one aspect of the fascination with biology is the contrast between the overwhelming detail of nature and its order .
deus creavit , linnus disposuit , was linnus famous , not so humble summary of the two sides and who cared for which .
the friendly and modest country gentleman darwin had little of linnus 's self - image , but shared his obsession with the nitty - gritty of flora and fauna , since early childhood .
the idea of evolution must have come forth as gratifyingly simple to him and to any biologist immersed in the richness of nature . to a mathematician
, it invites further reduction into a more or less axiomatic theory , where everything can be derived from first principles .
it is successful albeit ( of course ! ) criticized for simplistic genetics as well as ecology .
a critique more on grounds of principle is that it treats populations as infinite and even as a continuous matter .
recently , mlard , champagnat , lambert and co - authors have published a sequel of seminal articles [ 24 ] , and others , reformulating classical adaptive dynamics   in terms of population - size - dependent birth - and - death processes , then passing to various limits by letting first the carrying capacity grow , so that populations become infinite , then mutational steps shrink to zero .
the present paper presents another such oversimplified model , going back to the ideas of , and indeed in a certain sense the simplest possible , still retaining indispensable properties of a population under evolution . in our view , they are as follows . 

( 1)any such population is finite and consists of reproducing individuals.(2)individuals are characterized by their traits.(3)together with population size ( of the own species as well as competitors ) traits may influence reproduction.(4)but they do not determine it : there is a variation in the reproductive behaviour between individuals of the same kind and , seemingly , in the same circumstances.(5)at reproduction , there is a probability for mutation , resulting in a child with a trait different from its mother.(6)the resulting population survives or not , depending upon chance and fitness .
individuals are characterized by their traits . together with population size ( of the own species as well as competitors ) traits may influence reproduction .
but they do not determine it : there is a variation in the reproductive behaviour between individuals of the same kind and , seemingly , in the same circumstances . at reproduction
, there is a probability for mutation , resulting in a child with a trait different from its mother .
still , passage to limits of infinite populations may help to uncover underlying structure , or exhibit its aspects more clearly .
but this is a tool only and it should occur explicitly , so as not to hide difficulties , avoid mistakes and also render investigations into the accuracy of approximations feasible .
there is no all - embracing continuity of properties from the finite to the infinite ; infinite populations need not die out , even though their finite counterparts necessarily do , if bounded and allowing variation in reproduction between individuals .
usually , in models , furthermore , not only is the carrying capacity of the habitat large ( tending to infinity ) but also mutation steps small ( infinitesimal ) , and two limits thus involved , the order between them possibly significant .
a deterministic description of individual life remains an illusion : there is randomness in life .
the simplest stochastic reproduction conceivable assumed here is asexual binary splitting . following the adaptive dynamics tradition , we disregard mating
thus , we assume that each individual either gets two children in the next generation or none .
the habitat concerned has a carrying capacity k , and if there is only one population present , of size n , the probability of an individual successfully splitting is taken to be k/(k + n ) . given the population size , individuals reproduce independently .
if there are several morphs present , the number n is replaced by a linear combination of the various population numbers and coefficients expressing the strength of competition .
again , a gaussian form of these coefficients will be assumed in the last section .
the carrying capacity can be different for different species ; k can depend upon the trait considered .
then we write k(t ) , t being a real number , the trait , which can be thought of as abstract , like a reproductivity measure , or in a very concrete manner , like the length of a beak . in the last section
, we shall work with a specific gaussian form of dependence of carrying capacity upon trait . at reproduction
, mutation may occur with some small probability which we will assume to be neither trait nor population - dependent ; with a probability  , a newborn with a mother with trait t turns out to be a mutant , then with a random trait normally distributed around t. time structure is stripped to its bare bones : only the successive ( non - overlapping ! ) generation size are counted , or alternatively everybody is supposed to have the same life - span of one season .
clearly , when compared with the many - faceted phenomenon of evolution , such a mathematical sketch is nothing but a caricature .
still , though reproduction and time structure are the simplest possible , they retain a fundamental affinity to reality . other properties are more arbitrary , like the form of dependence on the environment or the normal distribution of mutational steps or in the form of certain coefficients .
the former has some historical precedence , in the papers cited and earlier , but nothing else to speak for it than its ability to yield a reproductive behaviour changing with the relation of population size to carrying capacity and prevalence of competing species .
the assumption of normal distributions is also ad hoc though a central limit style argument from many small effects might have passed through the minds of the first proponents of this distribution for mutational steps .
( in other contexts , it has been argued that mutational steps should follow extreme value distributions , since only the most advantageous mutations result in viable children . ) but actually , we shall not use more of the normal distribution of mutational steps than the fact that deviations are of the order of the standard deviation , a fact that is much more generally true .
and in many regards , this specific model is just a representative , and many of its properties , like those building on second moment properties , large deviation theory , quasi - stationarity and early growth and ultimate extinction , extend to more general structures . in the next two sections , neither the mutation probability nor
any mutation step distribution will occur , though in section 3 the fact that we consider two populations , one of which called the resident starts from its carrying capacity , the other called the mutant population starting from one individual , certainly reflects having a mutation probability in mind , which is taken to be small enough for the first population to have grown to its carrying capacity before the mutation occurs . in section 4 , these mechanisms will be made explicit .
the difference from the series of papers by champagnat , ferrire , mlard and lambert lies in our taking discrete time , non - overlapping generations , i.e. galton  watson - style processes as the simple  pure structure for first study .
the model population is , thus , a binary , population - size - dependent galton  watson branching process .
it starts from a positive integer number z0 = z and is then recursively given by 
  the random variable nj is the number of children of individual i in generation n. these are independent and identically distributed , given the population size zn , or indeed the whole past population history , z0 , z1 ,  , zn .
we refer to the process value zn as population size ( at time or generation n ) .
since reproduction is identically distributed for all individuals in the same generation and the distribution , given zn , is the same for all generations n , we shall often delete the suffices , at least when not referring to several individuals in one context .
the trait dependence of the carrying capacity k is suppressed , since we consider only one trait , and no mutations yet .
k should be thought of as a large number , and later we shall let it tend to infinity .
it is important to grasp this single trait process not only for its own sake but also for the analysis of the complete model with mutation and several traits .
obviously , it behaves like a subcritical branching process whenever the population is greater than k , like a supercritical process for all population sizes smaller than k , and it is critical if the size is precisely k ( then necessarily an even integer ) .
since thus the population has a bounded expectation , it follows that the extinction probability is p(zn  0 , as n  ) = 1 ( cf .
108 ff . ] or ) . ultimately , it dies out . under the level
therefore , it tends to increase , while this is the case , and it is prone to reach large values ( around k ) before ultimate extinction .
in fact , we shall show that the time to extinction is very large , with an overwhelming probability .
the population will settle at a quasi - stationary regime , in the sense that it will fluctuate around the value k for a time period that is exponentially long in k , until it finally dies out .
for a  0 , write ta for the first time the population reaches , or passes , a , from below or above , depending upon the starting position . for short , let t = t0 be the time of extinction . what is the relation between these two random variables for large a ? indeed
, it is only rarely that the process dies out before reaching a level dk , if 0 < d < 1 is small or the starting number z large .
let 0 < d < 1 . then for any 1  z  dk , 
  in this , and elsewhere , the probability or expectation indexed by z , pz , ez means the probabilities of such a population starting from size z0 = z. no index either indicates z = 1 , or else some completely different probability measure .
such assertions are proved by comparison with suitably chosen ( not population size dependent ) simple galton  watson branching processes , about which much is known . in the present case ,
consider such a binary splitting process with the probability of begetting zero children being d/(d + 1 ) .
call it n. since x/(1 + x ) is an increasing function of x , any k < dk yields 
  hence , as long as population size stays below dk , the probability of producing no offspring is smaller than the corresponding probability pertaining to this classical galton  watson process . therefore ,
clearly the probability that our process becomes extinct by time n , without crossing dk , is smaller than the corresponding probability for the binary galton  watson process n. the latter must be smaller than the galton 
[ 8 , p. 113 ] ) that   is the smallest root of the quadratic equation 

 which is simply d. hence , 
  thus , with a positive probability , the population will not die out but reach sizes at the order of the carrying capacity . sooner or later
whatever the starting number z , carrying capacity k and time ( generation ) n , 

 and 
 ez[t ]  e. the elementary inequality 

 yields 
  hence , 

 and the upper bounds on the probabilities follow by induction and another elementary inequality , 0 < 1  u < e for 0 < u < 1
. the second assertion follows by summation : 
  a corresponding lower estimate can be deduced but it is irrelevantly small . instead , note that we already know that with a positive probability the process will move from any starting point up to the vicinity of the carrying capacity . by large deviation theory
, it will stay there a very long time until by a random perturbation it leaves the area ( if it were deterministic , it would get stuck ) and by chance becomes extinct .
this is why the upper exponential bound derived indeed gives a survival time of the right order ; persistence time is exponential in the carrying capacity .
introduce the density process xn = zn / k , and note that the offspring distributions are in fact functions of the density , 
  we have that 

 where the nj are the offspring numbers in the nth generation , explicitly indexed , and are indeed independent and distributed like twice a bernoulli random variable with the parameter p(xn ) = 1/(1 + xn ) . writing m(xn ) = 2p(xn ) for the offspring mean when the density is xn = zn / k , and centring the variables of the sum in the usual central limit fashion , we obtain 
  with 

 and 

 we can write this 
  clearly , e[n + 1 ] = 0 and 
  hence , a central limit theorem argument shows that the noise term in equation ( 2 ) is of order 1/k .
it follows that the process xn , which does of course depend upon k , converges ( in various senses ) , as k   , to the deterministic sequence { xn } , defined by an initial value x0 and the recurrence xn + 1 = f(xn ) .
this is a sort of law of large numbers ; a corresponding central limit theorem is available in .
however , this convergence also pinpoints the difference between stochastic and deterministic systems . indeed , if x0 = 0 , then for all
n , xn = 0 since 0 is a fixed point of f. in its awkward way , this is how the deterministic system tells us that extinction is certain . for other outcomes ,
we need z0/k  x0 > 0 , as k  . in other words , the population should become comparable to k in order to get attracted to the stable fixed point 1 .
however , by proposition 1 , we know that this will occur with a high probability , and after that xn will be approximated , for large k , by xn with a positive starting value .
the process will be near 1 for late times n and this means that the original population size process zn will be near k. the deterministic dynamics being globally attractive , zn will aim at moving towards k ; only the event of extinction can prevent this .
now , due to the random nature of the density process , exhibiting small but persistent perturbations from xn , ultimately there will be an exit from the levels [ d k , d'k ] , 0 < d < 1 < d ' , due to large deviations of random terms .
wentzell theory and was developed in , but the present special setting , the problem of exit from a domain , was solved in . under some assumptions , it was shown that if the deterministic system has a stable fixed point , then the perturbed system will stay in the domain of attraction of this point for an exponentially long time , e in the special case of our model , where summands are ( twice ) binomial random variables , there is a direct proof , using a large deviation result due to janson , stating c explicitly .
since , in the present context , our interest in this process is rather as a representative of a broad class of population models , proof and results specific for that model will be published elsewhere . if the process starts from a small number z of individuals , then the time it takes to reach the level dk ( if it does not die out ) is at least of an order approximated by the binary galton 
watson process with mean 2/(1 + d ) and at most of an order corresponding to such a process with mean reproduction 2k/(k + 1 ) , i.e. of order log dk , since for any galton 
watson process n we will have 
  five population developments with k = 50 . by exponential growth , the conclusion will follow that the time to reach d k is of the order log d k ; for details cf . .
close to k , our process is near critical and approach will be slower ; it is well known that a non - extinct critical branching process tends to grow linearly , and our process being on the supercritical side will tend to reach the carrying capacity quicker than in linear time ( 1  d)k .
actually , it can be proved that linear time is correct . in conclusion , if the population does not die out before reaching high levels , the time until extinction is of the order e consider the distribution of xn , given that xn > 0 , for fixed k. as n   , this converges to a proper distribution function called the quasi - stationary distribution .
when in this setting k   , all probability concentrates at the point 1 .
this was shown in   and is also contained in the more general setting of . indeed , the existence of quasi - stationary distributions is a consequence of the krein 
simulations exhibit first the quick growth and ensuing quasi - stabilization of five populations starting from one ancestor ( figure 1 ) and then the pseudo - stable distribution ( figure 2 ) .
after some , probably a long , time , the population will experience its first mutation giving rise to a new population . by then
, the original resident population will be in its quasi - stationary stage ; mutation probabilities are thus taken to be small .
the new , mutant population has just one member , its ancestor . the basis of adaptive dynamics can then be said to be furnished by the branching process fact that the new population either dies out or else embarks on exponential growth , in which case the old resident dies out , or the two populations will coexist for a time span that is exponential in the carrying capacity .
the process starts from a pair ( z0 , z0 ) of positive integers , the first component denoting the size of the resident and the second that of the mutant population , when the mutation appears .
thus , z0 is in the vicinity of the carrying capacity and z0 = 1 .
transitions from generation n to n + 1 are described by 

 where nk , k = 1 , 2 ,  ,
are independent given the preceding generation zn and satisfy 

 and 
  now , this process has two competing types , but it is not genuinely multi - type , in the sense that none of them can produce children with the other trait .
the interaction coefficient  is supposed to be positive but less than one , and the same in both directions .
this means that interaction is taken as symmetric , an assumption that is not at all necessary . indeed , taking them
it may be noted that though the specific form of dependence is certainly ad hoc , it has some historical precedence , going back at least to .
we assume that the resident population starts exactly at its carrying capacity , which is a1k .
then the mean reproduction of the mutant in a generation with z mutants present will be 
  it follows that the mutant is subcritical throughout , if a2 < a1 , and indeed dominated by a subcritical binary galton 
watson process with the splitting probability a2/(a2 + a1 ) . by , the probability that the maximum m of the latter reaches j is of the order 
  hence , for large carrying capacities , the mutant population will die out directly , never coming close to k with overwhelming probability .
histogram of a population size for the last 500 of 10,000 generations with k = 50 .
if instead 

 and k large enough , the mutant population will instead start supercritically , and either die out as above or else grow at a geometric rate for some time , much as in the preceding section .
it is illustrated in the following figure , which shows five runs of a population system with  = 0.7 , a1k = 40 and a2k = 70 . in three cases the resident prevailed , the invader dying out very quickly , and in two runs , the invader took over ( figure 3 ) .
we proceed to have a look at the qualitative behaviour of the process and the domains of super- and subcriticality ( figure 4 ) .
they will be given in terms of the ( x1 = z1/k , x2 = z2/k)-plane in which the process has a supercritical , critical and subcritical reproduction , respectively .
solving the inequalities for the mean offspring number greater than one yields that these regions are five competitive population evolutions , k = 100 ,  = 0.7 , a1 = 0.4 and a2 = 0.7 . in two of them ( red and orange lines ) , the invader takes over .
note that the boundary is where one of the populations is critical ; it consists of two segments of straight lines .
the point at which both processes are critical is 
  since 0 <  < 1 , there are three possibilities for this point 
  the remaining case x1 < 0 , x2 < 0 can be ruled out , since it implies both that  > a1/a2 and that  > a2/a1 , and hence  > 1 . as an example , we give the following picture for x1 < 0 , x2 < 0 made with 
  and where we take k as the scale unit of the graph . the domain a above the two lines corresponds to subcriticality of both populations , the left intermediate domain b is where the mutant is subcritical and the resident supercritical and the right intermediate domain c corresponds to supercriticality of { zn } and subcriticality of { zn } , whereas the domain below the two lines corresponds to their being supercritical .
this is precisely the situation where coexistence may be realized . like in the single trait setup , the process { ( zn , zn ) } scaled by k
, the density process , can be approximated by a two - dimensional dynamical system for large values of k. the latter has a fixed point at ( x1 , x2 ) and is attracted to it .
combine this with the fact that the density process is non - negative to see that the co - existence case is when the fixed point lies in the positive quadrant x1 > 0 , x2 > 0 and corresponds to the condition 
  like in the single trait case , the two - trait process must die out .
but the process being supercritical for small population sizes , both populations will settle to a quasi - stationary regime , in the sense that their sizes will fluctuate around the values k x1 and k x2 for an exponentially long time ( in k ) , provided the new trait gets established , in the sense that its size becomes comparable to k. thus , the probability of this happening is needed .
estimates from below and from above the probability of more or less direct extinction are established in the following proposition , pointing at a1/a2 as a rough indicator .
we omit the suffixing starting point ( which is ( a1 
 k , 1 ) ) .
the probability of the mutant dying out while the resident remains above the level dk , 0 < d < a1 , is at least 
  similarly , the probability of the mutant population dying out before it reaches the level b k , b < a2 , while the resident remains below the level u k , u > a1 , is 
  in this b , u can be so chosen that u + b < a2 .
proof  on the set { t < tu k }  { t < tb k }  { n < t } , zn
< u k and zn < b k. hence , the probability of the new trait to reproduce successfully is 
  provided u + b < a2 , the splitting probability is larger than one - half . selecting u in the range a1
< u < a2/ allows us to choose 0 < b < a2  u < 1 .
this leads to the upper bound . bounding the splitting probability from above on the set { t < td k } yields 
  and the lower bound for the extinction probability of the mutant .
we proceed to bounding the probability of the mutant dying out , while the resident remains close to the carrying capacity .
for z > d k , the mutant survival probability 

 satisfies 
  we note that for large k , we will have that 

 say . for i
 1 and j  d k , 

 since the expression in the middle decreases with i. hence , 
  thus , for any k 
. a similar argument can be used to find an exponential bound for mutant survival probabilities , given that the resident does not surpass high levels .
like the single trait process , considered in section 2 , the present process behaves as a deterministic dynamical system perturbed by small noise , which is , however , not to be forgotten .
write the density process xn = zn / k , xn = zn / k and note that the offspring distributions , as given in the beginning of this section , are again functions of the density . as before , 
  denote the offspring mean by m(x ) when the density is x , m = ( m1 , m2 ) , 

 and write 

  the deterministic dynamics is given by 

 and 
  it is also easy to see that the noise term when the system is at x is of order 1/k .
hence , we can write equations ( 9 ) and ( 10 ) as 

  the function f in equation ( 11 ) that generates this system has two fixed points 0 and 
  this is exactly the point at which both reproductions are critical .
the determinant  of partial derivatives shows that 0 is repelling , since  = 4 there . at the point x , the determinant of partial derivatives is 

 which allows deriving conditions for stability  < 1 of that fixed point in terms of the parameters  , a1 , a2 .
indeed , the deterministic system has four equilibrium points ( 0 , 0 ) , ( a1 , 0 ) , ( 0 , a2 ) and ( x1 , x2 ) . as pointed out
the fixed point corresponding to ( x1 , x2 ) lies in the positive quadrant only if  is less than the maximum of a2/a1 and a1/a2 . in that case
if  > a1 , the point ( a1 , 0 ) is stable , so the second morph goes extinct . if  > a1/a2 , ( 0 , a2 ) is stable , and the former resident goes extinct
having investigated one lone species and the joint behaviour of a resident and a mutant , we return to the well - established special   model sketched in section 1 , determined by gaussian functions .
it is controlled by five parameters : k ,  , 1 , 2 and 3 , with the following interpretation .
that 1 <  < 1 , 

 and 
  a morph with trait t   has the carrying capacity 

 explaining the role of 1 , in what sense it describes how carrying capacity varies for different traits , with the maximum k(0 ) = k for the trait value t = 0 . in a polymorphic population , the competition between two morphs t1 and t2 is quantified by the coefficient 

 depending only on the difference |t1  t2| and reaching its maximum value (t , t ) = 1 for individuals with the same trait value
. mutation in a monomorphic population occurs at a rate of the order  mutations per individual and time unit . from the preceding analysis
, the mutation rate should be taken to satisfy e    1/k , the constant c defined in equation ( 3 ) , so as to avoid repeated mutations during the time while one mutant establishes itself , while guaranteeing mutations during the residence . in adaptive dynamics ,
this would be formulated as conditions on mutations not occurring in ecological time spans , but ( almost ) certainly in an evolutionary time scale .
given a mutation , the new trait value t2 then has the normal conditional density function 

 implying symmetry in the step distribution and that larger changes in trait value are less probable .
the process starts from a finite number of individuals of a finite number of different traits .
traits in the various generations are denoted by vectors ( t1 , t2 ,  , tp ) , where p can vary from generation to generation .
denote the number of individuals with trait t in the nth generation by zn(t ) . given the vector ( zn(t1 ) , zn(t2 ) ,  , zn(tp ) ) , individuals of the nth generation reproduce independently , splitting into two with the probability 

 otherwise begetting no child .
each child is a mutant with probability  , the mutant 's trait being normally distributed around its mother 's , and with the variance . this is an informal ( but precise ! ) description of a discrete - time stochastic process , whose states are counting measures on . if it starts from one individual at say t1 , it will develop according to section 2 ( with k(t1 ) replacing k ) , until the first mutation occurs . then we find ourselves in the circumstances of section 3 with  as defined by equation ( 15 ) , a1 = exp(t1/(21 ) ) , and a2 = exp(t2/(21 ) ) , k = k(0 ) , according to equation ( 14 ) . after some algebra from the inequality a2 > a1 ( equation ( 4 ) ) , we arrive at a simple condition for the mutants possibly being able to establish themselves : if t2 < t1 , then also t2 < t1 .
if t1 > 0 , the second condition is empty , whereas if t1 < 0 , t2 must be at a considerable distance left of t1 , which is not feasible due to the smallness of . we conclude that residents with traits on the right half - axis can only establish mutants to the left of them , and vice versa .
indeed , the situation is summarized by the condition 

 being necessary for establishment , whatever be the position of t1 , and equivalent to t2/t1  (  , 1 )  ( 1/ ,  ) .
this means that in the monomorphic phase , the trait value will move towards zero until it reaches a -neighbourhood of zero .
then the trait value may jump in both directions and moreover the first evolutionary branching becomes plausible .
indeed , the reverse invasion condition is 

 equivalent to t2/t1  (  ,  )  ( 1 ,  ) .
when both conditions are fulfiled , the pair of trait values ( t1 , t2 ) may coexist . in a slightly more precise wording , we start the process from a monomorphic population , say with trait t0 = t > 0 .
the population will either die out or ( quasi-)establish itself at its carrying capacity . in the latter case ,
sooner or later at a random time v1 , a successful mutant will replace it , and the trait will have moved to tv1 , where it will stay until the next successful mutation v2 , until finally it would have come so close to zero , so as to render branching possible , the trait being of the order of . then { tn } will take the value of two traits .
thus , we obtain a stochastic process with values in n  , about which many interesting questions can be posed . what is the time to the first successful mutation ?
when will the first branching occur , or rather , what is the distribution of the waiting time until a monomorphic population turns dimorphic , the distribution to the next change in number of morphs , etc .
how can the stochastic movement { tn } in trait space be described ? and how the degree of polymorphism ( the development of the number of coexisting species ) ? what limiting behaviours can be discerned ,
when k   and   0 ? for birth - and - death processes ,
many of these matters have , as pointed out , been discussed by mlard , champagnat et al .
as one of the referees pointed out , the basic method used here , domination by not - size - dependent branching processes during time periods when populations sizes satisfy certain inequalities , invites usage also in more realistic , age - dependent or general branching processes .
vatutin was supported by the ras programme  mathematical control theory and intas 03 - 51 - 5018 .
 first a population model with one single type of individuals is considered . 
 individuals reproduce asexually by splitting into two , with a population - size - dependent probability . 
 population extinction , growth and persistence are studied . 
 subsequently the results are extended to such a population with two competing morphs and are applied to a simple model , where morphs arise through mutation . 
 the movement in the trait space of a monomorphic population and its possible branching into polymorphism are discussed . 
 this is a first report . 
 it purports to display the basic conceptual structure of a simple exact probabilistic formulation of adaptive dynamics . 