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Adaptation to a Linear Environmental Gradient Moving in Time: Phenotypic Model

Following Pease et al. (1989), the change of the mean phenotype z can be written as This equation describes the effects of migration and selection on a population with density n and with quantitative trait (z) under selection. The first term represents migration, approximated by diffusion with variance σ². The second term describes gene flow from populations that vary in population density n. The third term describes the effect of selection on a normally distributed character z with additive genetic variance V_A (Fisher 1930; Lande 1976; is the mean [Malthusian] fitness, which is equal to in continuous time and in discrete time): ; here, V_A is additive genetic variance, V_P is the phenotypic variance, and h² is the narrow‐sense heritability ( ).

We assume that there is an optimal value θ(x, t) for the trait z(x, t) that is changing at steady and independent rates through space (x) and time (t): where b is the gradient of optimum in space and k is the rate of change of the optimum in time. The habitat is one‐dimensional, and the position on it is denoted by x.

Following Kirkpatrick and Barton (1997), one would naturally start with a simple population regulation, where population density just reflects mean fitness, because this readily leads to exact results. In appendix A, we explain how models with joint regulation converge to the equilibrium solution of this simple model.

It is more realistic to assume that there is a joint regulation of trait mean (eq. [1]) and population density (Kirkpatrick and Barton 1997). The population grows locally at rate , and migration is approximated by diffusion with variance σ²:

The average fitness gives the intrinsic rate of increase of the population, . Fitness and the intrinsic rate of increase depend on both the population density (ecological component, r_e) and adaptation in the trait (genetic component, r_g): , . The genetic component of fitness declines with the distance of the trait from the optimum: , (the last term arises as the phenotypic variance for any ). The variable V_S is the variance of stabilizing selection around the optimum; the strength of stabilizing selection is .

We assess both a “logarithmic model” of density dependence, where fitness is very high for low densities ( ): , and a “logistic model,” where the environmental growth rate is defined as , K reflects the carrying capacity of the environment, γ is the intensity of density‐dependent regulation, and r_m gives the maximum growth rate. The logarithmic density dependence converges to “simple” regulation (app. A) near equilibrium ( ), and as such it allows an exact solution. The logistic model has a more realistic growth rate at low density, which, in contrast to the logarithmic model, leads to a threshold of rates of change in space and time where the population becomes extinct.

It is useful to reduce the number of parameters by rescaling time, distance, and trait. Following Kirkpatrick and Barton (1997), we therefore introduce Time is scaled to the strength of density dependence, defined as (Kirkpatrick and Barton 1997), where is the density at carrying capacity (i.e., when the trait mean matches the optimum). For the logarithmic model, .

We thus have three parameters, A, B, and k*: The new parameters A, B, and k* describe three kinds of “load,” that is, the decrease of mean fitness due to the standing genetic variance, the spatial gradient, and the temporal change in the optimum. When time is scaled relative to the rate of return to the equilibrium (r*), is the standing genetic load due to variance around the optimum, B² is the load due to dispersal across the spatial gradient, and is the load due to temporal change in the optimum over the characteristic time . The load components are defined as the expected loss of the mean Malthusian fitness ( ) due to any of the above given factors.

In addition, we scale the population density so that it is equal to 1 when the trait mean matches the gradient: (in the logarithmic model, ; in the logistic model, ).

For joint regulation of trait mean and population density, we obtain (from eqq. [1] and [3]): where the environmental optimum is now . The scaled growth rate is, for the logarithmic model, and, for the logistic model, For the static case ( ), these equations correspond to equations (8), (9) and below in Barton (2001).

We search for an equilibrium solution in the form of a traveling wave, where transforms the spatial coordinate with the distance traveled over time T, and the speed of the traveling wave is c*. Now the scaled lag of trait mean behind the optimum is a function of a single variable, U: , where a* measures the lag load (Maynard Smith 1968) at the center of the range (or anywhere for a uniform solution). The logarithmic model now gives an exact solution with the population density of the form ; for the logistic model, we get only an approximate solution using in equation (7). Just as when the environmental gradient is fixed in time (Kirkpatrick and Barton 1997), we find two classes of solutions: uniform adaptation along the environmental gradient and, when the loss of fitness due to dispersal across the spatial gradient is sufficiently large ( ), a limited adaptation, where density declines away from the center. Both solutions with uniform and limited adaptation can be described jointly by the following formulas, which are exact for the logarithmic model but are only approximations for the logistic case. Later we discuss the cases of uniform and limited adaptation in detail.

At equilibrium, the lag load caused by the temporally changing optimum, relative to r*, is , and . Here, describes the degree of adaptation in the gradient of trait mean, which stays the same (or close to) that found in the static model. The trait mean adapts at a rate .

For limited adaptation, the range shifts at a rate for . When the gradients match ( ), the solution is uniform with respect to U; the range extends along the whole gradient and for any point density, . The rate of adaptation then simplifies to .

Whereas the above formulas apply to both the logarithmic model (exactly) and the logistic one (as an approximation), the equilibrium population density differs: for the logarithmic model, the density is ; for the logistic model, the density is . Here, is the inverse of the variance in population density along the scaled space U: is a measure of the width of the species range.

Uniform Adaptation

The formulas above simplify considerably when adaptation is uniform (see fig. 1, left), so that spatial gradients in the trait mean and the environment are equal ( ). The rate of adaptation in trait mean matches the rate of a temporal change of the environment, . The scaled trait mean, , lags behind the optimum by a*, leading to a load of . The scaled lag of the trait mean behind the optimum increases linearly with the scaled rate of temporal change: . In the original units, we recover ; this is the same for unstructured populations (Lande and Shannon 1996).

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Figure 1: Distribution of phenotypes (Ψ) for the logarithmic model with a spatially and temporally varying environment: left, uniform adaptation; right, limited range. Distance X is scaled relative to the standard deviation of dispersal distance, trait Z is scaled relative to the strength of selection, and time T is scaled relative to the rate of return to equilibrium; see equation (4). The top row shows the equilibrium density when the optimum is stable in time ( ); the bottom row shows the density when the optimum is changing at speed at time . Notice the decrease in density for uniform adaptation, where the lag behind the optimum is about . The environmental optimum is shown by the solid line; the dashed line depicts the trait mean. Other parameters, measuring genetic and dispersal load, are kept the same throughout this figure: and .

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The population density for uniform adaptation is for the logarithmic model and for the logistic model. When genetic variance is fixed, uniform adaptation is possible on arbitrarily steep spatial gradients: in principle, the only limitation comes from phenotypic load (with logistic growth rate, must be greater than the maximum growth rate, r_m). However, in fact, the uniform adaptation is prone to collapse when the spatial gradient is steep, , as we discuss later.

When the cost k*² due to the optimum changing in time is large relative to the genetic load scaled by the strength of density dependence A, a uniformly adapted population fails to survive. Under the logistic regulation, the population becomes extinct when the cost due to temporal change in the optimum is greater than . Logarithmic regulation is likely to be less relevant to extinction thresholds because the growth rate increases without bound as the density approaches 0.

When we scale the population density back to the original units, we uncover the trade‐off between the phenotypic load and the lag load , which decreases with the additive genetic variance V_A. When adaptation is uniform, the genetic component of fitness and the population density is highest when , as predicted for an unstructured population by Lande and Shannon (1996). The rate of temporal change, which leads to extinction in the logistic model, is in the original units , which is in agreement with the result for an unstructured population (see eq. [11] in Lynch and Lande 1993).

Limited Adaptation

Whereas uniform adaptation confirms the earlier predictions for panmictic populations, this is not the case when adaptation is limited ( ); see the right‐hand side of figure 1. This solution emerges when the spatial gradient, B, is sufficiently steep relative to the scaled genetic variance, A: . Now, as the gradient in trait mean is much shallower than the environmental gradient, the population density decreases away from the center, leading to a limited species range. As the optimum changes in time, the trait mean adapts more slowly than the environment changes ( ), and the species range shifts as the population moves toward the habitat to which it was previously adapted.

With logarithmic regulation, the gradient in trait mean, β*, is independent of the speed of movement of the optimum, k*. The gradient in trait mean β* stays shallower than the environmental gradient B following a cubic equation for : (see fig. 2). It follows that a solution with limited range exists when , which, for small A, occurs approximately when or in the original units (see fig. 3). Note that this approximate formula is the same as the exact result for the gradient in trait mean under simple population regulation (eq. [A2]). In contrast to the simple and logarithmic regulations, the approximation for the logistic model shows a weak dependence on the rate of temporal change, although the deviation is smaller than can be detected numerically (see fig. 5). It is worth noting that, although overall density declines with increased rate or cost of temporal change (k, k*), this does not directly reduce the width of species range: for example, two standard deviations of N(X) is , where and the degree of adaptation, φ, is independent of k*.

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Figure 2: Equilibrium values of scaled gradient in trait mean, β*, for logarithmic growth rate (thick lines) and simple regulation (thin lines), as functions of the scaled environmental gradient, B. B² is the load due to dispersal across the gradient, when time is scaled relative to the rate of return to equilibrium. The solution with always exists, and the solution with limited adaptation ( ) exists when the spatial gradient is steeper than the critical gradient B_c (dots on B‐axis; see also fig. 3). The thick line shows the solution for joint regulation with logarithmic density dependence, , . The equilibrium value for imperfect adaptation under joint regulation tends to the one with simple regulation as . When population density is just given by mean fitness, as under simple regulation, the gradient in trait mean for limited adaptation is (thin line). Equilibrium gradients in trait mean that are always unstable (both below and above equilibrium, ) are shown as dashed lines. The dots illustrate the critical gradients B_c for both simple and logarithmic regulation (see fig. 3).

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Figure 3: A solution on a limited range exists only for steep gradients, : black thick, dashed, and thin lines show the critical gradient, B_c, for logarithmic, logistic, and simple regulation, respectively. A critical gradient for simple regulation, (solid line), is also the approximation for the joint regulation for small values of A. is the load due to standing genetic variance when time is scaled relative to the rate of return to equilibrium. The exact formula for the logarithmic model is , and from the approximation for a logistic model (using a Gaussian density at equilibrium ) we obtain for small values of k. In the simple and logarithmic models, the critical gradient does not depend on the rate at which the optimum changes in time, and the dependence is weak for the logistic model. The blue dotted line is the estimated extinction gradient for logistic growth rate and imperfect adaptation, ; hence, the area between the dotted and dashed lines delimitates the region where a solution with limited range exists for the logistic model. In the logarithmic model, density as , so the extinction gradient depends on the (arbitrary) choice of density, N_tr, which would be deemed as subcritical. Extinction gradients are discussed in the text and in figure 6. Note that the solid lines for B_c in the figure are the same as in Barton (2001), but the dashed line for the logistic model differs because here we do not assume that B is large when estimating B_c.

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Figure 4: Scaled rate of adaptation, q*, and the speed of a traveling wave, c*, for the solution with limited range . The solid line is the exact solution for logarithmic growth rate, whereas the dashed line is an approximation. The scaled rate of adaptation is approximately . The scaled lag of trait mean behind the optimum at equilibrium is ; hence, (not shown). With uniform adaptation, the trait mean tracks the optimum, matching its rates of change in both space and time , and the scaled lag is (not shown). The scaled rate at which the point (e.g., the center of) population density moves through space is . The dotted line depicts the solution for uniform adaptation, where any point moves at speed . B measures the load due to dispersal across the gradient; A measures the load due to standing genetic variance. For the left column, ; for the right column, .

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Figure 5: As the optimum moves in time, the equilibrium gradient in trait mean, β*, stays close to that found in the static case if the population can persist. The equilibrium values with logistic regulation in a static environment ( ) are shown in black; solutions for the scaled rate of temporal change of are shown in red. Lines refer to analytic results, and dots refer to numerical solutions. The dashed line depicts the prediction for steep gradients: . In a static environment (black), the uniform solution (with ) always exists, the solution with limited adaptation ( ) exists when the spatial gradient B is steeper than the critical gradient (see figs. 2 and 3), and both are locally stable on the infinite range. In numerical solutions (dots), however, the adaptation collapses from the margins whenever the solution with limited adaptation exists (and even for slightly shallower spatial gradients, B: the gradient in trait mean, β*, is concave rather than constant, as assumed in the depicted analytical approximation). Note that, when gradient is too steep ( ), a population with limited adaptation cannot persist ( for decreases to for ; see text). A uniformly adapted population cannot adapt to a temporal change faster than . Parameters: , (black), and (red). The numerical solutions are run on a spatial lattice with spacing and the time step is (so that consistently, with the scaled model, migration is ), and there is no migration over the margins (reflective boundary conditions).

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The rate of adaptation increases with the standing genetic load (A): (approximately for small A; see fig. 4, top left). Unexpectedly, the rate of adaptation hardly depends on the effective gradient B (fig. 4, top right). The scaled lag of trait mean behind the optimum is simply . The population is centered at , and the rate at which the population moves in space is , which increases as the standing genetic load (A) increases and decreases as the effective gradient B gets steeper (fig. 4, bottom row).

Assuming that favorable habitat is available, the decline in density is now much less sensitive to k* than it was when adaptation was uniform. In contrast, the degree of maladaptation increases quite quickly with the dispersal load, B². The critical rate of change of environment in space (and time) when the population with logistic regulation becomes extinct is given by (where O(x^y) refers to terms of the “order of” xⁱ for ). In terms of the original units, we recover . The extinction gradient increases with the genetic load both because of the static term and because the decrease due to temporal change (last term) becomes smaller. Note that, for the static case ( ), this formula corrects a typographical error in Kirkpatrick and Barton (1997; eq. [A5]), as their scaling uses genetic load A rather than (as stated in their eq. [11a]; this was mentioned earlier by Case and Taper 2000).

Transition: The Critical Gradient

The critical rate (gradient) of change of the optimum in space, above which uniform adaptation may collapse, does not change significantly with the rate at which the optimum moves in time, and it depends only on standing genetic variance. For both models of density dependence that we assessed, the critical gradient is close to (see fig. 3), as shown previously for a static optimum by Kirkpatrick and Barton (1997) and Barton (2001). In the original units, limited adaptation emerges approximately when the change in spatial optimum over one dispersal range, bσ, relative to the standard deviation of genetic variance, , is larger than twice the square root of the strength of density dependence: .

Both uniform adaptation and limited‐range solutions are locally stable on an infinite range (see app. B); however, as the spatial gradient steepens above the critical gradient B_c, uniform adaptation becomes increasingly prone to collapse toward the limited adaptation due to perturbations of the uniform solution. Such a collapse is triggered by the imposed reflective boundary conditions at the margins of the available habitat: this effect can be important in nature when there is a rigid boundary to the habitat (e.g., a river), so that dispersal—and hence, maladaptive gene flow—is reduced at the margin. Density of a population adjacent to the boundary then increases, increased gene flow to the neighboring population (closer to the center of the range) leads to a drop in a density there, and, eventually, the whole population collapses to the regime with limited adaptation (see Kirkpatrick and Barton 1997; Polechová and Barton 2005). When the cost of temporal change (k*) is large, limited adaptation may constitute the only way a population can survive: this is because the lag of trait mean behind the optimum is actually smaller, because populations with limited adaptation move faster through space (see figs. 4, 6).

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Figure 6: Population density for the uniform population (where ) decreases as fast as the optimum moves: for the logarithmic model (solid black line), and ( ) for the logistic model (dashed black line, ). As the cost of temporal change, measured by k*, increases, local population density becomes higher for a population adapted to a limited range (where , blue lines), because the population can slide along the environmental gradient. The cost of temporal change (k*) when local population density becomes higher for the population with limited adaptation than for the uniform one increases with A (not shown). Parameters: , .

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Adaptation to a Linear Environmental Gradient Moving in Time: Population Genetic Model

In order to relax the assumption of fixed genetic variance, we need a model with explicit determination of a trait; for comparison with the phenotypic model above, the trait distribution should be (close to) a Gaussian. We chose a two‐allele model (Barton 2001) where the trait under selection is determined by n_l independent biallelic diploid loci of additive effect (with frequencies q_i and p_i and effects and ). The trait mean is , and variance at linkage equilibrium is .

The cline shape in a static environment ( ) has been derived by Barton (1999, 2001), with the assumption that clines have the same form and are distributed in space so that the trait mean matches the gradient. Substituting in equation (1) with for the logarithmic model as defined below, equation (3) gives the rate of change of allele frequency: where , μ is the mutation rate (which is assumed to be symmetric), and . Note that now the genetic variance changes with allele frequency, so we get an extra term arising from .

For the static case, Barton (2001) showed that at the spatially uniform equilibrium with no mutation, allele frequency (centered at ) has a form (we set ) where the width of the cline is . The variance contribution due to one locus is , obtained by integrating the variance formula over space, with . To match the spatially variable optimum θ, the additive genetic variance at equilibrium is , since there must be clines per unit distance (as each cline shifts the trait mean by 2α); the genetic variance is maintained by the dispersal across the gradient, and it is independent as to the number of loci. In the scaled model (see app. C), we get .

As the optimum changes in time, allele frequencies will need to move in space. We are again looking for a traveling‐wave solution where the allele frequency, , is solely a function of a new variable (and , ): and where the allele frequency (which was, at time , centered on ) has a form . Then, and , so with no mutation there is a spatially uniform solution for a given , where and .

The cline shape stays the same as it is in the static case: as the optimum changes in time, the clines uniformly shift in space at a rate c. We find the solution only with uniform adaptation, where the rate of change in the trait mean matches the change in the optimum ( ). Hence, we must have , and the lag of trait mean behind the optimum is . The number of clines required to match the optimum at any particular time stays the same as in the static case at , and hence the resulting variance stays at (without mutation and under linkage equilibrium); as in the static case (Barton 2001), it is independent of allelic effect or numbers of genes. The lag of the trait mean is therefore , in agreement with the prediction for the phenotypic model.

We test the robustness of the predictions by numerically iterating the two‐allele model, following joint evolution of clines (and hence that of mean and variance) and population density, as described by equations (C1) and (7). All clines evolve independently, and so there is no linkage disequilibrium. Initially, the population has no spatial adaptation: allele frequencies at time 0 are uniform in space and almost fixed at 0 or 1, with uniform distribution of deviations ranging from 0 to 0.01. Over time (see fig. 8), allele frequencies diversify across the range to match the optimum (fig. 9).

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Figure 7: Simulations (dots) and predictions (dashed lines) for the two‐allele model with logarithmic regulation match well. At the top, the lag of the trait mean behind the optimum matches the analytical solution shown by the dashed line , where (in the original units, ). In the middle, genetic variance stays close to prediction for a fixed gradient, (rescaling back to the original units, ). At the bottom, population density at equilibrium is . The dots show results of a stepping‐stone model on a spatial lattice with range and spacing δX. In the scaled model, the time step must be , where is the migration rate. Parameters: , number of loci , , , , , and (note that we display the density n(x) in the original units). Cline shapes and more details of the equilibrium solution are shown in figures 8 and 9.

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Figure 8: Illustration of the shape of allele frequencies (p) at time (top) and (bottom). At equilibrium, the cline shape is , where the scaled cline width is and the clines move across the space X at a speed . The allelic effect is scaled as , and the maximum scaled variance is . Fixing α* and A_m, and taking a higher number of loci than can fix in the static case, , does not lead to a higher number of diversified loci (and thus higher variance). Parameters are as they are in figure 7; .

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Figure 9: Scaled lag of the trait mean behind the optimum, a*, scaled variance V, and the population density n(X) vary periodically as the optimum is matched by a finite number of loci. Parameters are as they are in figure 7; , .

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The population evolves to be uniformly adapted, with the gradient in trait mean matching the optimum and lagging behind by ( in the original units), matching the predictions for the phenotypic model (see top rows of figs. 7, 9). Scaled genetic variance V stays very close to the prediction (above), (figs. 7, 9, middle rows). Because genetic variance does not increase above the static equilibrium when the optimum changes faster in time, population density decreases toward 0 when the loss of fitness due to temporal change is too large relative to the standing genetic variance (fig. 7). The variance does not increase above the static equilibrium even when we add mutation to the model. The rate of decrease of population density obviously quantitatively differs between the logistic and the logarithmic models; for the logistic model, population density declines faster with k*, leading to extinction at .

Parameters in Nature

What are plausible values for the parameters A, B, k*, and r*? First, consider A, a measure of the load due to genetic variance around the optimum. Since Lande and Arnold (1983) renewed interest in the quantitative genetics of wild populations, there have been hundreds of studies of the strength of stabilizing selection and additive genetic variation in nature. The observed distribution (Kingsolver et al. 2001) of the standardized quadratic selection gradient, γ_i, is wide and fairly symmetrical on the continuum of stabilizing ( ) to disruptive selection ( ), with the median (tilde) for stabilizing selection , but ranging from 1.5 to 0. This corresponds to rather than , which used to be the common assumed value (see Lande 1975; Johnson and Barton 2005). If we take heritability (which implies that , where the components of V_R are all nonadditive components of genetic variance), the median of per measured trait is for and, mostly, .

Burt (1995, 2000) reviewed evidence on the additive genetic variance for fitness and made an interesting argument concerning (in our notation) the scaled dispersal load, B². He pointed out that the total dispersal load can be estimated from transplant experiments in which individuals are moved from their native location or are fertilized by pollen from elsewhere. This dispersal load must be balanced against the increase in relative mean fitness due to selection, which equals the standardized additive variance in fitness ( ). Now suppose that we can identify a principal component (of measured traits) that explains most of the variance in the reproductive success due to dispersal across the spatial gradient. Then, B can be related to the combined genetic load (A) for this component. By dispersal of a distance σ, fitness decreases by . The decrease in fitness due to dispersal and mutation is at equilibrium, balanced by a corresponding increase via additive variance in fitness, V_W; if we ignore mutation, (where W is the fitness in discrete time, and ). Burt (1995, 2000) reviewed a few estimates for the total dispersal load and reported a median of and, mostly, .

How fast might environmental optima change through time? In reality, change may occur over all timescales, rather than as a simple linear change as assumed here. However, fast changes will average out, and slow changes will have a negligible effect; we are concerned with changes that occur over the joint evolutionary and ecological timescale. The load from a perfectly adapted population, due to a changing optimum over characteristic time , is . We can get an estimate of a load due to a temporally changing environment from the speed of advance of the range due to temporal change in the environment. This speed (in terms of dispersal ranges, as ), at which a point population density moves in space, is when or . We give an example of one well‐studied, fast‐advancing species: the butterfly Hesperia comma is advancing at a rate of km per generation due to rising temperature (Thomas et al. 2001), while its expected dispersal distance is km (as measured by Hill et al. 1996 for the first nine generations). Approximately, the load due to a temporally changing optimum is at equilibrium; using the medians for B and r*, we get an estimate of k* of about 2.7 (this is an overestimate, because the range will expand faster than would follow from a model with diffusive migration as a result of occasional long‐distance dispersal, and σ is necessarily going to be an underestimate to some extent, because migrants a long distance away will not be measured and dispersal may increase during expansion). Perhaps as a better approach, the load due to the temporal change in the optimum could be estimated directly by comparing reproductive success between progeny of a current and a significantly older generation (e.g., studying organisms with a long dormancy, using seeds or diapausing invertebrates as in Decaestecker et al. [2007]).

Finally, the characteristic time is given by the inverse of the strength of density dependence, , where r* is the rate of return toward the equilibrium at carrying capacity : . From Sæther et al. (2005), we see that values of , where λ^Δt is the growth rate per generation, lie mostly between 0 and 2.5. In appendix D, we show that D is approximately equivalent to r*, and thus, the median for r* is around 1. Also, for logistic growth, the intrinsic growth rate gives the upper bound for r*: Grosholz’s (1996) study provides for some invasive species: the range of to 10, with a median of again around 1 (see also Case and Taper 2000). The importance of strength of density dependence for limits to species range and behavior at the margins has also been discussed in a recent article by Filin et al. (2008).

The above rough overview gives the estimates for the measure of standing genetic load, A, at (generally, <2). The known (few) estimates of total dispersal load give at (generally, <0.3). From the anecdotal butterfly example, we see that some populations can respond to strong selection due to a temporally changing optimum, ; the indirectly estimated value should be taken with caution, perhaps on the order of 1. In the examples above, all loads are given relative to the characteristic time . The strength of density dependence, r*, is mostly <2.5, and above it is set to around 1. We return to the ways of relating a standing genetic load to the (total) dispersal and/or temporal loads in the next section. Ideally, we would like to get all three loads and the rate of return to the equilibrium, r*, estimated for one species, as they may well be correlated. We have not found such data, however, and the above paragraphs are intended to give insights into plausible ranges and possible estimation methods as well as to illustrate the meaning of the load parameters.

Limitations, Predictions, and Nature

Out of an infinite number of possible scenarios one could think of, we chose a particular simple model of uncorrelated temporal and spatial change and assessed adaptation in a single additive trait. The population dynamics are simple (e.g., no Allee effect) and deterministic, although genetic drift could have a qualitative effect on the results regarding species range (see Butlin et al. 2003; Alleaume‐Benharira et al. 2006); certainly, drift would further limit adaptation at low densities at the edge of the species range. Large asymmetries in carrying capacity can also impede adaptation deterministically in the subpopulation, with little contribution to the total reproduction (see Kawecki et al. 1997). Another obvious extension is to include the age structure of the population (see Charlesworth 1980): the spatial gradient can be thought of as “blurred” with the standard deviation as a function of the change in the environment over the average generation time, kΔt, which would lead to an increase in variance due to the temporal change in the optimum.

We find it surprising that, when the environmental optimum moves in time, genetic variance does not increase from the value maintained by dispersal across the spatial gradient. An optimum changing smoothly over sufficiently long time periods (relative to the width of the fitness function) can lead to a significant increase in variance when recombination is high (Bürger and Lynch 1995; Kondrashov and Yampolsky 1996; Bürger 1999, 2000; Waxman and Peck 1999). According to Bürger’s theoretical predictions based on dynamics of underlying cumulants of the genotypic distribution, genetic variance should increase with the lag behind the optimum multiplied by the skewness of the distribution. However, as the equations do not form a closed system, it is unclear how skewness increases as the optimum moves in time. (Both the lag and the skewness decrease with the variance.) We see no increase in variance; the distribution remains Gaussian, with skewness of 0.

Although we can think of genetic variance as being constrained due to interactions with other traits, we do not explicitly model interactions. It is well known that covariance between traits can pose a significant constraint on the response to selection on the trait mean (see Antonovics 1976; Grant and Grant 1995; Etterson and Shaw 2001), and it also influences the variance: if variance of a correlated trait decreases, for example, due to stabilizing selection, so then would the variance of our focal trait (see Pearson 1903; Lande and Arnold 1983; eq. [11]). Our model also does not include genotype‐by‐environment interactions, which can readily extend the range of favorable habitat (Nussey et al. 2005). Perhaps more importantly, the population genetic model for an additive trait follows only allele frequencies, and not genotypes. Therefore, these results only apply with linkage equilibrium, under weak selection. Sensitivity of our results toward specific scenarios, which violate our assumptions, can be numerically and experimentally tested.

A recent preliminary study by Bridle et al. (2009) reveals results that are reasonably consistent with some of the predictions of Kirkpatrick and Barton (1997), but as far as we are aware, no study provides the data needed to test any of the predictions above. What could be done in principle? The total dispersal load (B²) could be measured directly by making transplants between different places and finding the decrease in fitness with transplant distance, measured relative to the standard deviation of dispersal distance. Similarly, the load due to change through time ( ) could in principle be measured by comparing the fitnesses of individuals from the present population with the fitnesses of those in the past (e.g., using diapausing individuals from sediment caves, as in Decaestecker et al. 2007). For all loads, the time should be scaled relative to the rate of return to equilibrium, . The load due to additive genetic variance in specific traits ( ) can be measured by comparing the fitnesses of those at the population mean with the fitnesses of those that are some standard deviations from the mean; heritability would also need to be measured. However, unlike the other two loads, we do need to know about the traits: what matters is the loss of fitness with variance along a particular axis in trait space, defined by the change in mean through time and along the transect. However, finding this axis is a not a trivial task in the real world. If we know the trait axis, then we can readily measure the load due to genetic variance along the axis. If we do not know the trait axis, then we could measure the excess variance in fitness between clones of the parents and their offspring as a function of distance between the parents. Replicates of parents can be ideally obtained from an organism capable of both asexual and sexual reproduction; however, in principle, one could also reintroduce inbred lines. If the traits involved were additive, this would give us the total standing genetic load directly. Obtaining the load parameters allows us to assess the key predictions of the model; most obviously, we can ask whether there is evidence for a critical gradient, B_c, above which uniform adaptation tends to collapse.