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https://www.cell.com/trends/ecology-evolution/fulltext/S0169-5347(17)30325-7

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The biogeography of islands is a fascinating topic that extends to more than just island locations. The 'island' part of 'island biogeography' can refer to actual islands as well as isolated areas such as mountaintops, valleys, and other secluded spaces of unique biodiversity. These areas, surrounded by unsuitable habitat, often are very high in variety of species of flora and fauna that thrive there.

Island biogeography, or insular biogeography, is the study of places in the world that have different biogeographic and biodiversity levels than the areas that surround them. This field of study was created in the 1960s by two ecologists named Robert MacArthur and E.O. Wilson, who theorized what a newly created island might look like in terms of biodiversity.

Specific regions of island biogeography come about in a few ways. They can be areas that split off from a mainland continent many hundreds of thousands of years ago, or areas that limit geographic emigration because of the environment, distance to the next habitable region, and rate of extinction. Insular environments are created by the fact that few species can spread to other locations and other species have a hard time getting to the insular location or surviving once they are there.

Size is also a major factor when it comes to promoting an isolated biogeographical region. A smaller island environment will minimize the amount of species that can survive there, while a large environment will have more species that can live and thrive there. This is also known as the 'target effect;' the chances of a species immigrating and finding a larger island ecology zone is greater than the chances of them finding and surviving life on a smaller island area. Size also predicts rate of extinction; smaller isolated zones allow for faster species extinction while on larger isolated areas the opposite is true. After many years the island zone will achieve a balance of biodiversity and remain at a stable level of immigration and extinction.

Factors that influence the creation and longevity of island biogeography include the degree that the zone is isolated, or the distance between the next nearest landform and continental mainland; how long the area has been isolated from other regions around it and how close it is to achieving equilibrium (or how long it has maintained equilibrium); size; climate; baseline ecological makeup; current ecological makeup; location relative to wind and ocean current patterns; the chances of accidental species coming in contact with the area; and, finally, any human activity in the region that might affect the zone directly or indirectly.

Scientific studies have been done in the recent past, and many scientists have studied the fossil records of locations around the world to further the theory of island biogeography. More and more areas around the world are being discovered around the world, and new areas of island biodiversity are being created with the expansion of human settlement across every corner of the globe (for better or for worse).

Research in the Florida Keys involving the repopulation of islands was conducted by E.O. Wilson and colleagues to study the theory of island biogeography. They found that islands closer to the North American mainland repopulated their arthropod species faster than islands further away, which was in line with the original theory. Studies regarding the ecological implications of the island biogeography theory have also been conducted in regions around the world.

There are many implications for the scientific community that stem from studies of island biogeography. Implications for national parks, the future of plant and animal conservation, and the design of wildlife corridors are all affected by the results of studies regarding island biogeography. The study of island biogeography can influence the size of national parks (to improve biodiversity and mitigate the effects of immigration and extinction), affect where wildlife corridors are placed, where, and how big or small they are; and can help researchers pinpoint areas where plants and animals might be in danger of extinction or are simply conforming to their evolutionary and environmental journey.

The study of island biogeography might seem simple in theory, but its overarching implications can be understood and used for ecological conservation throughout the world. The continued study of isolated zones is needed, as is the continued understanding of how the evolutionary past has influenced isolated area of biodiversity today.
References
Wikipedia. Island Biogeography. Web 27 October 2014. http://en.wikipedia.org/wiki/Island_biogeography

Island Biogeography. 2014. Web 27 October 2014. http://www.islandbiogeography.org/

Dobkin, David S., Ehrlich, Paul R., Wheye, Darryl. Island Biogeography. Stanford University. 1988. Web 27 October 2014. https://web.stanford.edu/group/stanfordbirds/text/essays/Island_Biogeography.html

Related Resources

Island Biogeography of the Great Basin: In the Great Basin, habitat islands are formed though the Basin and Range topography in which north-south trending mountain ranges (some peaks in excess of 10,000 feet) are separated by low and broad desert valleys (as low as 282 feet below sea level, in Death Valley).

How Space and Time Affect Conservation Biology: Time and space are two extremely important concepts that are central to formulating theories and models in biotic conservation.

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The biogeography of islands is a fascinating topic that extends to more than just island locations. The 'island' part of 'island biogeography' can refer to actual islands as well as isolated areas such as mountaintops, valleys, and other secluded spaces of unique biodiversity.

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Environmental variability and the evolution of life-history traits

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Environmental variability and the evolution of life-history traits

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" The Chemical Basis of Morphogenesis" is an article that the English mathematician Alan Turing wrote in 1952. It describes how patterns in nature, such as stripes and spirals, can arise naturally from a homogeneous, uniform state. The theory, which can be called a reaction-diffusion theory of morphogenesis, has become a basic model in theoretical biology. Such patterns have come to be known as Turing patterns. For example, it has been postulated that the protein VEGFC can form Turing patterns to govern the formation of lymphatic vessels in the zebrafish embryo.

Reaction-diffusion systems have attracted much interest as a prototype model for pattern formation. Patterns such as fronts, spirals, targets, hexagons, stripes and dissipative solitons are found in various types of reaction-diffusion systems in spite of large discrepancies e.g. in the local reaction terms. Such patterns have been dubbed " Turing patterns".

Reaction-diffusion processes form one class of explanation for the embryonic development of animal coats and skin pigmentation. Another reason for the interest in reaction-diffusion systems is that although they represent nonlinear partial differential equations, there are often possibilities for an analytical treatment.

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" The Chemical Basis of Morphogenesis" is an article that the English mathematician Alan Turing wrote in 1952. It describes how patterns in nature, such as stripes and spirals, can arise naturally from a homogeneous, uniform state. The theory, which can be called a reaction-diffusion theory of morphogenesis, has become a basic model in theoretical biology.

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The cradles, museums, and graves of biodiversity

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Understand behaviour at one level in terms of ensembles of units at lower levels

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Understand behaviour at one level in terms of ensembles of units at lower levels

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In mathematics, a random walk is a mathematical object, known as a stochastic or random process, that describes a path that consists of a succession of random steps on some mathematical space such as the integers.

An elementary example of a random walk is the random walk on the integer number line, , which starts at 0 and at each step moves +1 or −1 with equal probability. Other examples include the path traced by a molecule as it travels in a liquid or a gas (see Brownian motion), the search path of a foraging animal, the price of a fluctuating stock and the financial status of a gambler: all can be approximated by random walk models, even though they may not be truly random in reality.

As illustrated by those examples, random walks have applications to engineering and many scientific fields including ecology, psychology, computer science, physics, chemistry, biology, economics, and sociology. Random walks explain the observed behaviors of many processes in these fields, and thus serve as a fundamental model for the recorded stochastic activity. As a more mathematical application, the value of π can be approximated by the use of a random walk in an agent-based modeling environment. The term random walk was first introduced by Karl Pearson in 1905.

Various types of random walks are of interest, which can differ in several ways. The term itself most often refers to a special category of Markov chains, but many time-dependent processes are referred to as random walks, with a modifier indicating their specific properties. Random walks (Markov or not) can also take place on a variety of spaces: commonly studied ones include graphs, others on the integers or the real line, in the plane or higher-dimensional vector spaces, on curved surfaces or higher-dimensional Riemannian manifolds, and also on groups finite, finitely generated or Lie. The time parameter can also be manipulated. In the simplest context the walk is in discrete time, that is a sequence of random variables indexed by the natural numbers. However, it is also possible to define random walks which take their steps at random times, and in that case, the position has to be defined for all times t ∈ [0,+∞). Specific cases or limits of random walks include the Lévy flight and diffusion models such as Brownian motion.

Random walks are a fundamental topic in discussions of Markov processes. Their mathematical study has been extensive. Several properties, including dispersal distributions, first-passage or hitting times, encounter rates, recurrence or transience, have been introduced to quantify their behavior.

A popular random walk model is that of a random walk on a regular lattice, where at each step the location jumps to another site according to some probability distribution. In a simple random walk, the location can only jump to neighboring sites of the lattice, forming a lattice path. In simple symmetric random walk on a locally finite lattice, the probabilities of the location jumping to each one of its immediate neighbors are the same. The best-studied example is of random walk on the d-dimensional integer lattice (sometimes called the hypercubic lattice) .[4]

If the state space is limited to finite dimensions, the random walk model is called simple bordered symmetric random walk, and the transition probabilities depend on the location of the state because on margin and corner states the movement is limited.

An elementary example of a random walk is the random walk on the integer number line, , which starts at 0 and at each step moves +1 or −1 with equal probability.

This walk can be illustrated as follows. A marker is placed at zero on the number line, and a fair coin is flipped. If it lands on heads, the marker is moved one unit to the right. If it lands on tails, the marker is moved one unit to the left. After five flips, the marker could now be on -5, -3, -1, 1, 3, 5. With five flips, three heads and two tails, in any order, it will land on 1. There are 10 ways of landing on 1 (by flipping three heads and two tails), 10 ways of landing on −1 (by flipping three tails and two heads), 5 ways of landing on 3 (by flipping four heads and one tail), 5 ways of landing on −3 (by flipping four tails and one head), 1 way of landing on 5 (by flipping five heads), and 1 way of landing on −5 (by flipping five tails). See the figure below for an illustration of the possible outcomes of 5 flips.

To define this walk formally, take independent random variables , where each variable is either 1 or −1, with a 50% probability for either value, and set and The series is called the simple random walk on . This series (the sum of the sequence of −1s and 1s) gives the net distance walked, if each part of the walk is of length one. The expectation of is zero. That is, the mean of all coin flips approaches zero as the number of flips increases. This follows by the finite additivity property of expectation:

A similar calculation, using the independence of the random variables and the fact that , shows that:

This hints that , the expected translation distance after n steps, should be of the order of . In fact,[6]

This result shows that diffusion is ineffective for mixing because of the way the square root behaves for large .[citation needed ]

How many times will a random walk cross a boundary line if permitted to continue walking forever? A simple random walk on will cross every point an infinite number of times. This result has many names: the level-crossing phenomenon, recurrence or the gambler's ruin. The reason for the last name is as follows: a gambler with a finite amount of money will eventually lose when playing a fair game against a bank with an infinite amount of money. The gambler's money will perform a random walk, and it will reach zero at some point, and the game will be over.

If a and b are positive integers, then the expected number of steps until a one-dimensional simple random walk starting at 0 first hits b or − a is ab. The probability that this walk will hit b before − a is , which can be derived from the fact that simple random walk is a martingale.

Some of the results mentioned above can be derived from properties of Pascal's triangle. The number of different walks of n steps where each step is +1 or −1 is 2n. For the simple random walk, each of these walks is equally likely. In order for Sn to be equal to a number k it is necessary and sufficient that the number of +1 in the walk exceeds those of −1 by k. It follows +1 must appear ( n + k)/2 times among n steps of a walk, hence the number of walks which satisfy equals the number of ways of choosing ( n + k)/2 elements from an n element set,[7] denoted . For this to have meaning, it is necessary that n + k be an even number, which implies n and k are either both even or both odd. Therefore, the probability that is equal to . By representing entries of Pascal's triangle in terms of factorials and using Stirling's formula, one can obtain good estimates for these probabilities for large values of .

If space is confined to + for brevity, the number of ways in which a random walk will land on any given number having five flips can be shown as {0,5,0,4,0,1}.

This relation with Pascal's triangle is demonstrated for small values of n. At zero turns, the only possibility will be to remain at zero. However, at one turn, there is one chance of landing on −1 or one chance of landing on 1. At two turns, a marker at 1 could move to 2 or back to zero. A marker at −1, could move to −2 or back to zero. Therefore, there is one chance of landing on −2, two chances of landing on zero, and one chance of landing on 2.

The central limit theorem and the law of the iterated logarithm describe important aspects of the behavior of simple random walks on . In particular, the former entails that as n increases, the probabilities (proportional to the numbers in each row) approach a normal distribution.

As a direct generalization, one can consider random walks on crystal lattices (infinite-fold abelian covering graphs over finite graphs). Actually it is possible to establish the central limit theorem and large deviation theorem in this setting.

As a Markov chain

A one-dimensional random walk can also be looked at as a Markov chain whose state space is given by the integers For some number p satisfying , the transition probabilities (the probability Pi,j of moving from state i to state j) are given by

In higher dimensions, the set of randomly walked points has interesting geometric properties. In fact, one gets a discrete fractal, that is, a set which exhibits stochastic self-similarity on large scales. On small scales, one can observe "jaggedness" resulting from the grid on which the walk is performed. Two books of Lawler referenced below are a good source on this topic. The trajectory of a random walk is the collection of points visited, considered as a set with disregard to when the walk arrived at the point. In one dimension, the trajectory is simply all points between the minimum height and the maximum height the walk achieved (both are, on average, on the order of ).

To visualize the two-dimensional case, one can imagine a person walking randomly around a city. The city is effectively infinite and arranged in a square grid of sidewalks. At every intersection, the person randomly chooses one of the four possible routes (including the one originally travelled from). Formally, this is a random walk on the set of all points in the plane with integer coordinates.

Will the person ever get back to the original starting point of the walk? This is the 2-dimensional equivalent of the level-crossing problem discussed above. In 1921 George Pólya proved that the person almost surely would in a 2-dimensional random walk, but for 3 dimensions or higher, the probability of returning to the origin decreases as the number of dimensions increases. In 3 dimensions, the probability decreases to roughly 34%. The mathematician Shizuo Kakutani was known to refer to this result with the following quote: "A drunk man will find his way home, but a drunk bird may get lost forever".

Another variation of this question which was also asked by Pólya is: if two persons leave the same starting point, then will they ever meet again? It can be shown that the difference between their locations (two independent random walks) is also a simple random walk, so they almost surely meet again in a 2-dimensional walk, but for 3 dimensions and higher the probability decreases with the number of the dimensions. Paul Erdős and Samuel James Taylor also showed in 1960 that for dimensions less or equal than 4, two independent random walks starting from any two given points have infinitely many intersections almost surely, but for dimensions higher than 5, they almost surely intersect only finitely often.

The asymptotic function for a two-dimensional random walk as the number of steps increases is given by a Rayleigh distribution. The probability distribution is a function of the radius from the origin and the step length is constant for each step.

A Wiener process is a stochastic process with similar behavior to Brownian motion, the physical phenomenon of a minute particle diffusing in a fluid. (Sometimes the Wiener process is called "Brownian motion", although this is strictly speaking a confusion of a model with the phenomenon being modeled.)

A Wiener process is the scaling limit of random walk in dimension 1. This means that if you take a random walk with very small steps, you get an approximation to a Wiener process (and, less accurately, to Brownian motion). To be more precise, if the step size is ε, one needs to take a walk of length L/ε ² to approximate a Wiener length of L. As the step size tends to 0 (and the number of steps increases proportionally), random walk converges to a Wiener process in an appropriate sense. Formally, if B is the space of all paths of length L with the maximum topology, and if M is the space of measure over B with the norm topology, then the convergence is in the space M. Similarly, a Wiener process in several dimensions is the scaling limit of random walk in the same number of dimensions.

A random walk is a discrete fractal (a function with integer dimensions; 1, 2, ...), but a Wiener process trajectory is a true fractal, and there is a connection between the two. For example, take a random walk until it hits a circle of radius r times the step length. The average number of steps it performs is r ².[citation needed ] This fact is the discrete version of the fact that a Wiener process walk is a fractal of Hausdorff dimension 2.[citation needed ]

In two dimensions, the average number of points the same random walk has on the boundary of its trajectory is r 4/3. This corresponds to the fact that the boundary of the trajectory of a Wiener process is a fractal of dimension 4/3, a fact predicted by Mandelbrot using simulations but proved only in 2000 by Lawler, Schramm and Werner.

A Wiener process enjoys many symmetries random walk does not. For example, a Wiener process walk is invariant to rotations, but the random walk is not, since the underlying grid is not (random walk is invariant to rotations by 90 degrees, but Wiener processes are invariant to rotations by, for example, 17 degrees too). This means that in many cases, problems on a random walk are easier to solve by translating them to a Wiener process, solving the problem there, and then translating back. On the other hand, some problems are easier to solve with random walks due to its discrete nature.

Random walk and Wiener process can be coupled, namely manifested on the same probability space in a dependent way that forces them to be quite close. The simplest such coupling is the Skorokhod embedding, but there exist more precise couplings, such as Komlós-Major-Tusnády approximation theorem.

The convergence of a random walk toward the Wiener process is controlled by the central limit theorem, and by Donsker's theorem. For a particle in a known fixed position at t = 0, the central limit theorem tells us that after a large number of independent steps in the random walk, the walker's position is distributed according to a normal distribution of total variance:

where t is the time elapsed since the start of the random walk, is the size of a step of the random walk, and is the time elapsed between two successive steps.

This corresponds to the Green's function of the diffusion equation that controls the Wiener process, which suggests that, after a large number of steps, the random walk converges toward a Wiener process.

In 3D, the variance corresponding to the Green's function of the diffusion equation is:

By equalizing this quantity with the variance associated to the position of the random walker, one obtains the equivalent diffusion coefficient to be considered for the asymptotic Wiener process toward which the random walk converges after a large number of steps:

Remark: the two expressions of the variance above correspond to the distribution associated to the vector that links the two ends of the random walk, in 3D. The variance associated to each component , or is only one third of this value (still in 3D).

A random walk having a step size that varies according to a normal distribution is used as a model for real-world time series data such as financial markets. The Black-Scholes formula for modeling option prices, for example, uses a Gaussian random walk as an underlying assumption.

Here, the step size is the inverse cumulative normal distribution where 0 ≤ z ≤ 1 is a uniformly distributed random number, and μ and σ are the mean and standard deviations of the normal distribution, respectively.

If μ is nonzero, the random walk will vary about a linear trend. If v s is the starting value of the random walk, the expected value after n steps will be v s + n μ.

For the special case where μ is equal to zero, after n steps, the translation distance's probability distribution is given by N(0, nσ ²), where N() is the notation for the normal distribution, n is the number of steps, and σ is from the inverse cumulative normal distribution as given above.

Proof: The Gaussian random walk can be thought of as the sum of a sequence of independent and identically distributed random variables, X i from the inverse cumulative normal distribution with mean equal zero and σ of the original inverse cumulative normal distribution:

but we have the distribution for the sum of two independent normally distributed random variables, Z = X + Y, is given by

(μ X + μ Y, σ ² X + σ ² Y) (see here).

(0, 2σ ²)

By induction, for n steps we have

Z ~ (0, nσ ²).

For steps distributed according to any distribution with zero mean and a finite variance (not necessarily just a normal distribution), the root mean square translation distance after n steps is

But for the Gaussian random walk, this is just the standard deviation of the translation distance's distribution after n steps. Hence, if μ is equal to zero, and since the root mean square(RMS) translation distance is one standard deviation, there is 68.27% probability that the RMS translation distance after n steps will fall between ± σ . Likewise, there is 50% probability that the translation distance after n steps will fall between ± 0.6745σ .

In disordered systems such as porous media and fractals may not be proportional to but to . The exponent is called the anomalous diffusion exponent and can be larger or smaller than 2.[17] Anomalous diffusion may also be expressed as σ r ² ~ Dt α where α is the anomaly parameter. Some diffusions in random environment are even proportional to a power of the logarithm of the time, see for example Sinai's walk or Brox diffusion.

The number of distinct sites visited by a single random walker has been studied extensively for square and cubic lattices and for fractals.[18][19] This quantity is useful for the analysis of problems of trapping and kinetic reactions. It is also related to the vibrational density of states,[20][21] diffusion reactions processes[22] and spread of populations in ecology.[23][24] The generalization of this problem to the number of distinct sites visited by random walkers, , has recently been studied for d-dimensional Euclidean lattices.[25] The number of distinct sites visited by N walkers is not simply related to the number of distinct sites visited by each walker.

The information rate of a Gaussian random walk with respect to the squared error distance, i.e. its quadratic rate distortion function, is given parametrically by

where . Therefore, it is impossible to encode using a binary code of less than bits and recover it with expected mean squared error less than . On the other hand, for any , there exists an large enough and a binary code of no more than distinct elements such that the expected mean squared error in recovering from this code is at most .

As mentioned the range of natural phenomena which have been subject to attempts at description by some flavour of random walks is considerable, in particular in physics and chemistry, materials science, biology and various other fields. The following are some specific applications of random walk:

In financial economics, the "random walk hypothesis" is used to model shares prices and other factors. Empirical studies found some deviations from this theoretical model, especially in short term and long term correlations. See share prices.

In population genetics, random walk describes the statistical properties of genetic drift

In physics, random walks are used as simplified models of physical Brownian motion and diffusion such as the random movement of molecules in liquids and gases. See for example diffusion-limited aggregation. Also in physics, random walks and some of the self interacting walks play a role in quantum field theory.

In mathematical ecology, random walks are used to describe individual animal movements, to empirically support processes of biodiffusion, and occasionally to model population dynamics.

In polymer physics, random walk describes an ideal chain. It is the simplest model to study polymers.

In other fields of mathematics, random walk is used to calculate solutions to Laplace's equation, to estimate the harmonic measure, and for various constructions in analysis and combinatorics.

In computer science, random walks are used to estimate the size of the Web.

In image segmentation, random walks are used to determine the labels (i.e., "object" or "background") to associate with each pixel. This algorithm is typically referred to as the random walker segmentation algorithm.

In brain research, random walks and reinforced random walks are used to model cascades of neuron firing in the brain.

In vision science, ocular drift tends to behave like a random walk. According to some authors, fixational eye movements in general are also well described by a random walk.

In psychology, random walks explain accurately the relation between the time needed to make a decision and the probability that a certain decision will be made.

Random walks can be used to sample from a state space which is unknown or very large, for example to pick a random page off the internet or, for research of working conditions, a random worker in a given country.[citation needed ]

When this last approach is used in computer science it is known as Markov chain Monte Carlo or MCMC for short. Often, sampling from some complicated state space also allows one to get a probabilistic estimate of the space's size. The estimate of the permanent of a large matrix of zeros and ones was the first major problem tackled using this approach.[citation needed ]

A number of types of stochastic processes have been considered that are similar to the pure random walks but where the simple structure is allowed to be more generalized. The pure structure can be characterized by the steps being defined by independent and identically distributed random variables.

A random walk of length k on a possibly infinite graph G with a root 0 is a stochastic process with random variables such that and is a vertex chosen uniformly at random from the neighbors of . Then the number is the probability that a random walk of length k starting at v ends at w. In particular, if G is a graph with root 0, is the probability that a -step random walk returns to 0.

Building on the analogy from the earlier section on higher dimensions, assume now that our city is no longer a perfect square grid. When our person reaches a certain junction, he picks between the variously available roads with equal probability. Thus, if the junction has seven exits the person will go to each one with probability one-seventh. This is a random walk on a graph. Will our person reach his home? It turns out that under rather mild conditions, the answer is still yes, but depending on the graph, the answer to the variant question 'Will two persons meet again?' may not be that they meet infinitely often almost surely.

An example of a case where the person will reach his home almost surely is when the lengths of all the blocks are between a and b (where a and b are any two finite positive numbers). Notice that we do not assume that the graph is planar, i.e. the city may contain tunnels and bridges. One way to prove this result is using the connection to electrical networks. Take a map of the city and place a one ohm resistor on every block. Now measure the "resistance between a point and infinity." In other words, choose some number R and take all the points in the electrical network with distance bigger than R from our point and wire them together. This is now a finite electrical network, and we may measure the resistance from our point to the wired points. Take R to infinity. The limit is called the resistance between a point and infinity. It turns out that the following is true (an elementary proof can be found in the book by Doyle and Snell):

Theorem: a graph is transient if and only if the resistance between a point and infinity is finite. It is not important which point is chosen if the graph is connected.

In other words, in a transient system, one only needs to overcome a finite resistance to get to infinity from any point. In a recurrent system, the resistance from any point to infinity is infinite.

This characterization of transience and recurrence is very useful, and specifically it allows us to analyze the case of a city drawn in the plane with the distances bounded.

A random walk on a graph is a very special case of a Markov chain. Unlike a general Markov chain, random walk on a graph enjoys a property called time symmetry or reversibility. Roughly speaking, this property, also called the principle of detailed balance, means that the probabilities to traverse a given path in one direction or the other have a very simple connection between them (if the graph is regular, they are just equal). This property has important consequences.

Starting in the 1980s, much research has gone into connecting properties of the graph to random walks. In addition to the electrical network connection described above, there are important connections to isoperimetric inequalities, see more here, functional inequalities such as Sobolev and Poincaré inequalities and properties of solutions of Laplace's equation. A significant portion of this research was focused on Cayley graphs of finitely generated groups. In many cases these discrete results carry over to, or are derived from manifolds and Lie groups.

In the context of random graphs, particularly that of the Erdős-Rényi model, analytical results to some properties of random walkers have been obtained. These include the distribution of first and last hitting times of the walker, where the first hitting time is given by the first time the walker steps into a previously visited site of the graph, and the last hitting time corresponds the first time the walker cannot perform an additional move without revisiting a previously visited site.

A good reference for random walk on graphs is the online book by Aldous and Fill. For groups see the book of Woess. If the transition kernel is itself random (based on an environment ) then the random walk is called a "random walk in random environment". When the law of the random walk includes the randomness of , the law is called the annealed law; on the other hand, if is seen as fixed, the law is called a quenched law. See the book of Hughes, the book of Revesz, or the lecture notes of Zeitouni.

We can think about choosing every possible edge with the same probability as maximizing uncertainty (entropy) locally. We could also do it globally - in maximal entropy random walk (MERW) we want all paths to be equally probable, or in other words: for every two vertexes, each path of given length is equally probable. This random walk has much stronger localization properties.

There are a number of interesting models of random paths in which each step depends on the past in a complicated manner. All are more complex for solving analytically than the usual random walk; still, the behavior of any model of a random walker is obtainable using computers. Examples include:

The self-avoiding walk of length n on Z^d is the random n-step path which starts at the origin, makes transitions only between adjacent sites in Z^d, never revisit a site, and is chosen uniformly among all such paths. In two dimensions, due to self-trapping, a typical self-avoiding walk is very short, while in higher dimension it grows beyond all bounds. This model has often been used in polymer physics (since the 1960s).

Long-range correlated time series are found in many biological, climatological and economic systems.

Random walk chosen to maximize entropy rate, has much stronger localization properties.

Random walks where the direction of movement at one time is correlated with the direction of movement at the next time. It is used to model animal movements.

Aldous, David; Fill, James Allen (2002). Reversible Markov Chains and Random Walks on Graphs. Archived from the original on 27 February 2019.

Ben-Avraham D.; Havlin S., Diffusion and Reactions in Fractals and Disordered Systems, Cambridge University Press, 2000.

Doyle, Peter G.; Snell, J. Laurie (1984). Random Walks and Electric Networks. Carus Mathematical Monographs. 22. Mathematical Association of America. arXiv:math.PR/0001057. ISBN 978-0-88385-024-4. MR 0920811.

Feller, William (1968), An Introduction to Probability Theory and its Applications (Volume 1). ISBN 0-471-25708-7

Hughes, Barry D. (1996), Random Walks and Random Environments, Oxford University Press. ISBN 0-19-853789-1

Norris, James (1998), Markov Chains, Cambridge University Press. ISBN 0-521-63396-6

Pólya G.(1921), "Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Strassennetz", Mathematische Annalen, 84(1-2):149-160, March 1921.

Révész, Pal (2013), Random Walk in Random and Non-random Environments (Third Edition), World Scientific Pub Co. ISBN 978-981-4447-50-8

Sunada, Toshikazu (2012). Topological Crystallography: With a View Towards Discrete Geometric Analysis. Surveys and Tutorials in the Applied Mathematical Sciences. 6. Springer. ISBN 978-4-431-54177-6.

Weiss G. Aspects and Applications of the Random Walk, North-Holland, 1994.

Woess, Wolfgang (2000), Random Walks on Infinite Graphs and Groups, Cambridge tracts in mathematics 138, Cambridge University Press. ISBN 0-521-55292-3

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In mathematics, a random walk is a mathematical object, known as a stochastic or random process, that describes a path that consists of a succession of random steps on some mathematical space such as the integers.

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Lorenz Equations

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If there was a simple meaning of the often-used scientific term "emergence," then 100-plus scientists wouldn't have spent four days presenting, debating and not infrequently disagreeing about what it was.

But as last month's organizers of the Earth-Life Science Institute's "Comparative Emergence" symposium in Tokyo frequently reminded the participants, those debates and disputes are perfectly fine and to be expected given the very long history and fungibility of the concept.

At the same time, ELSI leaders also clearly thought that the term can have resonance and importance in many domains of science, and that's why they wanted practitioners to be exposed more deeply to its meanings and powers.

Emergence is a concept commonly used in origins of life research, in complexity and artificial life science; less commonly in chemistry, biology, social and planetary sciences; and - originally - in philosophy. And in the 21 st century, it is making a significant comeback as a way to think about many phenomena and processes in the world.

So what is "emergence?" Most simply, it describes the ubiquitous and hugely varied mechanisms by which simple components in nature (or in the virtual or philosophical world) achieve more complexity, and in the process become greater than the sum of all those original parts.

The result is generally novel, often surprising, and sometimes most puzzling - especially since emergent phenomena involve self-organization by the more complex whole.

Think of a collection of ants or bees and how they join leaderless by the many thousands to make something - a beehive, an ant colony - that is entirely different from the individual creatures.

Think of the combination of hydrogen and oxygen gases which make liquid water. Think of the folding of proteins that makes genetic information transfer possible. Think of the processes by which bits of cosmic dust clump and clump and clump millions of times over and in time become a planetesimal or perhaps a planet. Think of how the firing of the billions of neurons in your brain results in consciousness.

All create complexity out of component parts, produce something irreducible from those original parts, and all have been resistant to a full explanation using the usual reductionist tools of the scientific world. This doesn't mean that something magical or divine is going on - rather, that either humans have not figured out what happens or that what happens is not comprehensible given our understanding of the laws of nature and physics.

Heady stuff, which is why the study and use of the concept of emergence has become increasingly widespread as a tool, or perhaps a pathway, to address complex problems.

Before going on, a little history is in order. Luis Campos, a historian of science at the University of New Mexico and the most recent Baruch S. Blumberg NASA/Library of Congress Chair in Astrobiology, described the concept of philosophical and scientific emergence - though not the word - as going back to the Greeks, at least.

The term "emergence" comes from the Latin verb emergo which means to arise, to rise up. Its meaning and importance have ebbed and flowed, with a major flowering in the United Kingdom in the 1920s.

It is seldom discussed, Campos said at the ELSI symposium, but the modern concepts of emergence flow to some extent from dialectics of Hegel, Marx and Engels, as well as the holism of South Africa's controversial leader and thinker, Jan Smuts.

Campos said the "emergence" of today is quite different from those iterations. His point, he said, is that major scientific approaches grow from the societal bedrock of their times - and today that means a willingness to consider possible limits to the strict scientific reductionism of the modern era.

In philosophical terms, and generally in the history of science, there are several basic forms of emergence.

The first involves surprising complexity that arises unexplained and remains so at a particular scale, though science may catch up some day and provide plausible explanations. The formation of stars and galaxies and black holes would all be considered emergent phenomena, but astrophysics is gradually learning many of the processes that allow immeasurable complexity to appear throughout the cosmos - all from the simplest of components.

The other involves complex phenomena that appear to be beyond the capacity of physical or natural laws to ever explain. The quantum world, for instance, includes the phenomenon called quantum entanglement - a physical sensation which occurs when

pairs or groups of particles cannot be described independently of the state of the other, even when the particles are separated by a large distance.

So while emergence may be, as information scientist Francis Heylighen of the Vrije Universiteit in Brussel argued at the ELSI symposium, "simple, common and natural," it also comes in innumerable forms that can seem mysterious or, as described in the quantum world, "spooky."

David Pines, a co-founder of the Santa Fe Institute, which specializes in complexity studies, illustrated the dimensions of the emergence debate when he wrote about it several years ago for the online publishing site "Medium."

"We live in an emergent universe," he wrote, "in which it is difficult, if not impossible, to identify any existing interesting scientific problem or study any social or economic behavior that is not emergent."

In the presence of claims like this, emergence has become a phenomenon where scientific consensus - or even agreement - can be difficult to achieve. Is it a trite "buzzword," as some argue? Or is it a profound and important pathway to understand underlying phenomena of the world that cannot be adequately described by reductionist, deterministic science?

Some of the modern pioneers in thinking about emergence come from the world of Artificial Life, or ALife. Using computer simulations, ALife researchers study essential properties of living systems such as evolution and adaptive behavior. Since the evolutionary clock cannot go backwards to see the what attributes are inevitable and what is more random, ALife analyzes these kinds of processes by simulating lifelike behaviors and patterns within computers.

As described at the ELSI symposium by University of Tokyo ALife and complex systems specialist Takashi Ikegami, decades of work in the ALife field have led to the conclusion that the pathways to life and consciousness are created by a cascade of emergent phenomena possessing the capability for "open-ended evolution."

And with computing power still increasing steadily, he said that the scale at which emergent properties can be identified and traced will similarly increase. As an example of how faster and larger computers can and will scale up ALife experiments and research, he told the story of the Rubik's cube and what became known as "God's algorithm."

For 30 years, mathematicians and others working to solve the famous cube puzzle concluded that the minimum number of moves needed to complete the task was 22. This was not based on the experience of some Rubik's cube fans, but rather of sustained mathmatical analysis.

But then in 2010 a teams of computer scientists and mathemeticians with access to Google's supercomputers found that the minimum number of moves for any of the 43 quadrillion Rubik's position was actually a very surprising 20. This result, Ikegami argues, is a reflection of the emergence of new technological capabilities in the last decade that are changing the world.

Some other classic artificial life simulations involve virtual birds or fish that are given some very simple rules to follow about how close individuals can approach each other and how they should steers in relation to other flock or schoolmates. Those simple instructions lead to the formation of virtual swarms and schools of enormous complexity that emerge from the individual-to-group transition.

An example of the unexpected behavior that can emerge: A flock splits to avoid an obstacle and then reunite once passed.

ALife is virtual, but emergence is everywhere in the natural world as well once you know what to look for. Bénard cells, for instance, are geometrically regular convection structures that spontaneously form in water or other thicker fluids when it is heated from below and/or cooled from above.

The cells are formed as the hotter fluid rises and the cooler sinks, a process that results in spontaneous self-organization into a regular pattern of cells. The cells would be considered to be emergent phenomena.

John Hernlund is a geophysicist and vice director ELSI, and was the lead science organizer for the conference.

"Bénard convection is a composite phenomenon that arises from the combination of simpler processes: thermal expansion, Archimedes principle, thermal conduction, and viscous resistance to shearing motion in a fluid," he said. "Nothing about those basic constituent processes alone would enable you to predict that their combination would yield the beautiful regular geometric patterns seen in Bénard cells, and this is why these are often used as an example of emergent behavior."

Astrophysicist Elizabeth Tasker, a professor and communicator for the Japanese Space Agency JAXA, chaired an early session on emergence which focused on how the universe and planets were formed. She said her field has generally not described that 13.7 billion year process that followed the Big Bang in terms of emergent phenomenon, but that over the week she gradually saw the usefulness and did so in particularly compelling terms.

"As an astrophysicist," she said. "I began to see the history of the universe as a manga, with examples of emergence forming the individual frames: matter strewn around the cosmos was drawn by gravity into structures that became galaxies, gas collapsed until fusion birthed a star, rocky boulders accreted and then began to melt and circulate to produce plate tectonics.

"Each manga frame represented the introduction of a new property of the universe, one that could not be exhibited by the individual pieces that had created it. A lone gas molecule could not show the spinning spiral of a galactic disc, nor begin to fuse elements within a star. Likewise, the rocky pieces that formed a planet could not start the circulation of plate tectonics by themselves. Neighboring interactions created a system that could spawn an entirely new process."

While emergence is most often understood in terms of physical systems, and then virtual systems that try to capture their patterns and laws, the concept also has a presence in the social and economic sciences as well.

Alex Penn of the University of Surrey, in fact, introduced emergence in the social spheres and humanities as emergent on a scale similar to what Daniel Pines said about science. "Basically, almost everything of significance that arises in the social world is emergent," she said.

Those phenomena ranged from the rise of different kinds of economic levels of development to political movements, cultures and down to the raising of children and residential segregation.

That housing segregation has been studied sufficiently that it has been determined that when a self-identifying group becomes less than one-third of the neighborhood population, they will begin to move to move out to be closer to those they identify with. Within a relatively short period of time, a mixed neighborhood will emerge into a segregated community via self-organization.

Other examples of emergence put forward from the social and cultural worlds include traffic jams, improv jazz performances and marriage. (A married person remains themselves, but also become part of a larger and more complex entity, from which novelty and surprise are sure to arise.)

Since the emergence symposium was taking place at an institute focused on how our planet formed and how life later appeared, it was only natural that emergence in that realm was a frequent topic. How a world without biology became one with biology is emergence writ large, with simplicity transforming to complexity innumerable times in innumerable ways.

The ELSI model of that process is often presented as a kind of hourglass, where evolution of all kinds - in minerals, in prebiotic polypeptides, in simple organisms - creates new systems that take over and expand until they reach a bottleneck that stops the advance. Only a newly evolved version of the entity can make it through to the next step.

Participants including evolutionary biologist Simonetta Gribaldo of the Institut Pasteur argued that there is good reason to conclude that these processes were playing out wherever they could on the early Earth, doubtless resulting in many versions of prebiotic and proto-life before the Last Universal Common Ancestor (LUCA). In other words, the path to and beyond LUCA was not only messy but had many failed efforts along the way.

It fell to geochemist and geomicrobiologist Karyn Rogers of Rensselaer Polytechnic Institute to put it all into the context of emergence.

Long active in origin of life science, she said that she has become very uncomfortable with that description of what the field is trying to understand. Like many others, she does not see the "origin of life" as a singular phenomenon and so she moved to "origins of life."

But "origins," too, didn't seem right because it seemed to focus too narrowly on the compounds and processes directly associated with life. She (and others) saw a larger picture.

Rogers said she doesn't think scientists can talk about how biology appeared and prospered without also talking about the the mineral and geochemical world from which it arose

Her conclusion, then, about how to describe what happened on prebiotic Earth so long ago:

"It's the emergence of life," she said at the end of the symposium, not the "origin".

"One of interesting things about emergence is that it's that smudgy place in between that we can't quite describe. We know it's necessary to go from an abiotic planet to a biotic planet, but can't quite really wrap our minds on it.

"It's not even a series of events, it's a series of events that led to life plus all the stuff around it as well that didn't lead to life...all the surrounding processes. I think you need all that for life to emerge."

She said she came into the symposium convinced that the "emergence of life" was the way to see the our biological beginnings, but that hearing about the many other understandings of emergence led her to think she didn't understand the concept and would have to revise her views. But by the end she was back to emergence, but with a difference.

Perhaps, she said, the group could consider having "emergence" and "smudginess" become synonyms for that cross-discipline arena of simple-to-significantly more-complex transitions they had been discussing all week.

The idea was embraced by many of the other scientists.
A Livestream of the symposium is available here.

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If there was a simple meaning of the often-used scientific term "emergence," then 100-plus scientists wouldn't have spent four days presenting, debating and not infrequently disagreeing about what it was. But as last month's organizers of the Earth-Life Science Institute's "Comparative Emergence"

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Background. Interacting species sometimes impose reciprocal natural selection on each other. This process of reciprocal evolutionary change driven by natural selection is called coevolution. It shapes interactions between pairs of species, small groups of species, and large webs of species. It also affects how species radiate into groups of descendent species. As the number of potentially coevolving species increases, so does the number of ways in which coevolution can shape an interaction.

In the early years of the study of coevolution, mathematical theory and empirical studies often analyzed coevolution of a single population of one species interacting with a single population of another species. Real species, though, are often collections of populations distributed across different environments. Species do not coevolve; local populations coevolve and what we perceived at the species level is the sum of those local coevolutionary processes.

The geographic mosaic theory of coevolution was developed as a framework for envisioning the coevolutionary process in real populations and species. It has been an attempt to incorporate the minimum components of population biology needed for an ecologically and evolutionarily realistic theory of coevolution and evolving interactions in general.

See the books listed on the Home page for detailed summarizes of geographic mosaic theory, coevolutionary biology in general, and, even more generally, the relentlessness of evolutionary change.

Assumptions: Geographic mosaic theory is based on several observations long known to biologists. These observation are taken as assumptions in the development of geographic mosaic theory:

1. Species are often collections of genetically distinct populations

2. Interacting species often differ in their geographic ranges

3. interactions among species differ among environments in their ecological outcomes

The hypothesis: From these assumptions, geographic mosaic theory argues that coevolution proceeds by natural selection acting on three sources of variation that affect interactions among species. These three sources of variation can be formally partitioned as genotype by genotype by environment interactions (GxGxE).

Geographic selection mosaics: The structure of natural selection on interactions differs among environments (e.g., high vs. low temperatures, high vs. low nutrient conditions; a surrounding species web that is species-rich vs. species poor). This variation occurs because genes are expressed in different ways in different environments (GxE interactions) and how one species affects the evolutionary fitness of another species depends upon the environment in which the interaction occurs.

For example, an interaction may be antagonistic in one environment and mutualistic in another environment; or it may be antagonistic in all environments but selection may favor different traits in different environments).

Coevolutionary hotspots: The i ntensity of reciprocal selection differs among environments. Interactions are subject to reciprocal selection only within some local communities. These coevolutionary hotspots are embedded in a broader matrix of coevolutionary coldspots, where local selection is non-reciprocal or where only one of the participants occurs.

For example, an interaction may be mutualistic or antagonistic in some environments (coevolutionary hotspots) but commensalistic in other environments (coevolutionary coldspots).

Trait remixing: The overall genetic structure of coevolving species continually changes through new mutations, genomic alterations, gene flow among populations, differential random genetic drift among populations, and extinction of local populations that differ in the combinations of coevolving traits they harbor. New genetic material on which natural selection can act can result from simple genetic mutations, chromosomal rearrangements, hybridization between populations, or whole genome duplications (polypoloidy). These processes contribute to the shifting geographic mosaic of coevolution by continually altering the spatial distributions of potentially coevolving genes and traits.

The combination of these processes continually changes the distribution of genotypes within any local population and the distribution of genotypes among populations.

NOTE: Some descriptions of geographic mosaic theory collapse this "trait remixing" part of geographic mosaic theory to gene flow. That is an incorrect characterization. The point of trait remixing is that through a combination of genetic, genomic, and ecological processes, the available distribution of coevolving traits on which natural selection can act continues to change over time within and among populations.

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Background. Interacting species sometimes impose reciprocal natural selection on each other. This process of reciprocal evolutionary change driven by natural selection is called coevolution. It shapes interactions between pairs of species, small groups of species, and large webs of species. It also affects how species radiate into groups of descendent species.

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Abstracting just enough to still be able to reproduce patterns

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Abstracting just enough to still be able to reproduce patterns

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Understanding patterns in terms of the processes that produce them is the essence of science

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Understanding patterns in terms of the processes that produce them is the essence of science

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MacArthur's 4 ingredients to biogeographical patterns

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MacArthur's 4 ingredients to biogeographical patterns

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- Structure of the environment
- Morphology of the species
- Economics of species behaviours
- Dynamics of population changes

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Anyone who has ever gone 'birding' in North America has likely come across a wave of warblers. You might have recounted how very different they all appear in coloration and pattern on the outside, but how they all seem to be doing exactly the same thing: flitting about looking for bugs and caterpillars. These warblers bring up one of the age-old questions in ecology: How do closely related species coexist despite sharing the same resources? In other words, why don't we see that one warbler outcompetes all others (Gause 1934)?

In the greatest detail, Macarthur documents the variation in habitat use, territoriality, nesting site, time of activity, foraging direction, style, and zone of the five warbler species. He shows that all species vary to some degree in how they capture resources, although there is some overlap in use. His best example, in my opinion, is his documentation of foraging behavior on coniferous trees where some warblers use the top and outer most branches to capture prey, where as others use the interior branches or the basal region. He also has some of the coolest figures ever to show these foraging patterns; see the top of this post for an example of an artistic adaptation of one of his original figures.

MacArthur's detailed observation of the variation in natural history of the five warblers has painted a picture of these species easily coexisting despite obvious overlaps in resource use. It is one of our greatest examples of niche partitioning and one of our most visual representations of the niche concept (see also Grinnell 1917). For this MacArthur's paper has been described as a foundational paper in ecology. I also feel it is foundational for biodiversity research because of the importance of niche partitioning in the study of the origin of species diversity as well as the importance of niche partitioning (complementarity) as a mechanism for how multiple species can be more efficient at performing ecosystem functions and services relative to sole species. For a more in-depth review of the significance of Macarthur's warblers see Kaspari (1998).

References
Gause, G. F. 1934. The struggle for existence. Hafner, London
Grinnell, J. 1917. The niche-relationship of the California thrasher. The Auk 34: 427-433
Kaspari, M. 1998. Knowing your warblers: Thoughts on the 50th anniversary of Macarthur. Bulletin of the Ecological Society of America. 448-458

January 23, 2013

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Resource partitioning: A warbler's eye view

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Anyone who has ever gone 'birding' in North America has likely come across a wave of warblers. You might have recounted how very different they all appear in coloration and pattern on the outside, but how they all seem to be doing exactly the same thing: flitting about looking for bugs and caterpillars.

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Upon witnessing that two species of barnacles seemed to have the adults of their species segregated into horizontal bands on a rocky intertidal shore but that the youth of the higher species could be found in the lower band with the other species, Connell set out to study the primary reason for that segregation. His theory was that the completion for space between the two species bore at least some responsibility for the separation. This theory was supported by the following studies; two species will either compete for resources with one species becoming more dominant in an area (Beauchamp and Ullyott 1932) (Kenny and Stevenson 1956), equal distribution of one species is due to that species competing primarily with itself (Holme 1950) (Clark and Evans 1954), and if two species with similar needs are living in the same area it is because they are not competing for resources (Lack 1954) (MacArthur 1958).

The method Connell used to test this theory was to map the locations of the barnacle species Chthamalus Stellatus, hereafter referred to at C.S., in the period of the year before what he hypothesized to be C.S.'s competitor Balanus Balanoides, hereafter referred to as B.B.. After mapping the locations of C.S. it was possible to control the height above or below mean tide level so that the effects of competition could be seen in environments where both C.S. and B.B. were primarily observed. One half of all clusters of C.S. growth were kept from being interfered with by B.B.. The growth and mortality rates in each case were recorded.

The results showed that C.S. was fully capable of growing to maturity at the levels on which B.B. was typically dominant, implying that the competition for space is what was preventing C.S.'s proliferation at the lower levels. In fact, while the hypothesis was that competition was at least somewhat responsible for the distribution of the two species, the study found that predation by carnivorous aquatic snails, battery by waves, and intraspecies crowding combined were not much more likely to be responsible for the death of an individual C.S. than was crowding of some sort by B.B..

I read this essay about two hours ago and have been idly trying to come up with an example of competitors coexisting. None come to mind except in the case of lions and tigers and bears coexisting in Oz. Given this, I can't find fault in Connell's reasoning. His experiment took into account as many variables as I could think of and the information gained fulfilled his hypothesis without assumption. In the case of this essay the limited resource in question was space, but it is easy to imagine that if the same experiment were performed on species competing for water or a nutrient source like meat the results would show that at least some of the reason for animal dispersion was due to interspecific competition. As a follow up to this study, I would be interested to see if there were more studies done on intraspecific competition and what its causes may be.

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The Influence of Interspecific Competition and Other Factors on the Distribution of the Barnacle Chthamalus Stellatus. Joseph Connell (1961)

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Upon witnessing that two species of barnacles seemed to have the adults of their species segregated into horizontal bands on a rocky intertidal shore but that the youth of the higher species could be found in the lower band with the other species, Connell set out to study the primary reason for that segregation.

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I recently received an old library copy of " Geographical Ecology: Patterns in the Distribution of Species" by Robert MacArthur (1972). It's the last book that MacArthur wrote before his early death to cancer. It is an ambitious book that connects repeated ecological patterns to mechanisms as broad as the earth's rotations (producing climate as we experience it) and as focused as organismal behaviour.

But honestly, the thing that has struck me most so far as I read is the timelessness and wisdom in MacArthur's introduction. Issues ranging from focusing on questions versus systems, the value of repeated patterns, complexity, and what generality really means, aren't at all new.
"To do science is to search for repeated patterns, not simply to accumulate facts, and to do the science of geographical ecology is to search for patterns of plant and animal life that can be put on a map. The person best equipped to do this is the naturalist who loves to note changes in bird life up a mountainside, or changes in plant life from mainland to island, or changes in butterflies from temperature to tropics. But not all naturalists want to do science; many take refuge in nature's complexity as a justification to oppose any search for patterns. This book is addressed to those who do wish to do science. Doing science is not such a barrier to feeling or such a dehumanizing influence as is often made out. It does not take the beauty from nature. The only rules of scientific method are honest observations and accurate logic. To be great it must also be guided by a judgment, almost an instinct, for what is worth studying. No one should feel that honest and accuracy guided by imagination have any power to take away nature's beauty.
Science should be general in its principles. A well-known ecologist remarked that any pattern visible in my birds but not in his Paramecium would not be interesting, because, I presume, he felt it would not be general. The theme running through this book is that the structure of the environment, the morphology of the species, the economics of species behaviour, and the dynamics of population changes are the four essential ingredients of all interesting biogeographic patterns. Any good generalization will be likely to build in all these ingredients, and a bird pattern would only be expected to look like that of a Paramecium if birds and Paramecium had the same morphology, economics, and dynamics, and found themselves in environments of the same structure."
--Robert MacArthur

It's interesting that an introduction written in 1972 is so relevant that it could have been written today. The pessimistic view is that ecology is just iterating through the same problems and solutions, or progress is slow. Or maybe classic books remain as classics because their authors understood and explored the issues at the core of the science and had the benefit of being there in the formative years. It's fun to see that when MacArthur thanks particularly four friends who influenced his work most, he means G. Evelyn Hutchinson, E.O. Wilson, Richard Levins, and Jared Diamond. I suppose any book influenced by the combination of all these scientists and written by MacArthur will always have something interesting to say.

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MacArthur's words still resonate 40 years on

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I recently received an old library copy of " Geographical Ecology: Patterns in the Distribution of Species" by Robert MacArthur (1972). It's the last book that MacArthur wrote before his early death to cancer. It is an ambitious book that connects repeated ecological patterns to mechanisms as broad as the earth's rotations (producing climate as we experience it) and as focused as organismal behaviour.