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Paper text:
charles darwin explained the evolution of living organisms on the basis of the observation that only the species best adapted to their surroundings survive in a competitive environment .
this idea , paraphrased as `` survival of the fittest '' , has since become the cornerstone of evolutionary theories .
the idea has been refined over the years , and we now understand its genetic underpinnings .
hereditary transmission of genetic information is not perfect , and occasional errors in dna replication produce mutations of living organisms .
the mutated organism is in essentially the same environment as the original one , and both have to compete for the available resources .
the resources are limited , and so the organism that can use them more efficiently obtains an edge over its competitors .
the net result is that if the mutation improves the ability of the organism to survive , the mutated organism grows in number , otherwise it fades away . in this mechanism ,
the mutations are not conscious adaptations .
they occur randomly  as a matter of fact most of them fail  but once in a while they modify the organism in the right manner and improve the chances of its survival . if the intermediate steps are glossed over , the outcome looks like a solution to the optimisation problem , with the organisms adapting to the selection pressures exerted by the environment .
all this is fairly logical , and can be illustrated by many examples ( see for instance @xcite ) .
my aim here is to quantify this mechanism using a simple evolutionary model .
more specifically , i consider the situation where a number of species are dependent on a common physical resource , say food .
( this leaves out predator - prey conflicts where one species may become another s resource . )
when the resource is available in plenty , there is hardly any competition , and all species prosper .
the rate of growth of a species is then proportional to its population , and so all populations grow exponentially .
this typically happens in the early stage of evolution of a new species .
no resource is unlimited , however , even if it is continuously regenerated ( e.g. food availability ultimately depends on sunlight ) .
a time comes when competition for the resource is inevitable . in the later stage of evolution , therefore , one species can prosper only at the expense of another , and eventually populations stabilise .    such a situation can be described by a set of linear evolution equations .
( quad - ratic evolution equations are more appropriate in a predator - prey setting . )
let the index @xmath0 label a set of coexisting species in a given environment , and let @xmath1 denote their populations at time @xmath2 .
then the discrete time evolution of the populations can be written as a matrix equation ( generically @xmath3 is not symmetric ) , @xmath4 the fact that the next generation populations must arise from the present generation ones , even in case of mutations , makes this equation homogeneous .
the diagonal terms @xmath5 represent the individual rates of growth , while the off - diagonal terms @xmath6 represent interactions between species . as mentioned above ,
when the resource is available in plenty , the populations evolve according to : @xmath7 the interesting situation is the competitive stage , where the populations become interdependent due to mutual interactions and the nature of the interactions ( para - metrised by @xmath6 ) determines how they evolve .
to analyse the competitive stage , let us choose normalisations such that a unit population of any species consumes the same amount of resource , and the total resource available at any time is one .
then we have the conservation laws : @xmath8 the constraint on @xmath9 arises because the total population supported by a fixed resource is fixed . when applied to both sides of eq.(1 ) , it yields the constraint on @xmath10 as a consistency condition .
the continuity of evolution in time implies that @xmath11 the matrix @xmath12 , describing the change in populations , clearly shows that what is gained by some species is lost by some others .
obviously @xmath6 can take positive as well as negative values . as a matter of fact ,
situations of both positive and negative interactions occur routinely in biological systems ( e.g. catalysis and inhibition , symbiosis and parasitic behaviour , defence mechanisms and cancer , etc . ) .
situations of this type , where there is no net gain or loss , have been formally called zero sum games . in the conventional form of zero sum games , a multitude of strategies
are available to the players , and competition amongst the players leads them to choose a stable strategy .
this stable strategy turns out to be a `` mini - max solution '' , i.e. each player chooses the strategy that maximises his minimum gain @xcite .
no player can improve his performance by unilaterally departing from this stable strategy , and the explicit solution can be obtained by linear programming methods .
the model described by eqs.(1 ) and ( 3 ) is somewhat different .
first , in the evolutionary context , the strategies are not actively chosen but are created by random mutations . also competition for the resource
is guaranteed , because mutations produce closely related species in the same environment .
it is then of importance to study the dynamics of the system to understand how the incorrect strategies are eliminated and the correct ones are selected .
that requires not just the evaluation of the stable solution but also the manner in which it is approached .
second , the quantitative gains and losses of various species depend not only on the choice of strategy but also on their current populations . in a sense
, the more populous species gets more chances to grab the resources , and that alters the final outcome of the competition .
let us first look at the simpler situation when @xmath13 .
eqs.(1 ) and ( 3 ) then describe stochastic evolution of probability distributions , which has been extensively studied ( see for instance @xcite ) . the linear evolution is a markov chain , which in the generic case is ergodic and converges to a unique stationary distribution .
( if the matrix @xmath3 has a block - diagonal structure , then each block can be studied independently . ) to see this , consider the asymptotic behaviour of @xmath14 as @xmath15 . at every iteration
, @xmath16 is the weighted average of @xmath17 @xmath18 with weights @xmath19 .
the process of repeated averaging converges to the fixed point , @xmath20 with all rows identical , @xmath21 has only one non - vanishing eigenvalue , equal to one .
it follows that @xmath3 has the leading eigenvalue @xmath22 corresponding to the stationary distribution , and remaining eigenvalues @xmath23 corresponding to transient distributions .
the left and right eigenvectors for the leading eigenvalue are , @xmath24 the left and right eigenvectors satisfy the orthogonality relation , @xmath25 for @xmath26 .
all transient distributions are orthogonal to @xmath27 and necessarily contain negative components .
the next - to - leading eigenvalue @xmath28 provides the rate of convergence towards the stationary distribution , and can be obtained my maximising @xmath29 , with the vectors @xmath30 and @xmath31 orthogonal to @xmath32 and @xmath27 respectively , and @xmath33 .
the convergence is monotonic because @xmath34 , @xmath35 .
all these properties belong to altruistic evolution , since @xmath36 while @xmath37 . in the competitive case
, some of the @xmath38 , and the consequences need to be analysed .
the presence of both positive and negative contributions produces cancellations .
monotonic convergence is no longer automatic ; oscillations and instabilities may occur in stead .
an important restriction is imposed by the physical requirement that no population can become negative .
( note that there is no such requirement on the components of the individual eigenvectors of @xmath3 . )
the evolution must be modified whenever it drives some @xmath9 negative .
the correct procedure is to stop the evolution at the instance @xmath9 becomes zero , eliminate @xmath39 row and column from the matrix @xmath3 , and then continue evolution in the reduced dimensional space .
obviously , @xmath9 can be driven negative only if some @xmath6 is negative .
the reduction of dimensionality , therefore , decreases the number of negative @xmath10 and increases the stability of the system .
the inverse process , increasing dimensionality of the space , occurs when chance mutation creates a new species .
it is known that even with @xmath13 , eq.(1 ) can not be evolved backward in time _
indefinitely_. this is true in spite of the fact that generic stochastic matrices are positive definite and @xmath40 exists .
if backward evolution is attempted , some @xmath9 is driven negative at some stage , and beyond that point interpretation of @xmath9 as populations ( or probabilities ) is lost .
evolution further back in time is possible only by modifying @xmath3 .
this is easy to see because eigenvalues of @xmath40 are the reciprocals of those for @xmath3 , and with @xmath41 , the growing transient distributions drive some @xmath9 out of their physically allowed range @xmath42 $ ] .
it is useful to compare this evolution problem to quantum algorithms .
both are constrained by conservation laws .
while eq.(3 ) preserves the linear norm of the vector , quantum evolution preserves the quadratic norm .
the most general evolution preserving the quadratic norm is described by orthogonal transformations for real variables , and by unitary transformations for complex variables .
orthogonal transformations are generated by antisymmetric matrices ( generating matrices parametrise infinitesimal group transformations in the neighbourhood of identity ) , and they inevitably contain negative matrix elements  the negative elements are an automatic consequence of the underlying conservation laws .
( only orthogonal transformations without any negative matrix elements are simple permutation matrices . )
although complete understanding of quantum evolution requires use of complex numbers , classical language extended to include negative probabilities can explain unusual features of certain simple systems ( e.g. quantum correlations of two spins violating bell s inequalities ) @xcite .
recent developments in algorithms for quantum computation ( see for instance @xcite ) offer a hint of how evolution may change in presence of negative matrix elements .
classical algorithms based on boolean logic can be expressed in terms of permutation matrices .
quantum algorithms exploit two features to beat them , superposition of states ( which is a generic property of waves ) and quantum entanglement ( which is not relevant here ) .
superposition means letting multiple states be in the same place at the same time , and it can reduce the spatial degrees of freedom required for the algorithm exponentially . cleverly designed destructive interference amongst superposed states can reduce the the time required to execute the algorithm by eliminating unwanted states .
coexistence of species in biological evolution is not quite the same as superposition ; it allows simultaneous evolution of all species but without reducing the spatial degrees of freedom . but destructive interference is still an option available to reduce the evolution time .
the known quantum algorithms reduce the execution time compared to their classical counterparts , at least by a constant factor if not polynomially or exponentially .
it is important to note that biological evolution occurs over long time scales , and even a tiny change in the rate of growth  fraction of a percent  matters because that can translate into exponential changes in populations over a long time .
the lessons learnt from quantum computation thus suggest that negative values of @xmath10 may create destructive interference and help the species reach their asymptotic populations faster , i.e. competition should beat altruism in picking a winner amongst the contenders .
let us now explicitly analyse how evolution changes , when the range of @xmath6 is extended to include negative values .
clearly , extending the range of @xmath10 can not make the evolutionary process any less powerful .
the stationary eigenvalue of @xmath3 , @xmath43 , which follows just from the averaging properties as described above , remains unaffected .
but other eigenvalues and eigenvectors change , and modify evolution .
an example with two interacting species illustrates the possibilities . in this case , the solution of eq.(1 ) in terms of the ( unnormalised ) right eigenvectors of the evolution matrix is : @xmath44 @xmath45 @xmath46 are small , @xmath47 .
the evolutionary behaviour depends on their signs : + ( a ) @xmath48 : the asymptotic population is proportional to @xmath32 . @xmath49 and
the transient part proportional to @xmath50 fades away .
the two species coexist , and their population ratio is stable against small perturbations . such a behaviour does not occur in quantum algorithms .
+ ( b ) @xmath51 : both @xmath32 and @xmath50 contain negative components , and so the non - negative population vector has to be a mixture of the two .
in course of evolution , the population @xmath9 with @xmath38 is monotonically driven to zero .
thereafter evolution has to continue in the reduced dimensional space .
+ ( c ) @xmath52 : the eigenvalue @xmath53 , and dominance of @xmath50 drives one of the @xmath9 to zero . the relative size of initial populations ( i.e. comparison of @xmath54 vs. @xmath55 ) determines which @xmath9 is driven to zero .
afterwards evolution has to continue in the reduced dimensional space .
such a dependence on the initial populations is quite distinct from the `` mini - max analysis '' .
these results for two interacting species have a topological interpretation .
the three regimes can be looked upon as minimisation of a function over an interval , when it ( a ) has a single minimum , ( b ) is monotonic , and ( c ) has a single maximum .
the relation between linear evolution and gradient of a quadratic form is generic , and the features exhibited in the above example can be expected to generalise to more complicated multi - species systems . specifically , whenever some @xmath38 : + ( 1 ) the non - stationary eigenvectors play an important part in evolution .
+ ( 2 ) having any @xmath38 makes the stationary eigenvector unstable , and drives the system towards reduced dimensionality .
+ ( 3 ) evolution one by one eliminates species @xmath9 with some @xmath56 , till the reduced dimensional system no longer has any @xmath56 .
+ ( 4 ) the elimination of species takes place in finite time , which depends on initial populations and @xmath10 , but is roughly @xmath57 .
this is in contrast to exponentially decaying tails of transient parts in an altruistic evolution .
+ ( 5 ) the surviving population is given not by the stationary eigenvector of the original system , but by the stationary eigenvector of the reduced dimensional system .
the simple model presented above provides a quantification of features anticipated in darwinian evolution .
the crucial ingredient has been the limited availability of a resource leading to destructive interference .
chance mutation may introduce a species that snatches away the resource from another one .
this always produces an instability , which eliminates the weaker species in a finite time .
the stronger survivors are stable until the next mutation instability .
it is worthwhile to observe that the elementary components of biological systems are so simple and cheap that they can be produced in large numbers even with limited resources . as a result ,
biological systems often exhibit wastefulness , e.g. millions of eggs and pollen grains are produced when a few would have sufficed to propagate the species in a secure environment .
such an overkill actually strengthens the competition and enforces survival of the fittest .    to summarise ,
the analysis presented in this article is straightforward , and points out certain similarities between competitive evolution , zero - sum games and quantum computation .
the latter two possess a sound mathematical framework , and so even a simple analogy with them can help us understand better the behaviour of highly complex biological systems .
the analogies are not perfect , however , and the effect of the differences has to be incorporated properly in the results .
my conclusion is that limited availability of resources leads to competition , which eliminates the weaker species in a finite time .
the arguments presented here can also be applied to other competitive situations , e.g. economic and social interactions .
competition for available resources is natural amongst coexisting species , and the fittest contenders dominate over the rest in evolution . 
 the dynamics of this selection is studied using a simple linear model . 
 it has similarities to features of quantum computation , in particular conservation laws leading to destructive interference . 
 compared to an altruistic scenario , competition introduces instability and eliminates the weaker species in a finite time .