest. It is also worth mentioning that the model uses the boundary conditions of the toroid: the right edge of the mesh is adjacent to the left edge, and the top connects to the bottom. This ensures that all places in the grid have exactly eight neighbors.
Connell and Jensen envisioned much wider exclusion zones, and therefore a larger list of species. Much more computation is required to implement such a model. A recent article by Tall Levy and others describes such a simulation. They found that the number of surviving species and their spatial distribution remain fairly stable over long periods of time (200 billion tree change events).
Can the Connell-Jensen mechanism work in temperate forests? As in the tropics, trees at higher latitudes have specialized enemies, and some of them are notorious - carriers of Dutch elm disease and endothian cancer of the bark of edible chestnut, ash emerald narrow-bodied goldfish, gypsy moth caterpillars that destroy oak foliage. The hemlock near my house withstands the fierce attacks of the hermes beetle sucking the tree juices. That is, the forces driving the growth of diversity and anti-clustering according to the Connell-Jensen model are also present here. However, the observed spatial structure of the northern forests is slightly different. Social distancing does not persist here. The distribution of trees is usually a little heaped, representatives of the same species gather in small groves.
Infection-driven diversification is a fascinating idea, but like the rest of the aforementioned theories, it has some plausibility issues. In the case of the formation of ecological niches, we need to find a unique limiting resource for each species. In the case of neutral drift, we need to make sure that selection is truly neutral and assigns equal fitness to trees that look very different. In the Connell-Jensen model, we need a specialized pest for each species, powerful enough to suppress all adjacent shoots. Could it be true that every tree has its own nemesis?
There are also reasons to doubt the stability of the model, its resistance to destruction. Suppose an invasive species emerges in a rainforest — a tree new to the continent with no nearby enemies. What happens then? The program shown below offers us the answer. Run it and then click on the Invade button .
You can press Invade several times, because the newly arrived species can become extinct even before it has time to gain a foothold. Also note that I slowed down this simulation because it would otherwise have finished in a moment.
In the absence of enemies, the invader can ignore the rules of social distancing, occupying any space vacated in the forest without looking back at the neighbors. Once the invader is in most of the sites, the rules for distance are less onerous, but by then it is too late for other species.
Another half-serious thought about Connell-Jensen's theory is that in the war between trees and their enemies, humanity has definitely chosen one side. We would love to destroy all these insects, fungi and other tree-killing pests if we knew how to do it. Everyone would be happy to bring back the elms and chestnuts, and save the eastern hemlock before it's too late. On this issue, I share the beliefs of the hippies hugging trees. But if Connell and Jensen are right, and their theory applies to temperate forests, then killing all enemies will actually cause a devastating collapse of tree diversity. Without pressure from pests, competitive exclusion will take over, leaving forests of the same tree species everywhere.
The diversity of species in forests is now comparable to the diversity of theories in biology. In the article, I examined three ideas - the formation of ecological niches, neutral drift and social distancing; it seems that in the minds of ecologists they all coexist. And why not? Each of the theories has been successful in the sense that it is able to overcome competitive suppression. Moreover, each theory makes clear predictions. In the case of niche formation, each species must have a unique limiting resource. Neutral drift generates unusual population dynamics in which species continually appear and disappear, despite the fact that the total number of species remains stable. Spatial anti-clustering follows from social distancing.
How do we choose the winner among these (and possibly other) theories? The scientific tradition says that the last word should remain with nature. We need to do experiments, or at least go out into the field and make systematic observations, and then compare those results with theoretical predictions.
There have been quite a few experimental tests of competitive crowding out already. For example, Thomas Park and his colleagues conducted a series of experiments that lasted for ten years with two closely related species of flour beetle. Ultimately, one species or the other always prevailed. In 1969, Francisco Ayala wrote of a similar experiment with fruit flies, in which he observed coexistence under conditions that were believed to make it impossible. His work sparked controversy, but ultimately the result did not lead to the abandonment of theory, but to the improvement of the mathematical description of what repression applies to.
Wouldn't it be wonderful to do such experiments with trees? Unfortunately, they are not easy to grow in vitro. And conducting research on many generations of organisms that have lived longer than we do is a difficult task. In the case of the flour beetle, Park had time to observe over a dozen years over 100 generations. In the case of trees, a similar experiment could take 10,000 years. But field biologists are a resourceful people, and I'm sure they'll come up with something. In the meantime, I would like to say a few more words about the theoretical, mathematical and computational approaches to this problem.
Ecology became a serious mathematical discipline in the 1920s, following the work of Alfred James Lotka and Vito Volterra. To explain their methodology and ideas, we can start with a familiar fact: organisms reproduce themselves, causing population growth. In mathematical terms, this observation turns into a differential equation:
dxdt=αx,
which states that the instantaneous rate of change in the population x proportional to yourself x- the more representatives of the species, the more of them will appear. Proportionality constantαcalled the own rate of reproduction; this coefficient is observed when nothing restricts the growth of the population and does not interfere with it. The equation has a solution giving usx as a function of t:
x(t)=x0eαt,
Where x0- the original population. This is a recipe for unlimited exponential growth (assuming thatαpositive). In the finite world, such growth cannot last forever, but this should not worry us now.
Let's introduce the second kindy, subject to the same type of growth law, but having its own reproduction rate β... Now we can wonder what happens if the two species interact in some way. Lotka and Volterra (working independently of each other) answered it with the following pair of equations:
dxdt=αx+γxydydt=βy+δxy
The solutions of this system are not as simple as solutions for unlimited growth of one kind. What happens depends on the signs and values of four constants:α, β, γ and δ... In all cases, we can takeα and βas positive, because otherwise the species will become extinct. This leaves four combinations forγ and δ, coefficients of cross-terms xy:
- (γ+,δ-): a predator-prey system in which x Is a predator; x gains an advantage and y suffers when the piece xy large (i.e. both species are abundant).
- (γ-,δ+): the "predator-prey" system, where y - a predator.
- (γ+,δ+): a symbiotic or mutualistic system in which the presence of each species benefits both the other and oneself.
- (γ-,δ-): competition in which each species slows down the growth of the other.
Any time t the state of a system of two types can be represented as a point on the plane x,y, the coordinates of which are the levels of the two populations. For some combinations of parametersα, β, γ, δthere is a point of stable equilibrium. When the system reaches this point, it remains a point, and returns to that point after any slight fluctuations. Other equilibria are unstable: the slightest deviation from the equilibrium points leads to a significant shift in population levels. And the really interesting cases don't have a fixed point; the state of the system describes a closed loop in the planex,y, constantly repeating the entire cycle of states. These cycles correspond to fluctuations in the levels of the two populations. Similar fluctuations have been observed in many predator-prey systems. In fact, it was the curiosity caused by the periodic increase and decrease in populations in Canada's fur and Adriatic fisheries that inspired Lotka and Volterra to work on this problem.
The 1960s and 70s brought new discoveries. Studies of equations very similar to the Lotka-Volterra system have revealed the phenomenon of "deterministic chaos", in which a point representing the state of the system moves along an extremely complex trajectory, but this wandering is not accidental. This discovery was followed by a lively debate about complexity and stability in ecosystems. Can you find chaos in natural populations? Is a community of many species and with many interrelationships more or less stable than a simpler one?
If you look at it as abstract mathematics, there is great beauty in these equations, but sometimes comparing mathematics with biology is difficult. For example, when the Lotka-Volterra equations were applied to species competing for resources, the resources themselves were not represented in the model. The mathematical structure describes something more like a predator-predator interaction - two species that eat each other.
Even the organisms themselves are represented in these models only ghostly. Differential equations are defined on the set of real numbers and give us levels or densities of populations, but not of individual plants or animals - those discrete objects that we count as whole numbers. The choice of the type of numbers is not so important for large populations, but it leads to oddities when the population falls, say to 0.001 - one milli tree. Using finite difference equations instead of differential equations avoids this problem, but the calculations become more confusing.
Another problem is that the equations are strictly deterministic. With the same input data, we will always get exactly the same results, even in a chaotic model. Determinism rules out the possibility of modeling something like neutral ecological drift. But even this problem can be solved using stochastic differential equations containing a source of noise or uncertainty. In this type of model, the responses are not numbers, but probability distributions. We don't recognize the populationx at the moment tand the probability P(x,t)in a distribution with a defined mean and variance. Another approach, called the Markov Chain Monte Carlo (MCMC), uses a source of randomness to obtain samples from such distributions. However, the MCMC method takes us into the world of computational, not mathematical models.
In general, computational techniques allow direct mapping between model elements and modeled entities. We can "lift the lid", look inside and find trees and resources, birth and death there. Such computational objects are not very flexible, but discrete and always finite. A population is not a number or a probability distribution, but a set of individual objects. I find these models intellectually less demanding. It takes inspiration and intuition to create a differential equation that conveys the dynamics of a biological system. Creating a program to implement several basic elements of forest life - a tree dies, another takes its place - is much easier.
The six small models included in this article mainly serve as visualizations; for the most part, their computational cost is in drawing colored dots on the screen. But the implementation of larger and more ambitious models is quite possible, an example of this was the work of Taal Levy and colleagues mentioned above.
However, while computational models are easier to create, they can be more difficult to interpret. If you run the model once and the view will die out in itX, then how much information can you get from this? Not really. In the next run, it may turn out thatX and Ycoexist. To draw correct conclusions, you need to collect statistics on a large number of runs, that is, the answer again takes the form of a probability distribution.
The concreteness and clarity of Monte Carlo models are usually a virtue, but they also have a dark side. If the differential equation model can be applied to any “large” population, then such a vague description may not be applicable in a computational context. The number needs to be given a name, let this choice be arbitrary. The size of my forest models (625 trees) was chosen for convenience only. On a larger scale, let's say100×100, we need to wait millions of steps of time before something interesting happens. Of course, the same problem arises with experiments in the laboratory and in the field.
Both types of model are always at risk of oversimplification. Sometimes important aspects are thrown out of the models. In the case of forest models, it bothers me that trees have no life history. One tree dies, and another immediately appears in its place, fully grown. Also, the models do not include pollination and seed dispersal, as well as rare events (hurricanes and fires) that can change the shape of an entire forest. Do we learn more if all of these aspects of forest life took place in equations or algorithms? Perhaps, but where is the line near which we need to stop?
My acquaintance with models in ecology began with John Maynard Smith's book Models in Ecology, published in 1974. I recently reread it, learning more than the first time. Maynard Smith points out the difference between simulations, which are useful for answering questions about specific problems or situations, and models, which are useful for testing theories. He recommends the following: "A good simulation should contain as much detail as possible, and a good model should contain as little detail as possible."
Additional reading
Ayala, FJ 1969. Experimental invalidation of the principle of competitive exclusion. Nature 224: 1076-1079.
Clark, James S. 2010. Individuals and the variation needed for high species diversity in forest trees. Science 327: 1129-1132.
Connell, JH 1971. On the role of natural enemies in preventing competitive exclusion in some marine animals and in rain forest trees. In Dynamics of Populations , PJ Den Boer and G. Gradwell, eds., Wageningen, pp. 298-312.
Gilpin, Michael E., and Keith E. Justice. 1972. Reinterpretation of the invalidation of the principle of competitive exclusion. Nature 236: 273-301.
Hardin, Garrett. 1960. The competitive exclusion principle.Science 131 (3409): 1292-1297. (A story of the first forty years of ecology. Someone needs to describe the next sixty.)
Hubbell, Stephen P. 2001. The Unified Neutral Theory of Biodiversity and Biogeography . Princeton, NJ: Princeton University Press.
Hutchinson, GE 1959. Homage to Santa Rosalia, or why are there so many kinds of animals? The American Naturalist 93: 145-159.
Janzen, Daniel H. 1970. Herbivores and the number of tree species in tropical forests. The American Naturalist 104 (940): 501-528.
Kricher, John. C. 1988. A Field Guide to Eastern Forests, North America. The Peterson Field Guide Series. Illustrated by Gordon Morrison. Boston: Houghton Mifflin.
Levi, Taal, Michael Barfield, Shane Barrantes, Christopher Sullivan, Robert D. Holt, and John Terborgh. 2019. Tropical forests can maintain hyperdiversity because of enemies. Proceedings of the National Academy of Sciences of the USA 116 (2): 581–586.
Levin, Simon A. 1970. Community equilibria and stability, and an extension of the competitive exclusion principle. The American Naturalist , 104 (939): 413-423.
MacArthur, RH, and EO Wilson. 1967. The Theory of Island Biogeography. Monographs in Population Biology. Princeton University Press, Princeton, NJ.
May, Robert M. 1973. Qualitative stability in model ecosystems. Ecology , 54 (3): 638-641.
Maynard Smith, J. 1974.Models in Ecology. Cambridge: Cambridge University Press.
Richards, Paul W. 1973. The tropical rain forest. Scientific American 229 (6): 58–67.
Schupp, Eugene W. 1992. The Janzen-Connell model for tropical tree diversity: population implications and the importance of spatial scale. The American Naturalist 140 (3): 526-530.
Strobeck, Curtis. 1973. N species competition. Ecology , 54 (3): 650-654.
Tilman, David. 2004. Niche tradeoffs, neutrality, and community structure: A stochastic theory of resource competition, invasion, and community assembly. Proceedings of the National Academy of Sciences of the USA 101 (30): 10854-10861.
Wallace, Alfred R. 1878. Tropical Nature, and Other Essays. London: Macmillan and Company.