Density-dependent selection and the limits of relative fitness

Selection is commonly described by assigning constant relative fitness values to genotypes. Yet population density is often regulated by crowding. Relative fitness may then depend on density, and selection can change density when it acts on a density-regulating trait. When strong density-dependent selection acts on a density-regulating trait, selection is no longer describable by density-independent relative fitnesses, even in demographically stable populations. These conditions are met in most previous models of density-dependent selection (e.g. “K-selection” in the logistic and Lotka-Volterra models), suggesting that density-independent relative fitnesses must be replaced with more ecologically explicit absolute fit-nesses unless selection is weak. Here we show that density-independent relative fitnesses can also accurately describe strong density-dependent selection under some conditions. We develop a novel model of density-regulated population growth with three ecologically intuitive traits: fecundity, mortality, and competitive ability. Our model, unlike the logistic or Lotka-Volterra, incorporates a density-dependent juvenile “reproductive excess”, which largely decouples density-dependent selection from the regulation of density. We find that density-independent relative fitnesses accurately describe strong selection acting on any one trait, even fecundity, which is both density-regulating and subject to density-dependent selection. Pleiotropic interactions between these traits recovers the familiar K-selection behavior. In such cases, or when the population is maintained far from demographic equilibrium, our model offers a possible alternative to relative fitness.


Introduction 24
There are a variety of different measures of fitness, such as expected lifetime reproductive 25 ratio R 0 , intrinsic population growth rate r, equilibrium population density/carrying capacity 26 (often labeled "K") (Benton and Grant, 2000), and invasion fitness (Metz et al., 1992). In 27 addition, "relative fitness" is widely used in evolutionary genetics, where the focus is on 28 relative genotypic frequencies (Barton et al., 2007, pp. 468). The justification of any measure 29 of fitness ultimately derives from how it is connected to the processes of birth and death 30 which drive selection (Metcalf and Pavard 2007;Doebeli et al. 2017;Charlesworth 1994, pp. 31 178). While such a connection is clear for absolute fitness measures like r or R 0 , relative 32 fitness has only weak justification from population ecology. It has even been proposed that 33 relative fitness be justified from measure theory, abandoning population biology altogether 34 (Wagner, 2010). Given the widespread use of relative fitness in evolutionary genetics, it 35 is important to understand its population ecological basis, both to clarify its domain of 36 applicability, and as part of the broader challenge of synthesizing ecology and evolution.

37
For haploids tracked in discrete time, the change in the abundance n i of type i over a 38 time step can be expressed as ∆n i = (W i − 1)n i where W i is "absolute fitness" (i.e. the 39 abundance after one time step is n i = W i n i ). The corresponding change in frequency is In continuous time, the Malthusian parameter r i 41 replaces W i and we have dn i dt = r i n i and dp i dt = (r i − r)p i (Crow et al., 1970). Note that the advantages of different types relative to each other. For instance, in continuous time 49 s = r 2 − r 1 is the selection coefficient of type 2 relative to type 1. Assuming that only 2 and 50 1 are present, the change in frequency can be written as 51 dp 2 dt = sp 2 (1 − p 2 ). (1) Thus, if r 1 and r 2 are constant, the frequency of the second type will grow logistically with 52 a constant rate parameter s. We then say that selection is independent of frequency and 53 density. The discrete time case is more complicated. Defining the selection coefficient by 54 W 2 = (1 + s)W 1 , and again assuming 1 and 2 are the only types present, we have We will refer to both the continuous and discrete time selection equations (1) and (2) through-56 out this paper, but the simpler continuous time case will be our point of comparison for the 57 rest of this section.

58
In a constant environment, and in the absence of crowding, r i is a constant "intrinsic" The logistic model dn 1 dt = r 1 (1 − n 1 +n 2 K 1 )n 1 and dn 2 dt = r 2 (1 − n 1 +n 2 K 2 )n 1 with r 1 = r 2 and K 2 > K 1 . (b) The constant-N , relative fitness description of selection. and how it succeeds or fails when selection is not weak, or N is not stable. For instance, 122 in wild Drosophila, strong seasonally-alternating selection happens concurrently with large 123 "boom-bust" density cycles (Messer et al., 2016;Bergland et al., 2014). Are we compelled to 124 switch to a more ecologically-detailed model of selection based on Malthusian parameters or 125 birth/death rates in this important model system? And if we make this switch, how much 126 ecological detail do we need? 127 Here we argue that the simplified models of density-regulated growth mentioned above 128 are potentially misleading in their representation of the interplay between selection and den-about the regulation of population density (e.g. Nicholson 1954), and also has a long history 135 in evolutionary theory, particularly related to Haldane's "cost of selection" (Haldane, 1957;136 Turner and Williamson, 1968 cess. This requires that we model a finite, density-dependent excess, which is substantially 144 more complicated than modeling either zero (e.g. logistic) or infinite (e.g. Wright-Fisher) 145 reproductive excess. Nei's "competitive selection" model incorporated a finite reproductive 146 excess to help clarify the "cost of selection" (Nei, 1971;Nagylaki et al., 1992), but used an 147 unusual representation of competition based on pairwise interactions defined for at most two 148 different genotypes, and was also restricted to equal fertilities for each genotype.

149
In models with detailed age structure, it is often assumed that the density of a "crit-150 ical age group" mediates the population's response to crowding (Charlesworth, 1994, pp. 151 54). Reproductive excess is a special case corresponding to a critical pre-reproductive age 152 group. A central result of the theory of density-regulated age-structured populations is that 153 selection proceeds in the direction of increasing equilibrium density in the critical age group 154 (Charlesworth, 1994, pp. 148). This is a form of the classical K-selection ideas discussed 155 above, but restricted to the critical age group (juveniles, in this case). The interdepen-156 dence of pre-reproductive selection and reproductive density is thus overlooked as a result Our model is inspired by the classic discrete-time lottery model, which was developed by 161 ecologists to study competition driven by territorial contests in reef fishes and plants (Sale, 162 1977;Chesson and Warner, 1981), and which has some similarities to the Wright-Fisher 163 model (Svardal et al., 2015). Each type is assumed to have three traits: fecundity b, mortality 164 d, and competitive ability c. In each iteration of the classic lottery model, each type produces 165 a large number of juveniles, such that N remains constant (infinite reproductive excess).

166
Competitive ability c affects the probability of winning a territory, and behaves like a pure 167 relative fitness trait. Thus, fitness involves a product of fertility and juvenile viability akin 168 to standard population genetic models of selection (e.g. Crow et al. 1970, pp. 185). We 169 relax the large-juvenile-number assumption of the lottery model to derive a variable-density 170 lottery with a finite, density-dependent reproductive excess.

171
The properties of density-dependent selection in our model are strikingly different from 172 the classical literature discussed above. The strong connection between crowding and selec-173 tion for greater equilibrium density is broken: selection need not affect density at all. And 174 when it does, the density-independent discrete-time selection equation (2) is almost exact 175 even for strong selection, provided that any changes in density are driven only by selection 176 (as opposed to large deviations from demographic equilibrium), and that selection occurs on 177 only one of the traits b, c, or d. On the flip side, the constant relative fitness approximation 178 fails when strong selection acts concurrently on two or more of these traits, or when the 179 population is far from demographic equilibrium.

181
Assumptions and definitions 182 We restrict our attention to asexual haploids, since it is then clearer how the properties 183 of selection are tied to the underlying population ecological assumptions. We assume that 184 Chance to win territory Random Dispersal Death Lottery competition Figure 2: One iteration of our model. Propagules are dispersed by adults at random (only those propagules landing on unoccupied territories are shown). Some territories may receive zero propagules. Lottery competition then occurs in each territory that receives more than one propagule (only illustrated in one territory). In a given territory, type i has probability proportional to c i x i of winning the territory, where c i measures competitive ability and x i is the number of i propagules present. In the illustrated territory, more black propagules are present, but white is a stronger competitor and has a higher probability of winning. Adult deaths make new territories available for the next iteration (red crosses).
reproductively mature individuals ("adults") require their own territory to survive and re-185 produce. All territories are identical, and the total number of territories is T . Time advances 186 in discrete iterations, each representing the time from birth to reproductive maturity. In a 187 given iteration, the number of adults of the i'th type will be denoted by n i , the total number 188 of adults by N = i n i , and the number of unoccupied territories by U = T −N . We assume 189 that the n i are large enough that stochastic fluctuations in the n i (drift) can be ignored, 190 with T also assumed large to allow for low type densities n i /T 1.

191
Each iteration, adults produce propagules which disperse at random, independently of 192 distance from their parents, and independently of each other (undirected dispersal). We density of type i per unoccupied territory. The total propagule density per unoccupied 199 territory will be denoted L = i l i .

200
We assume that adults cannot be ousted by juveniles, so that recruitment to adulthood where G is the number of types present, and type i is expected 206 to win a proportion c i x i / i c i x i of these, type i's expected territorial acquisition is given by Here the sum only includes territories with at least one propagule present. Note that ∆ + n i 208 denotes the expected territorial acquisition. Fluctuations about ∆ + n i (i.e. drift) will not be 209 analyzed in this manuscript. Note that drift can become important if U is not sufficiently 210 large even though n i and T are large (by assumption); we do not consider this scenario on 211 biological grounds, since it implies negligible population turnover.

212
Adult mortality occurs after lottery recruitment at a constant, type-specific per-capita 213 rate d i ≥ 1, and can affect adults recruited in the current iteration, such that the new 214 abundance at the end of the iteration is (n i + ∆ + n i )/d i (Fig. 2). In terms of absolute fitness, 215 this can be written as Here ∆ + n i n i is the per-capita rate of territorial acquisition, and 1/d i is the fraction of type 217 i adults surviving to the next iteration. Note that our model of mortality differs from 218 the classic lottery model (Chesson and Warner, 1981), where mortality affects adults only 219 and occurs after propagule production but before juvenile recruitment. In the latter case, 220 selection on mortality exhibits some density-dependence, although this reflects the fact that 221 newly recruited adults are guaranteed to reproduce before dying, which is not interesting for 222 our purposes here. Our mortality model ensures that selection on d i is density-independent, 223 allowing us to more clearly separate different sources of density-dependence and density 224 regulation.

225
Connection to the classic lottery model

226
In the classic lottery model (Chesson and Warner, 1981), unoccupied territories are assumed 227 to be saturated with propagules from every type (l i → ∞ for all i). From the law of large 228 numbers, the composition of propagules in each territory will not deviate appreciably from 229 the mean composition l 1 , l 2 , . . . , l G . Type i is thus expected to win a proportion c i l i / i c i l i 230 of the U available territories, where c = i c i m i / i m i is the mean competitive ability for a randomly selected propagule.

232
Note that all unoccupied territories are filled in a single iteration of the classic lottery model,

233
whereas our more general model Eq.
(3) allows for territories to be left unoccupied and hence 234 also accommodates low propagule densities.

236
Analytical approximation of the variable-density lottery 237 Here we evaluate the expectation in Eq. (3) where Comparing Eq. (6) to Eq. (5), the classic lottery per-propagule success rate c i /cL has 251 been replaced by three separate terms. The first, e −L , accounts for propagules which land 252 alone on unoccupied territories; these propagules secure the territories without contest. The  6), the classic lottery model, and simulations. The vertical axis is per-propagule success rate for all propagules ∆ + n i /m i , and for the three separate components in Eq. (6). Two types are present with c 1 = 1, c 2 = 1.5 and l 2 /l 1 = 0.1. Simulations are conducted as follows: x 1 , x 2 values are sampled U = 10 5 times from Poisson distributions with respective means l 1 , l 2 , and the victorious type in each territory is then decided by random sampling weighted by the lottery win probabilities c i x i /(c 1 x 1 + c 2 x 2 ). Dashed lines show the failure of the classic lottery model at low density. the growth of a rare invader in a high density population and determines invasion fitness).

256
The third term, A i c i /c, represents competitive victories in territories where two or more i 257 type propagules are present. The relative importance of these three terms varies with both 258 the overall propagule density L and the relative propagule frequencies l i /L. If l i 1 for all 259 types, we recover the classic lottery model (only the A i c i /c term remains, and A i → 1/L). for the high-frequency type, which depends instead on high density territorial victories. Fig. 3 265 also shows the breakdown of the classic lottery model at low propagule densities.

266
In the special case that all types are competitively equivalent (identical c i ), Eq. (6) takes a simpler form, where we have used the fact that L = bN/T to make the dependence on b and N explicit (b 269 is the population mean b). Eq. (7) happens to be exact even though it is a special case of the total number of such territories, and type i is expected to receive a fraction l i /L of these.

273
By similar reasoning, the total number of territories acquired is given by This formula is also exact, but unlike Eq. (7), it also applies when the c i differ between types. to co-existence are tangential to the aims of the current manuscript and will not be pursued 290 further here.

291
The strength of b-selection declines with increasing density. When types differ in b only 292 (b-selection), Eq. (6) simplifies to Eq. (7), and absolute fitness can be written as declines with increasing density: the advantage of 295 having greater b gets smaller the fewer territories there are to be claimed (Fig. 4).

296
In the case of c-selection, Eq. (6) The strength of c-selection thus peaks at an interme-298 diate density (Fig. 4), because most territories are claimed without contest at low density 299 (R 1 , R 2 , A 1 , A 2 → 0), whereas at high density few unoccupied territories are available to be 300 contested (T − N → 0).

301
Selection on d is independent of density, because the density-dependent factor 1 + ∆ + n i n i 302 in Eq. (4) is the same for types that differ in d only.

303
The response of density to selection; c-selection versus K-selection 304 We now turn to the issue of how density changes as a consequence of selection in our variable-305 density lottery, and in more familiar models of selection in density-regulated populations.

306
In the latter, selection under crowded conditions typically induces changes in equilibrium 307 density (see Introduction). In our variable-density lottery model, however, the competitive 308 ability trait c is not density-regulating, even though c contributes to fitness under crowded 309 conditions. Consequently, c-selection does not cause density to change. In this section we 310 compare this c-selection behavior with the previous literature, which we take to be exempli-  Figure 4: The density-dependence of selection in our variable-density lottery between an adaptive variant 2 and a wildtype variant 1 with at equal frequencies.
Here b 1 = 1, d 1 = 2 and c 1 = 1. For b-selection we set b 2 = b 1 (1+ ), and similarly for c and d, with = 0.1. d-selection is density-independent, b-selection gets weaker with lower territorial availability, while cselection initially increases with density as territorial contests become more important, but eventually also declines as available territories become scarce.
MacArthur considered two types (with densities n 1 and n 2 ) in a constant environment subject to density-dependent growth, The outcome of selection is determined by the relationship between the nullclines f 1 (n 1 , n 2 ) = 315 0 and f 2 (n 1 , n 2 ) = 0. Specifically, a type will be excluded if its nullcline is completely 316 contained in the region bounded by the other type's nullcline.

317
MacArthur used the four intersection points of the nullclines with the axes, defined by f 1 (K 11 , 0) = 0, f 1 (0, K 12 ) = 0, f 2 (K 21 , 0) = 0 and f 2 (0, K 22 ) = 0, to analyze each type's exclusion or persistence. Note that only K 11 and K 22 are equilibrium densities akin to the K parameter in the logistic model; the other intersection points, K 12 and K 21 , are related to competition between types. For instance, in the Lotka-Volterra competition model we have f 1 (n 1 , n 2 ) = r 1 (1 − α 11 n 1 − α 12 n 2 )n 1 f 2 (n 1 , n 2 ) = r 2 (1 − α 22 n 1 − α 21 n 2 )n 2 where α 11 = 1/K 11 and α 22 = 1/K 22 measure competitive effects within types, while α 12 = 318 1/K 12 and α 21 = 1/K 21 measure competitive effects between types. Hence, "fitness is K" 319 in crowded populations (MacArthur and Wilson, 1967, pp. 149) in the sense that selection 320 either favors the ability to keep growing at ever higher densities (moving a type's own nullcline In contrast, density trajectories for c-selection in our variable-density lottery converge on a line of constant equilibrium density (Fig. 5b). This means that once N reaches demographic 328 equilibrium, selective sweeps behave indistinguishably from a constant-N relative fitness 329 model (Fig. 1b). Thus, for c-sweeps in a constant environment, the selection factor (W 2 − 330 W 1 )/W in Eq.
(2) is density-independent. This uncoupling of density from c-selection arises 331 due to the presence of an excess of propagules which pay the cost of selection without affecting 332 adult density (Nei, 1971).
where δ i is per-capita mortality due to crowding. Starting from a type 1 population in 338 equilibrium, a variant with δ 2 = δ 1 (1 − ) has density-dependent selection coefficient s = 339 δ 1 N , which will change over the course of the sweep as N shifts from its initial type 1 340 equilibrium to a type 2 equilibrium. The equilibrium densities at the beginning and end of

345
In our variable density lottery, b regulates density and is subject to density-dependent 346 selection, yet b-sweeps are qualitatively different from δ sweeps in the above example. Greater 347 b means not only that more propagules contest the available territories, but also that a greater involves one of these traits obeys the density-independent relative fitness description of se-370 lection almost exactly (that is, (W 2 − W 1 )/W in Eq.
(2) is approximately independent of 371 density). This density-independence breaks down when strong selection acts on more than 372 one of b, c and d (Fig. 8). The c and d traits exemplify the two distinct directions in which 373 density and selection can interact: selection may depend on density, and density may change 374 in response to ongoing selection (Prout, 1980). The combination of both is necessary to 375 invalidate the constant-s approximation. Remarkably, the b trait demonstrates that the 376 combination is not sufficient; the density-dependence of b-selection effectively disappears 377 over equilibrium-to-equilibrium b-sweeps.

378
The distinctive properties of selection in the variable-density lottery arise from a repro-  1) and (2)) will likely occur in populations 416 far from demographic equilibrium e.g. as a result of a temporally-variable environment. This 417 is because extremely strong selection is needed to change population density by an amount 418 comparable to environmental varability (see Fig. 6). By contrast, temporally-variable envi-419 ronments can dramatically alter frequency trajectories for individual sweeps (e.g. Fig. 9.5 in to treat the x i = 1 case separately when allowing for low propagule densities. We thus start 553 by separating the right hand side of Eq. (3) into three components 554 ∆ + n i = ∆ u n i + ∆ r n i + ∆ a n i .
The relative magnitude of these components depends on the propagule densities l i . The first component, ∆ u n i , accounts for territories where only one focal propagule is present (x i = 1 556 and x j = 0 for j = i; u stands for "uncontested"). The proportion of territories where this 557 occurs is l i e −L , and so The second component, ∆ r n i , accounts for territories where a single focal propagule is 559 present along with at least one non-focal propagule (r stands for "rare"). The number of 560 territories where this occurs is where p(x|x i =1,X i ≥1) denotes the expectation with respect to the conditional probability 562 distribution p(x|x i = 1, X i ≥ 1) of propagule abundances in those territories where exactly 563 one focal propagule, and at least one non-focal propagule, landed.

564
The final contribution, ∆ a n i , accounts for territories where two or more focal propagules 565 are present (a stands for "abundant"). Similar to Eq. (14), we have To derive Eq. (6) we approximate the expectations in Eq. (14) and Eq. (15) by replacing 567 x i and the x j with "effective" mean values as follows Here r and a are the effective means, which are defined in the following subsection.
The effective means r and a 571 The decomposition Eq. (12) is exact and involves no additional assumptions. However this 572 decomposition complicates our approximation procedure because the separate components 573 in Eq. (12) must be approximated in a consistent manner.

574
To illustrate this consistency requirement, suppose that two identical types (same b, c 575 and d) are present, the first with small density l 1 1 and the second with large density 576 l 2 1. In this case, uncontested territories make up a negligible fraction of U ; the first 577 type's territorial acquisition is almost entirely due to ∆ r n 1 ; and the second type's territorial 578 acquisition is almost entirely due to ∆ a n 2 . For consistency, the approximate per-capita 579 growth rates in (16) and (17) must be equal ∆ r n 1 /m 1 = ∆ a n 2 /m 2 . Even small violations 580 of this consistency condition would mean exponential growth of one type relative to the 581 other. This behavior is pathological, because any single-type population can be arbitrarily 582 partitioned into identical rare and common subtypes. Thus, predicted growth or decline 583 would depend on an arbitrary assignment of rarity.

588
Similarly, j x j a ≈ l 2 in Eq. (17), and so ∆ a n 2 /m 2 ≈ 1/l 2 . Thus, the rare type would be 589 predicted to decline in frequency even though it has identical traits.

590
This pathological behavior occurs because the expected total density of propagules in the 591 respective groups of territories are different: As a result, the rare type's behavior is approximated as though it experiences more intense 593 lottery competition than the common type, which cannot be the case since the two types are 594 identical. The effective means must thus be taken in a way that ensures that the expected 595 total propagule density is the same in Eq. (16) and Eq. (17). 596 We achieve this as follows. For nonfocal types j = i, we separately evaluate the X-597 dependence of the conditional dispersal probabilities to ensure that X has the same distri-598 bution for both r and a . Specifically, we assume that X follows a Poisson distribution 599 with rate parameter L, conditional on X ≥ 2; this distribution will be denoted P (X|X ≥ 2).

600
However, for the focal type i, we use the exact conditional dispersal distributions p to cal-601 culate the effective mean, 602 x i r = 1, As we will see, these effective means are straightforward to calculate analytically, and ensure 603 that the expected total propagule density x i + j =i x j is the same in Eq. (16) and Eq. (17).

604
Starting with Eq. (16), we only need to evaluate x j r since x i r = 1. To evaluate the X-dependence separately, we first hold X fixed to obtain The right hand side is obtained by observing that the sum on the left is the expected number of propagules with type j that will be found in a territory which received X − 1 nonfocal propagules in total. We then take the expectation with respect to P (X|X ≥ 2) to give where the last line follows from P (X|X ≥ 2) = 1 1−(1+L)e −L P (X) and ∞ X=2 P (X)(X − 1) = ∞ X=1 P (X)(X − 1) = L − 1 + e −L . Substituting Eqs. (16) and (20) into Eq. (14), we obtain where R i is defined in Eq. (7).

607
Turning now to Eq. (17), from Eq. (18) the mean focal abundance is For nonfocal types j = i, we have analogously to Eq. (19), Again taking the expectation with respect to P (X|X ≥ 2) yields Combining these results with Eqs. (15) and (17), we obtain where A i is defined in Eq. (7).
It is easily verified from Eqs. (20), (22) and (24) that the total expected propagule 610 density is the same in in Eq. (16) and Eq. (17) i.e. x i r + j =i x j r = x i a + j =i x j a = 611 X P (X|X≥2) . As a result, Eq. (6) satisfies the consistency requirement (see Fig. 9).

613
Having derived the approximation Eq. (6), we now evaluate its domain of validity. Eq. (6) 614 relies on ignoring the fluctuations in x i and x j , such that we can replace them with 615 constant effective mean values. To justify this, we show that the standard deviations 616 σ p(x|x i =1,X i ≥1) ( j =i c j x j ) and σ p(x|x i ≥2) ( j c j x j ) are small compared to the corresponding 617 means j =i c j x j p(x|x i =1,X i ≥1) and j c j x j p(x|x i ≥2) in Eqs. (16) and (17). This result means 618 that using the exact distributions p(x|x i = 1, X i ≥ 1) and p(x|x i ≥ 2) for the effective means 619 would produce an accurate approximation of the components in (12) (though, as we have 620 seen, not a consistent one). It is then clear that the effective means derived in the previous 621 section will also give an accurate approximation since their magnitudes are similar to the 622 exact means; this is obvious from the fact that the expected total number of propagules is 623 of order max{L, 2} in both cases. 624 We first consider the means and standard deviations in Eq. (16).
We have x j p(x|x i =1,X i ≥1) = l j /C, where C = 1 − e −(L−l i ) , and the corresponding variances and covariances are given by and Note that 1 − 1/C is negative because C < 1. Decomposing the variance in j =i c j x j , we obtain Eq. (29) shows that, when the c j have similar magnitudes (their ratios are of order one),

627
Eq. (16) is an excellent approximation. The right hand side of Eq. (29) is then approximately , which is small for both low and high nonfocal densities. The 629 worst case scenario occurs when L − l i is of order one, and it can be directly verified that 630 Eq. (16) is then still a good approximation (see Fig. 9).

631
Turning to Eq. (17), all covariances between nonfocal types are now zero, so that for the focal type we have where D = 1 − (1 + l i )e −l i , and Similarly to Eq. (29), the right hand side of Eq. (31) is small for both low and high nonfocal 636 densities provided that the c j have similar magnitudes. Again, the worst case scenario occurs 637 when l i and L − l i are of order 1, but Eq. (17) is still a good approximation in this case 638 (Fig. 9).

639
In both Eqs. (29) and (31), the standard deviation in j =i c j x j can be large relative to 640 its mean if some of the c j are much larger than the others. Specifically, in the presence of 641 a rare, strong competitor (c j l j c j l j for all other nonfocal types j , and l j 1), then the 642 right hand side of Eqs. (29) and (31) can be large and we cannot make the replacement 643 Eq. (16). Fig. 9 shows the breakdown of the effective mean approximation when there are 644 large differences in c. have the same equilibrium density (for a related discussion on the density-and frequency-650 dependence of selection in the Lotka-Volterra model, see (Smouse, 1976;Mallet, 2012)).

651
We assume equal effects of crowding within types α 11 = α 22 = α intra and N = 1/α intra and check whether it is then possible for dN dt to be zero in the sweep (n 1 , n 2 = 0). Substituting  6) with simulations, and also with the naive r = p(x|x i =1,X i ≥1) and a = p(x|x i ≥2) approximation, as a function of the relative c difference between two types. Eq. (6) breaks down in the presence of large c differences. The inset shows the pathology of the naive approximation -growth rates for rare and common types are not equal in the neutral case c 1 = c 2 . Simulation procedure is the same as in Fig. 3, with U = 10 5 .
To get some intuition for Eq. (33), suppose that a mutant arises with improved competitive 653 ability but identical intrinsic growth rate and equilibrium density (r 1 = r 2 and α 11 = α 22 ).

654
This could represent a mutation to an interference competition trait, for example (Gill, 655 1974). Then, according the above condition, for N to remain constant over the sweep, the 656 mutant must find the wildtype more tolerable than itself by exactly the same amount that 657 the wildtype finds the mutant less tolerable than itself.

658
Even if we persuaded ourselves that this balance of inter-type interactions is plausible 659 in some circumstances, when multiple types are present the requirement for constant N 660 becomes 661 ij r i (α intra − α ij )p i p j = 0, which depends on frequency and thus cannot be satisfied in general for constant inter-type 662 coefficients α ij . Therefore, Lotka-Volterra selection will generally involve non-constant N .
In section "Density-regulating traits under strong selection" we argued that the density-

668
For simplicity, we introduce the notation D = N/T and assume that D is small. We can thus make the approximation 1 − e −bD ≈ bD and f (b, N ) ≈ b(1 − D). We expect this to be a conservative approximation based on the worst case scenario, because N is most sensitive to an increase in b in this low-density linear regime. We first calculate the value of f (b, N ) at the halfway point in a sweep, where the halfway point is estimated with simple linear averages for b and N . The sweep is driven by a b variant with b 2 = b 1 (1 + ), and we denote the initial and final densities by D 1 and D 2 respectively, where we have . We obtain Dividing by d 1 − 1, the proportional deviation in f (N ) at the midpoint of the sweep is where we have used the Taylor expansion 1 1+ = 1 − + 2 − 3 + . . ..

669
By contrast, for a δ sweep in Eq. (11), the density-dependent term N increases by a 670 factor of 1 1− = 1 + + 2 + . . .. Thus, the deviations in f (N ) are an order of magnitude smaller than those shown in Fig. (6).