while advances in free trade and globalization increase the movement and accelerate the accumulation of invasive species ( lockwood et al .
2005 ) , it is still unclear how introduced populations can successfully establish . as elton ( 1958 ) pointed out , for every successful invasion that occurs ,  there are enormously more invasions that never happen , or fail quite soon or even after a good many years ( page 109 ) .
this modern biological paradox can not readily be reconciled , especially in the characteristic case where the founder population is small , as  such populations are definitely in a precarious position ( mayr 1965 ; page 42 ) .
introductions of populations at low density and/or small size are often faced with inverse density dependent effects , attributed to demographic stochasticity or reduced cooperative interactions ( courchamp et al .
allee ( 1931 ) first proposed that under these conditions , populations may suffer a decrease in the per - capita rate of increase , from here on referred to as the allee effect . upon arrival in a novel environment , individuals need to overcome a series of challenges in order to reduce the population 's risk of extinction .
the time period in which this occurs is generally considered the initial establishment phase , and is thought to be a common feature and general pattern of invasion and the process of growth and expansion ( shigesada and kawasaki 1997 ; sakai et al .
the occurrence of a lag phase that precedes a noticeable increase in population growth and density can result from ecological and/or evolutionary phenomena ( sakai et al .
small populations that undergo logistic growth slowly increase through the initial phase of the exponential curve , leading to the perception of a time lag .
where this time lag is more pronounced , populations may be recovering from inverse density dependent effects ( i.e. , allee effects ) .
individuals may suffer a reduction in fitness at low densities for many reasons ( reviewed by courchamp et al .
even when the initial population is large , rapid dispersal required for expansion could be suicidal as the population density decreases , thereby enhancing inverse density dependent effects ( lewis and kareiva 1993 ; drake et al .
component allee effect where the population size and density affects the mean overall fitness or some component of individual fitness , respectively ( stephens et al . 1999 ; gascoigne et al .
it is often difficult to decipher the exact mechanism that manifests allee effects ( and it is not always the case that component allee effects lead to demographic allee effects ) .
nevertheless , we focus our attention on population level demographic allee effects with the underlying assumption that a component allee effect led to the demographic allee effect . essentially , it is the case that introduced individuals may be maladapted to small population sizes where their survival and reproductive ability are significantly impacted , and these impacts on individual fitness combine to produce an overall decrease in abundance ( i.e. , demographic allee effects ) . as allee effects impact individual fitness , the underlying traits that influence these effects (
i.e. , component allee effects ) may be subject to natural selection ( courchamp et al . 2008 ) . whereas propagule pressure is an emerging explanation for the establishment of invasive species ( lockwood et al . 2005 ) , it relies on an obvious relationship between the number and size of introduction events and the probability of success , as safety in numbers helps combat allee effects and stochastic extinction . in the event that propagule size is not large enough to overcome inverse density dependent effects , a population
traits that may be responsible for reproductive success and survival at small population densities and sizes include mate finding cues ( pheromones and vocal / visual signals ) , dispersal / aggregation behavior , habitat preferences , mating synchronicity , and gamete morphology and performance ( see courchamp et al .
direct evidence for the evolution of these traits as functional adaptations to allee effects is limited , but we can infer an adaptive evolutionary origin of these traits from studies of sexual selection ( courchamp et al . 2008 ; gascoigne et al .
an evolutionary response to intensive selection pressure imposed by density dependent survival and reproduction relies on genetic variants for adaptive evolution .
according to neutral quantitative genetic theory , a loss of genetic variation is expected from population bottlenecks and founder effects ( nei et al .
however it is not neutral variation that matters , but rather evolvability depends on the variation relevant to selection .
maintenance or even increases of this ( additive ) genetic variation has been theoretically and empirically observed following a bottleneck or in small founder populations ( bryant et al .
1986 ; goodnight 1988 ; willis and orr 1993 ; cheverud and routman 1996 ; briggs and goldman 2006 ; turelli and barton 2006 ) .
additionally , many recent studies suggest that there is actually no significant reduction in genetic diversity in most successful invaders ( lee 2002 ; allendorf and lundquist 2003 ; roman and darling 2007 and references therein ) , and that evolution can occur on contemporary timescales ( reznick and ghalambor 2001 ; carroll et al . 2007 ; kinnison and hairston 2007 ) .
our purpose here is to explore the feasibility of small populations that may adaptively respond to overcome allee effects in order to establish , given any amount of genetic variation . in this paper
, we present evidence for the enhanced potential for growth and spread of a small introduced population of organisms faced with allee effects when adaptation occurs .
we model the growth and spread of this population according to a reaction - diffusion equation , and allow evolution to influence inverse density dependent effects through a genetic subsystem that provides the opportunity for successful invasion when otherwise ( under the current , ecological paradigm ) the population would go extinct .
the deterministic model that we explore in this paper broadly describes population dynamics with density - mediated growth ( i.e. , an allee effect ) and diffusive dispersal .
this type of demographic model has been used as a compact and tractable representation of invasion ( e.g. , skellam 1951 ; lewis and kareiva 1993 ) .
specifically , it has been applied to systems such as introductions of nonindigenous freshwater and marine species through ballast water discharge ( drake et al .
2005 ; drury et al . 2007 ) . using this approach , drake et al .
( 2005 ) report acceptable volumes of discharge for various organisms ( with differing reproductive rates ) for a range of invasion risk tolerances . here
, we consider populations that are introduced below the invasion risk threshold , but nonetheless succeed if evolutionary dynamics are considered in conjunction with the ecological system .
the growth and spread of an introduced population of organisms is represented by a reaction - diffusion equation described by ( lewis and kareiva 1993 ) : which determines the rate of change in the local population density relative to the carrying capacity [ u which denotes u(x , t ) ] over time , at a point in space .
this equation models the growth of the population ( the first term on the right hand side of eqn .
1 ) at a spatial location that is subject to an allee effect in addition to migration ( which depends on the second term on the rhs of eqn .
1 ) . the diffusion coefficient ( d ) scales the rate of population spread , in this case across a one - dimensional habitat x. the reproductive rate is regulated by r , and a ( which is a function of space and time , derived below ) is the local critical density or allee threshold that determines if population growth is positive or negative ( fig .
growth dynamics of the model population ( a ) , and the solution of eqn .
( 1 ) without diffusion ( b ) with reproductive rate , r = 0.6 , and allee threshold , a = 0.3 .
trajectories for population size ( b ) are given for initial densities from 0 to 1 in increments of 0.05 .
there are many variations of single - species models of population dynamics that exhibit allee effects ( see table 1 of boukal and berec 2002 ) , however the growth function of eqn .
the behavior of this verhulst ( 1838 ) logistic model modified to include a nonlinear cubic term ( based on the fitzhugh - nagumo equations ; fitzhugh 1960 ; nagumo et al .
1962 ) , is bistable , with equilibria at u = 0 ( extinction ) , u = a ( unstable threshold ) , and u = 1 ( carrying capacity ) .
figure 1a shows this behavior in terms of the growth of the population ( change in population density with respect to time ) versus the population density . at densities below the critical threshold ( a )
there is negative population growth declining to extinction ( from here on the population is considered extinct below a cutoff density of 0.0001 , as a declining population trajectory will only asymptotically approach zero in a deterministic model ; gomulkiewicz and holt 1995 ) ; otherwise the population will reach carrying capacity .
1b with the graph of the solution of the growth function ( population size versus time ) at various initial densities . when diffusion is added to this model of population growth ,
there are two critical elements that emerge based on the solution to the partial differential equation ( pde ) .
the first is the wave speed , which is determined by the allee threshold ( a ) . as we are considering a strong allee effect in this model ( i.e. , 0 < a < 1 , where the population below this threshold exhibits negative growth versus reduced positive growth from a weak allee effect )
, there exists a unique wave speed of the invasion front that is a result of being 
pushed from the inside out , as opposed to being  pulled by the leading edge ( lewis and kareiva 1993 )
. this velocity can be derived through the solution of the pde ( 1 ) :   ( lewis and kareiva 1993 ; murray 1993 ) .
this result suggests that in order for a wave to maintain a positive velocity of advance , the magnitude of the allee threshold ( a ) must be less than half of the maximum value of the population density relative to the carrying capacity .
in addition to the velocity of the wave front , the region occupied by the invading population must exceed a certain critical size for positive growth to occur ( skellam 1951 ; kierstead and slobodkin 1953 ) .
this phenomenon is clearly explained by okubo ( 1980 ) by noting that whereas reproduction takes place within a region or patch , diffusion takes place at the boundaries resulting in a loss of organisms , thus reducing the density within the patch .
this tradeoff in the ratio of inner region volume to outer surface area will either allow a population to grow or decline with an inverse relationship of diffusivity to rate of growth .
this relationship gives a minimum region within which reproduction can not compensate for loss due to diffusion , especially when allee effects influence population growth .
thus , lewis and kareiva ( 1993 ) derive a minimum size condition ( i.e. , the radius of the initial beachhead ) based on the wave speed that is required for the population to establish and radially expand .
we address this critical size threshold qualitatively , as the analytical solution ( i.e. , ; lewis and kareiva 1993 ) is for two - dimensional spread , while we work with a simpler one - dimensional model .
the minimum critical radius is proportional to the ratio of diffusivity ( i.e. , diffusion coefficient , d ) to the reproductive rate ( controlled by r ) .
the inclusion of diffusion in the model provides a spatially explicit understanding of how all of the components interact to affect invasion / establishment success .
the diffusion process has been extensively analyzed in invasion processes ( e.g. , fisher 1937 ; skellam 1951 ; okubo 1980 ) . in order to incorporate evolutionary factors that may influence invasion success ,
this genetic subsystem is coupled to the ecological model to explore the effects of selection and genetic variance on traits that may increase a population 's likelihood of survival .
specifically , we allow the allee threshold to become a dynamic parameter that is considered to be a fitness related trait ( e.g. , a trait impacting the component allee effect ) . from here on , except in the absence of evolution , referring to the allee threshold implies that that value is the initial value , as it changes over time .
this quantitative trait influences an organism 's ability to survive and reproduce in a small population .
the results reveal the possibility that an introduced population that would fail to persist in the ecological context of this model has the potential to succeed through evolutionary means . including evolution within the context of ecological invasions can serve to provide more robust predictions for management strategies .
therefore , it is important to investigate the possibility of evolution in the analysis of invasions .
the framework that is used to link evolutionary change with ecological processes involves developing a relationship between the fast , ecological and slow , evolutionary timescales in order to make these rates comparable ( kondrashov and khibnik 1996 ) . in the coupled evolutionary ecology model , the reaction - diffusion eqn .
( 1 ) describes the change in the population density over time and is tied into a genetic subsystem that allows the organismal response to population density to evolve in terms of the selection gradient and genetic variance .
as the population dynamics vary across space , the genetic subsystem describes the rate of change of the trait mean ( i.e. , the allee parameter ) at each location x by : ( pease et al .
1989 ; garca - ramos and kirkpatrick 1997 ; kirkpatrick and barton 1997 ; hare et al .
the first term on the right hand side reflects the force of local directional selection , where the selection gradient for frequency - independent selection is the rate of change of the mean malthusian fitness function ( i.e. , per - capita growth rate : ) with respect to the trait , a ( lande 1976 ; falconer 1989 ) .
thus , , where we assume that individual fitness approaches the population mean fitness , as most individuals are close to the average phenotype ( webb 2003 ) .
this suggests that the genetic variance (  ) is small ( and constant in this model ) .
this small parameter for the genetic variance can be used to couple the fast ecological timescale , t , with the slow evolutionary timescale ,  = t ( kondrashov and khibnik 1996 ; webb 2003 ) . combing these two components of genetic variance and selection ,
quantifies the effect of natural selection on the local mean value of the quantitative trait ( the allee parameter ; lande 1976 ; falconer 1989 ) . in order to account for the influence of migration on the trait 's local mean , the latter two terms in eqn .
the middle term takes into account asymmetrical gene flow caused by the variation of density across space ( pease et al .
1989 ; garca - ramos and kirkpatrick 1997 ; kirkpatrick and barton 1997 ; hare et al .
this captures the influence of the mean trait value ( i.e. , genetic contribution ) from more abundant populations to less abundant neighboring locations due to the spatial gradient , as more individuals migrate from areas with relatively high population densities .
the last term mirrors the diffusion term from the ecological model , and describes the homogenizing effect of random dispersal .
we solved the spatially explicit system numerically using matlab 7.0 ( 2004 , the mathworks , natick , ma , usa ) using a finite difference method to incorporate diffusion and gene flow ( adapted from garvie 2007 ) . by iterating eqns ( 1 ) and ( 2 ) forward in time
, the population density and allee threshold at each location are updated with diffusion following growth and selection , respectively , while incorporating the spatial gradient .
the simulated populations , with and without evolution , behaved as we expected from the model eqns ( 1 ) and ( 2 ) , and adequately approximate / represent the critical conditions that govern this dynamical system .
the growth and spread of an introduced population of organisms is represented by a reaction - diffusion equation described by ( lewis and kareiva 1993 ) : which determines the rate of change in the local population density relative to the carrying capacity [ u which denotes u(x , t ) ] over time , at a point in space .
this equation models the growth of the population ( the first term on the right hand side of eqn .
1 ) at a spatial location that is subject to an allee effect in addition to migration ( which depends on the second term on the rhs of eqn .
1 ) . the diffusion coefficient ( d ) scales the rate of population spread , in this case across a one - dimensional habitat x. the reproductive rate is regulated by r , and a ( which is a function of space and time , derived below ) is the local critical density or allee threshold that determines if population growth is positive or negative ( fig .
growth dynamics of the model population ( a ) , and the solution of eqn .
( 1 ) without diffusion ( b ) with reproductive rate , r = 0.6 , and allee threshold , a = 0.3 .
trajectories for population size ( b ) are given for initial densities from 0 to 1 in increments of 0.05 .
there are many variations of single - species models of population dynamics that exhibit allee effects ( see table 1 of boukal and berec 2002 ) , however the growth function of eqn .
the behavior of this verhulst ( 1838 ) logistic model modified to include a nonlinear cubic term ( based on the fitzhugh - nagumo equations ; fitzhugh 1960 ; nagumo et al .
1962 ) , is bistable , with equilibria at u = 0 ( extinction ) , u = a ( unstable threshold ) , and u = 1 ( carrying capacity ) .
figure 1a shows this behavior in terms of the growth of the population ( change in population density with respect to time ) versus the population density . at densities below the critical threshold ( a )
there is negative population growth declining to extinction ( from here on the population is considered extinct below a cutoff density of 0.0001 , as a declining population trajectory will only asymptotically approach zero in a deterministic model ; gomulkiewicz and holt 1995 ) ; otherwise the population will reach carrying capacity .
1b with the graph of the solution of the growth function ( population size versus time ) at various initial densities . when diffusion is added to this model of population growth ,
there are two critical elements that emerge based on the solution to the partial differential equation ( pde ) .
the first is the wave speed , which is determined by the allee threshold ( a ) . as we are considering a strong allee effect in this model ( i.e. , 0 < a < 1 , where the population below this threshold exhibits negative growth
versus reduced positive growth from a weak allee effect ) , there exists a unique wave speed of the invasion front that is a result of being 
pushed from the inside out , as opposed to being  pulled by the leading edge ( lewis and kareiva 1993 )
. this velocity can be derived through the solution of the pde ( 1 ) :   ( lewis and kareiva 1993 ; murray 1993 ) .
this result suggests that in order for a wave to maintain a positive velocity of advance , the magnitude of the allee threshold ( a ) must be less than half of the maximum value of the population density relative to the carrying capacity .
in addition to the velocity of the wave front , the region occupied by the invading population must exceed a certain critical size for positive growth to occur ( skellam 1951 ; kierstead and slobodkin 1953 ) .
this phenomenon is clearly explained by okubo ( 1980 ) by noting that whereas reproduction takes place within a region or patch , diffusion takes place at the boundaries resulting in a loss of organisms , thus reducing the density within the patch .
this tradeoff in the ratio of inner region volume to outer surface area will either allow a population to grow or decline with an inverse relationship of diffusivity to rate of growth .
this relationship gives a minimum region within which reproduction can not compensate for loss due to diffusion , especially when allee effects influence population growth .
thus , lewis and kareiva ( 1993 ) derive a minimum size condition ( i.e. , the radius of the initial beachhead ) based on the wave speed that is required for the population to establish and radially expand .
we address this critical size threshold qualitatively , as the analytical solution ( i.e. , ; lewis and kareiva 1993 ) is for two - dimensional spread , while we work with a simpler one - dimensional model .
the minimum critical radius is proportional to the ratio of diffusivity ( i.e. , diffusion coefficient , d ) to the reproductive rate ( controlled by r ) .
the inclusion of diffusion in the model provides a spatially explicit understanding of how all of the components interact to affect invasion / establishment success .
the diffusion process has been extensively analyzed in invasion processes ( e.g. , fisher 1937 ; skellam 1951 ; okubo 1980 ) .
in order to incorporate evolutionary factors that may influence invasion success , we develop a quantitative genetic subsystem .
this genetic subsystem is coupled to the ecological model to explore the effects of selection and genetic variance on traits that may increase a population 's likelihood of survival .
specifically , we allow the allee threshold to become a dynamic parameter that is considered to be a fitness related trait ( e.g. , a trait impacting the component allee effect ) . from here on , except in the absence of evolution , referring to the allee threshold implies that that value is the initial value , as it changes over time .
this quantitative trait influences an organism 's ability to survive and reproduce in a small population .
the results reveal the possibility that an introduced population that would fail to persist in the ecological context of this model has the potential to succeed through evolutionary means . including evolution within the context of ecological invasions can serve to provide more robust predictions for management strategies .
therefore , it is important to investigate the possibility of evolution in the analysis of invasions .
the framework that is used to link evolutionary change with ecological processes involves developing a relationship between the fast , ecological and slow , evolutionary timescales in order to make these rates comparable ( kondrashov and khibnik 1996 ) . in the coupled evolutionary ecology model , the reaction - diffusion eqn .
( 1 ) describes the change in the population density over time and is tied into a genetic subsystem that allows the organismal response to population density to evolve in terms of the selection gradient and genetic variance . as the population dynamics vary across space , the genetic subsystem describes the rate of change of the trait mean ( i.e. , the allee parameter ) at each location x by : ( pease et al .
1989 ; garca - ramos and kirkpatrick 1997 ; kirkpatrick and barton 1997 ; hare et al .
the first term on the right hand side reflects the force of local directional selection , where the selection gradient for frequency - independent selection is the rate of change of the mean malthusian fitness function ( i.e. , per - capita growth rate : ) with respect to the trait , a ( lande 1976 ; falconer 1989 ) .
thus , , where we assume that individual fitness approaches the population mean fitness , as most individuals are close to the average phenotype ( webb 2003 ) .
this suggests that the genetic variance (  ) is small ( and constant in this model ) .
this small parameter for the genetic variance can be used to couple the fast ecological timescale , t , with the slow evolutionary timescale ,  = t ( kondrashov and khibnik 1996 ; webb 2003 ) . combing these two components of genetic variance and selection ,
quantifies the effect of natural selection on the local mean value of the quantitative trait ( the allee parameter ; lande 1976 ; falconer 1989 ) . in order to account for the influence of migration on the trait 's local mean , the latter two terms in eqn .
the middle term takes into account asymmetrical gene flow caused by the variation of density across space ( pease et al .
1989 ; garca - ramos and kirkpatrick 1997 ; kirkpatrick and barton 1997 ; hare et al .
this captures the influence of the mean trait value ( i.e. , genetic contribution ) from more abundant populations to less abundant neighboring locations due to the spatial gradient , as more individuals migrate from areas with relatively high population densities .
the last term mirrors the diffusion term from the ecological model , and describes the homogenizing effect of random dispersal .
we solved the spatially explicit system numerically using matlab 7.0 ( 2004 , the mathworks , natick , ma , usa ) using a finite difference method to incorporate diffusion and gene flow ( adapted from garvie 2007 ) . by iterating eqns ( 1 ) and ( 2 ) forward in time
, the population density and allee threshold at each location are updated with diffusion following growth and selection , respectively , while incorporating the spatial gradient .
the simulated populations , with and without evolution , behaved as we expected from the model eqns ( 1 ) and ( 2 ) , and adequately approximate / represent the critical conditions that govern this dynamical system .
the dynamics of the evolutionary ecology model can be interpreted using the idea of fast and slow timescales ( kondrashov and khibnik 1996 ; webb 2003 ) .
earlier , we assumed that the genetic variation (  ) was small ( to use mean fitness as a proxy for individual fitness ) , which can subsequently be taken advantage of for our analysis of the coupled evolutionary ecological dynamics . when  = 0 , the situation without evolution , the genetic subsystem is frozen and the population moves towards a stable equilibrium of the ecological subsystem ( carrying capacity or extinction ) depending on its initial density ( greater than or less than the allee threshold respectively ; fig .
when  > 0 but small , the allee threshold evolves relatively slowly and influences the ecological system . whenever the population is below its carrying capacity ( u = 1 for each spatial coordinate x when space is explicit ) , eqn .
( 2 ) is negative , and decreases the mean allee threshold ( a ) , as the intensity of selection is density dependent .
thus , fitness increases as allee effects are suppressed , and selection drives the allee threshold towards zero .
if the population density is greater than the allee threshold , but still below the carrying capacity , it will progress towards carrying capacity more rapidly than it would without evolution as a decreases ; as the rate at which the population density changes ( eqn .
1 ) is proportional to the difference between u and a. the ecological dynamics are reversed when the population density is below the allee threshold as the population declines towards extinction , but more slowly than it does without evolution . when u < a , eqn .
( 1 ) is negative , and the population density approaches extinction more rapidly with a constant ( as the difference between u and a increases ) , than it does with evolution as a decreases ( revealing a more pronounced time lag to extinction ) . during this time lag ,
as the population slowly declines , the opportunity for evolution to overcome inverse density dependent effects occurs .
if the rate of evolution is fast enough , the allee threshold can fall below the population density , causing the rate of change of population density to become positive ( where u > a ) and the population grows and can successfully invade .
the chance that evolution can rescue the population from extinction depends on the relative rates of genetic change in the quantitative trait ( i.e. , allee threshold ) and of population decline ( gomulkiewicz and holt 1995 ) ; hence the amount of genetic variance greatly impacts the ability to adapt and survive .
a nonspatial example of this process , referred to as evolutionary rescue ( gomulkiewicz and holt 1995 ) , is shown in fig .
2 , where a population is introduced below the allee threshold . without evolution , the population declines to extinction ( fig . 2a , solid line ) as the allee threshold remains constant ( fig . 2b , solid line ) . when the population can evolve ( fig .
2 , dotted line ) , it declines at first until it can overcome the magnitude of inverse density dependence , and is then able to successfully establish . as it is difficult to measure the allee effect empirically ( tobin et al .
2007 ) , we use an extreme value that exaggerates density dependent effects in order to investigate the  worst case scenario ( a = 0.3 , where the population exhibits deterministic decline when its density is less than 30% of its carrying capacity ) . when evolution is included , we used a small value for the genetic variance ,  = 0.02 , in order to remain consistent with fast - slow dynamics , unless otherwise indicated .
comparison of an invading population introduced at a density below the allee threshold , a = 0.3 ( u = 0.25 , r = 0.6 ) .
the solid line represents the nonspatial system ( d = 0 ) described by eqns ( 1 ) and ( 2 ) without evolution (  = 0 ) which results in extinction ( a ) and a constant allee threshold ( b ) .
the dotted line indicates population growth ( a ) when evolution (  = 0.02 ) acts to reduce the allee threshold ( b ) . in general , there is a range of parameter space that permits persistence for a population below the allee threshold in the nonspatial model with evolution ( instead of simple decline to extinction ) .
as genetic variance increases , we are essentially relaxing the assumption of fast - slow timescales and allow evolution to occur more rapidly .
3 where initial population densities below the allee threshold require a minimum amount of genetic variance in order to avoid extinction . in this case
, the rate of reproduction , r , also influences the potential for evolutionary rescue , as it impacts both population growth and rate of evolution ( eqns 1 and 2 , respectively ) .
as we relax the assumption of fast - slow timescales , the behavior remains qualitatively the same as that described analytically under a strict fast - slow timescale assumption .
parameter combinations of reproductive rate : r ; genetic variance :  , and initial population density : u , that result in extinction or evolutionary rescue . in this nonspatial scenario ,
initial population densities greater than the allee threshold ( a = 0.3 ) always succeed , thus the focus is on the parameter space that allows for evolutionary rescue ( i.e. , where the population growth changes from negative to positive ) . as the reproductive rate increases from 0.1 to 1
, there is less genetic variance needed for a population to evolve to overcome inverse density dependence as increased reproduction will contribute to suppressing allee effects .
nonetheless , the additional complexities result in qualitatively similar behavior to the nonspatial model . in this case , not only will evolution influence population growth , it affects the wave speed and the critical size threshold , rmin .
as the population overcomes allee effects with a decreasing allee threshold , the wave speed accelerates and the critical patch size becomes smaller .
thus , in addition to the initial density of the introduced population and the genetic variance , the initial radius or patch size of the initial invasion area , the ratio of diffusion to reproduction , and gene flow will factor into successful establishment and give rise to a wider range of interactions between the ecology and evolution of this system .
( 1 ) ( without the evolutionary subsystem ) in one - dimensional space , with an initial population density below the allee threshold , declines to extinction ( fig .
this is contrasted by the results when the evolutionary subsystem is included . with the initial population density below the allee threshold , fig .
the same type of rescue occurs for a population that starts near carrying capacity , but occupies an initial spatial size below that which is necessary for a population to successfully establish .
figure 5a shows a rapidly declining population that goes extinct . under the same circumstance , but where evolution of the allee threshold occurs , fig .
5b shows the population density at first beginning to shrink and then growing and expanding .
in addition to the time evolution of population density across space in figs 4 and 5 , the evolution of the mean trait value across space illustrates how gene flow and the density dependent selection gradient influences its rate of change and distribution ( figs 4c and 5c ) . as the intensity of selection is density dependent ( and we assume constant genetic variance ) , locations with smaller populations can evolve the trait value more rapidly compared to other areas where allee effects may not be as strong and experience weaker selection .
the trait distribution over time , figs 4c and 5c , therefore reflect the population density distribution , but are also influenced by the trait values of the migrants . as individuals disperse out to new locations and push the boundaries of the species range , their trait values are averaged to determine the demographic allee threshold for that spatial coordinate . this demographic allee threshold combines with their local population density to influence individual fitness and population growth ( where the distance between the density and mean trait value is the initial degree of maladaptation ) .
diffusive dispersal of an introduced population at an initial density ( bold dashed line ) below the initial allee threshold , a = 0.3 ( u = 0.25 , r = 1 , d = 0.1 ) across a linear , one dimensional habitat .
the population collapses over time to extinction ( a ) where there is no evolution (  = 0 ) , and succeeds ( b ) after an initial decline with evolution (  = 0.02 ) .
( c ) the evolution of the mean value of the allee threshold across space ( where the initial distribution is given by the bold dashed line ) . the population density distribution and corresponding trait values ( i.e. , allee threshold ) are plotted at equal time increments ( every 20 of 1200 model iterations ) .
population density of a diffusion dispersed population across one dimensional space . the initial population density ( bold dashed line )
is near carrying capacity ( u = 0.95 , a = 0.3 , r = 1 , d = 0.5 ) , but introduced below the minimum radius of area determined to be critical for invasion success .
( a ) is collapsing to extinction without evolution (  = 0 ) , whereas ( b ) shows success of an invader with evolution (  = 0.02 ) after initial decline .
( c ) the evolution of the mean value of the allee threshold across space ( where the initial distribution is given by the bold dashed line ) .
the population density distribution and corresponding trait values ( i.e. , allee threshold ) are plotted at equal time increments ( every 20 of 1200 model iterations ) .
we explored when evolutionary rescue occurred across a range of parameter values for the spatially explicit model . according to drake et al .
( 2005 ) , variability among locations and over time makes it unreasonable to determine precise estimates for the diffusion coefficient , d. we therefore explored a range of values , and present those that best illustrate breadth of behavior .
the parameter that controls the reproductive rate , r , was also varied substantially , but as the spatial dynamics depend on the ratio of diffusion to rate of reproduction ( resulting in a measure of length ) ; we fixed r and varied d , unless otherwise noted .
this was justified as the results of the spatial simulations are qualitatively identical for equivalent ratios .
the effects of the critical patch size , initial population density , ratio of diffusion to growth and genetic variation on evolutionary rescue and population dynamics are shown in fig .
the ratio of the diffusion coefficient ( d ) to the reproductive rate ( r ) determines whether the population will expand or collapse according to the initial radius of the introduced population .
the areas under the curves denote combinations of genetic variance and initial population density that result in extinction .
areas above the curves are combinations of genetic and/or demographic conditions that produce inevitable persistence .
the parameter space between the vertical dashed lines refers to the different ways population survival is influenced . to the left of the initial allee threshold
, the initial population density will either go extinct due to density dependent effects ( below the d / r curve ) , or given enough genetic variation , will be evolutionarily rescued ( above the d / r curve ) .
the area to the right of the initial allee threshold [ and between the dashed lines in ( b ) and ( c ) ] is the case where the initial population density is greater than the allee threshold but due to the initial spatial size and the ratio of diffusion to reproduction , the population may go extinct without sufficient genetic variance ( below the d / r curve ) , otherwise it will evolve to overcome the critical patch size effect . for initial population densities
thus , the area between the dashed lines in ( b ) and ( c ) truly delineates evolutionary rescue when d / r = 1 .
the rightmost vertical line moves slightly to the left to the point of intersection of the d / r curve and the x - axis for other values of d / r .
graphs ( a ) , ( b ) , and ( c ) represent different radii of the linear habitat that the introduced population initially occupies .
when the size of the initial population is too small ( i.e. , a radius of 1 ) , a population at carrying capacity ( i.e. , u = 1 ) will go extinct without evolution due to the relative effect of diffusion to reproduction ( fig .
if evolution occurs rapidly enough ( i.e. ,  > 0.02 ) , the population can overcome inverse density dependent effects and compensate for the loss due to diffusion and rebound from low densities . when the initial radius of the population is increased ( fig .
6b , c ) , the chance of survival and establishment ( growth and expansion ) of populations above or below the allee threshold increases with initial density and genetic variance .
therefore , the initial radius of the population can significantly impact the likelihood of evolutionary rescue for populations with the same amount of genetic variance .
this is demonstrated further in fig . 7a , where the rate of recovery ( i.e. , the inverse of the time lag before growth becomes positive and the population reaches carrying capacity ) for a population near carrying capacity depends on its initial size / radius and genetic variance . where size and variance are small
, rescue never occurs . as these parameters increase , the rate of recovery gradually becomes faster until it essentially plateaus ( although with greater variance and initial radius , the rate of recovery may slow slightly if the initial spatial extent is large enough for the population to experience early growth before diffusion causes decline prior to recovery ) .
if the population occupies a large enough spatial extent , it will succeed without evolution ( where the genetic variance is zero ) , however the lag time may be more pronounced depending on the ratio of diffusion to reproduction through the tradeoff between growth and spread ( e.g. , if spread is relatively fast compared to reproduction , d / r = 1 ) . the population density may thus initially decline across space until reproduction can sufficiently overcome the loss due to diffusion , and the population can grow to carrying capacity .
similar to the nonspatial case , a population ( greater than the allee threshold ) above the spatial threshold will grow to carrying capacity more rapidly with evolution than without .
rate of recovery in terms of the inverse of the time lag before population growth becomes positive , where one  timestep equals 24 iterations of the model . in ( a )
, the initial population density is near carrying capacity ( u = 0.95 , a = 0.3 , d / r = 1 ) , and the initial radius and genetic variance ,  , varies .
where the rate of recovery is zero , the population goes extinct as it initially occupies an area smaller than the critical patch size ( in this case , a radius of 1.4 ) or does not have sufficient genetic variance to evolve quickly enough to be rescued prior to extinction . increasing the genetic variance and initial radius will decrease this time lag until the population no longer experiences any negative growth ( in this case , for initial radii  3.8 and   0.036 ; for initial radii > 2.7 , the rate of recovery slows slightly due to early growth followed by a transient decline that precedes ultimate recovery ) .
when the initial population density varies ( indicating the initial degree of maladaptation where a = 0.3 ) , ( b ) shows the rate of recovery with the initial radius fixed ( as in fig .
5a where the radius = 1 and d / r = 1 ) . in this case
, extinction will occur without evolution not only for an initial density below the initial allee threshold , but for any density as the initial radius is below the critical patch size .
hence , a nonzero rate implies evolutionary rescue and a zero rate means extinction . as shown in fig .
7b , when evolutionary rescue is possible , the initial level of maladaptation ( a0u0 ) and the genetic variance (  ) also determine the rate at which evolutionary rescue proceeds .
figure 7b uses parameters ( i.e. , radius and ratio of d to r ) for a population that would decline and go extinct without evolution regardless of the initial density .
hence , it is clear that the amount of time required for a population to begin growing depends on its initial level of maladaptation ( to both the critical density and spatial thresholds ) and/or genetic variance . as the rate at which this rescue occurs depends on the amount of genetic variance ( eqn .
2 ) , it may take an extremely long time ( as  0 , the rate of recovery  0 ) for the allee threshold to fall below the population density . in this circumstance , as the population density becomes very close to zero , the rate of change of the allee threshold is greater than that of the population density ( as u  0 , u/t 0 and a/t  -2ra )
. thus , theoretically , rescue would always occur ( gomulkiewicz and holt 1995 ) . however , to maintain biological realism , when solving this system numerically , we always considered the population extinct when the maximum density ( across space , when diffusion is included ) becomes reasonably close to zero ( i.e. , u = 0.0001 ; we chose this protocol instead of the total population across space due to the diffusion dynamics based on the gaussian dispersal kernel and the  pushed wave front behavior ) .
overall , the numerical results qualitatively hold for a wide range of dimensional parameter values and initial conditions with and without diffusion and in one- and two - dimensional space .
results for two - dimensional space are not shown as they are qualitatively similar to the simpler , one dimensional model .
from these results , it is apparent that adaptations that enable organisms to overcome the negative effects of low densities can allow the population to rebound from a trajectory toward extinction to grow to reach carrying capacity .
current management strategies ( e.g. , reducing population density or size ) are based on ecological theory ( e.g. drake et al . 2005 ) , but this evolutionary ecology model suggests that adaptive evolution can enable successful establishment and that ecological considerations alone may not be sufficient . under the assumptions of an allee effect and diffusive dispersal , the idea of ecological size thresholds fits well with the ecological evidence that a large founding population is a primary cause of successful establishment ( lockwood et al .
however , by incorporating evolution , we see that the situation is not quite this simple because ecological size thresholds and genetic variance can interact to determine successful establishment . as
the ratio of diffusion to reproduction decreases , the spatial constraint on population growth becomes weaker , and less genetic variance is needed to rescue populations with densities below the allee threshold . as the initial spatial radius of introduction increases ,
furthermore , the rate of this rescue depends on the initial genetic load or maladaptation ( i.e. , how far the population density is from the allee threshold ) , as well as the amount of genetic variance .
because bottlenecks during founding events do not always result in highly reduced genetic variability , even small founding populations may have sufficient genetic variation to evolve to overcome allee effects and establish , contrary to solely ecologically based predictions .
additionally , we can draw several general insights about how dispersal impacts selection and evolution of allee effects in an invasion context .
as species are transported from their native environment into novel habitats or simply disperse on their own , it is clear that the genetic composition of the local population can influence the rate of evolution and adaptation to the new local conditions . given enough genetic diversity , local populations can adapt to their local environment , but dispersal may hinder survival across ecological clines as dispersers tend to be maladapted to the new local environment . essentially , local population persistence depends on the race between the rate of evolution and the degree of maladaptation ( gomulkiewicz and holt 1995 ) . in this case
, gene flow will play a major role in determining the outcome . as kirkpatrick and barton ( 1997 ) and garca - ramos and kirkpatrick ( 1997 )
demonstrate , individuals moving from one selection regime from the center of their species range to the periphery can introduce enough maladaptation that the new area becomes a sink environment . on the other hand , holt et al .
( 2003 , 2004 ) show that immigration can have a positive influence on adaptation to sink environments , in some circumstances .
resolving the disparity between these perspectives requires understanding what is contributing to the severity of maladaptation and the population 's ability to overcome it . in our model ,
dispersal impacts survival ecologically through the critical patch size , and genetically , as individuals may move from areas where they are well adapted ( i.e. , the population density is greater than the allee threshold or mean trait value ) to sink regions , where they are maladapted .
as individuals disperse across space , they may be contributing positively in an ecological sense to the quality of their new local environment ( by increasing the local population density ) .
however , dispersers are more likely to come from higher density areas where allee effects , and hence selection , are locally weak .
these dispersers potentially introduce more maladaptation to their new location , because they increase the average phenotype ( allee threshold ) in the new location where density is likely to be lower .
interestingly , the evolutionary impacts of migration in this model do not dramatically influence the dynamics .
changes to the local mean phenotype through local selection and simple mixing ( i.e. , diffusion ) actually slightly hastens the evolutionary rescue effect over a model that considers only the impact of local selection .
as the selection intensity is density dependent and proportional to u 1 for each point in space , the peripheral individuals faced with stronger selection with lower trait values have a small positive influence on the more dense neighboring populations .
the gradient term accounts for asymmetric gene flow due to differential migration from areas of relatively high population densities .
however , this term does not alter the overall evolutionary dynamics based on local selection any more than adding the diffusion term , as the negative effects of gene flow and the local rate of evolution ( which is relatively fast , based on the selection intensity ) essentially cancel each other out . in this context ,
similar to that of holt et al . ( 2003 , 2004 ) , the immigrants simply contribute to the local population density , which helps prevent extinction long enough for evolutionary rescue to occur locally ( i.e. , positive population growth ; note that whereas holt et al .
( 2003 ) attribute the main effect of immigration to the contribution of variation , this is not the case in our model , as we assume constant genetic variance ) .
( 2004 ) where immigration has a demographic effect on increasing fitness that can essentially outweigh the 
the primary determinant of invasion success depends on positive population growth at the center of the introduced range .
this result comes from the allee effect [ and the solution to the pde ( 1 ) ] by forcing a  pushed travelling wave front ( lewis and kareiva 1993 ) , where the wave speed causes population expansion , contraction , or propagation failure ( i.e. , pinning ; keitt et al
. 2001 ) . intuitively , aggregation - like behavior emerges based on the strength of the allee effects .
individuals that disperse too far from the whole are likely to die before they can  pull others in their vicinity . in this
regard , growth occurs from the inside out , where the population seemingly spills out and overflows to expand its range .
consequently , in this study , and for biological invasions that exhibit similar dynamics , it is more important to focus on the center of the invader 's range and whether the initial beachhead can survive ( through evolutionary rescue ) , than the fate of peripheral populations at the wave front when determining the importance of evolution on invasion success .
the overall dynamics are qualitatively similar in the parameter space that allows for evolutionary rescue to occur .
even though gene flow and spatial structure do not dramatically influence the establishment of an introduced population , additional invasion criteria need to be considered .
when analyzing the model behavior in a spatially explicit context , there is an ecological tradeoff between growth and spread that affects establishment and the rate of recovery . as previously mentioned , reproduction needs to compensate for the loss due to diffusion . including evolution and suppressing allee effects
, actually contributes to the acceleration of the wave front ( i.e. , enhancing dispersal speed ) . a population then can more rapidly disperse as it evolves , and may become more of an invasion threat as long as this range expansion does not reduce their density too quickly . whereas this increasing wave speed can lead to a slightly longer lag phase prior to positive population growth , the population will likely be inevitably rescued because this effect primarily influences the dynamics at the periphery and is offset by the reduction in the critical invasion area ( rmin ) .
although there is no range contraction ( as there is always a positive wave velocity with unbounded expansion due to the parameter values and absence of environmental heterogeneity or range limitations ; filin et al .
2008 ) , as the critical patch size ( rmin ) becomes smaller with the decreasing allee threshold , rescue occurs more readily at the range center as the critical patch size threshold criteria weakens and the behavior approaches that of the nonspatial model .
this may seem like an oversimplification of the global dynamics ; however these conclusions are valid in the context of this investigation which focuses on the establishment phase rather than subsequent range expansion and spread . recognizing that evolution can significantly affect the establishment success of invasive species is becoming more widely accepted , influencing the ways in which invasion biologists conduct their research ( see the other articles in this issue ) .
specifically , adaptations that diminish allee effects and evolutionary responses to density dependence are beginning to emerge as viable explanations for sustaining vulnerable populations at low density and size ( gascoigne et al .
as it is difficult to conclusively support this claim empirically ( as the origin of the adaptation or the associated cost may be unknown ; courchamp et al . 2008 ; gascoigne et al .
2009 ) , mathematical models that incorporate evolution and compare the effects of various strategies ( e.g. , mitigating component allee effects ) can help decipher the mechanisms that both limit and facilitate population growth .
two such models that incorporate adaptations to component mate - finding allee effects compare the efficiency and survival of populations at various densities that attract mates with or without a sexual pheromone ( jonsson et al .
another study suggests that broadcast spawners that evolve their gamete morphology and performance under sperm limitation ( at low density ) bear a cost of decreased fitness at high density due to hybridization and competition ( levitan 2002 ) .
in these cases , particular strategies are shown to influence population viability in addition to an associated tradeoff , whereas our investigation provides broad , albeit simplistic , results dealing with generalized demographic allee effects and evolution . in order to understand how the results of this simplistic model extend to more realistic and complex evolutionary scenarios , spatially explicit
, individually - based stochastic simulation of the introduced populations should be developed to investigate more closely the mechanisms that allow these population level dynamics to emerge .
in particular , tracking the mean value of a component allee effect is sufficient to illustrate how evolution can overcome inverse density dependence and result in invasion .
however , this approach may not be sufficient to make the specific quantitative predictions necessary for management of invasive species .
this is due to the simplifying assumption of constant genetic variance based on mutation - selection balance ( lande 1976 ) .
complex simulations could relax this assumption and permit genetic variation to change via mutation , selection , and drift , in tandem with the demographic processes in a heterogeneous environment , and explicitly investigate the costs associated with avoiding allee effects .
hence , future models should incorporate how propagule pressure ( size and frequency of introduction events ) impacts genetic variation and how more realistic genetic architectures contribute to the evolutionary trajectory of invasive species .
although there is still much more work to be done to elucidate the factors that determine establishment success of founder populations , this theoretical approach has the promise to provide evidence in support of our working hypothesis that adaptive evolution can mitigate allee effects and be an important driver of biological invasions .
 the mechanisms that facilitate success of an invasive species include both ecological and evolutionary processes . investigating the evolutionary dynamics of founder populations can enhance our understanding of patterns of invasiveness and provide insight into management strategies for controlling further establishment of introduced populations . 
 our aim is to analyze the evolutionary consequences of ecological processes ( i.e. , propagule pressure and threshold density effects ) that impact successful colonization . 
 we address our questions using a spatially - explicit modeling approach that incorporates dispersal , density dependent population growth , and selection . 
 our results show that adaptive evolution may occur in small or sparse populations , providing a means of mitigating or avoiding inverse density dependent effects ( i.e. , allee effects ) . 
 the rate at which this adaptation occurs is proportional to the amount of genetic variance and is a crucial component in assessing whether natural selection can rescue a population from extinction . 
 we provide theoretical evidence for the importance of recognizing evolution in predicting and explaining successful biological invasions . 
Introduction
Model description
The ecological system
The evolutionary subsystem
Results
Discussion