Significance Ecocultural niche modeling and radiocarbon dating suggest a causal role for interspecific competition in the extinction of Neanderthals. Most archaeologists argue that the advantage to modern humans lay in a higher culture level (a sizable minority dispute this view). Competition between the two species may have occurred when a modern human propagule entered a region occupied by a larger Neanderthal population. We present a model for this replacement, stressing the importance of the founder effect. Our findings shed light on the disappearance of the Neanderthals, showing that endogenous factors such as relative culture level, rather than such extrinsic factors as epidemics or climate change, could have caused the eventual exclusion of a comparatively larger population by an initially smaller one.

Abstract Archaeologists argue that the replacement of Neanderthals by modern humans was driven by interspecific competition due to a difference in culture level. To assess the cogency of this argument, we construct and analyze an interspecific cultural competition model based on the Lotka−Volterra model, which is widely used in ecology, but which incorporates the culture level of a species as a variable interacting with population size. We investigate the conditions under which a difference in culture level between cognitively equivalent species, or alternatively a difference in underlying learning ability, may produce competitive exclusion of a comparatively (although not absolutely) large local Neanderthal population by an initially smaller modern human population. We find, in particular, that this competitive exclusion is more likely to occur when population growth occurs on a shorter timescale than cultural change, or when the competition coefficients of the Lotka−Volterra model depend on the difference in the culture levels of the interacting species.

Neanderthals are a human species (or subspecies) that went extinct, after making a small contribution to the modern human genome (1, 2). Hypotheses for the Neanderthal extinction and their replacement by modern humans, in particular as recorded in Europe, can be classified into those emphasizing competition with modern humans and those arguing that interspecific competition was of minor relevance. Among the latter are the climate change (3) and epidemic/endemic (4) hypotheses. However, an ecocultural niche modeling study has shown that Neanderthals and modern humans exploited similar niches in Europe (5), which, together with a recent reassessment of European Paleolithic chronology showing significant spatiotemporal overlap of the two species (6), suggests a major role for interspecific competition in the demise of the Neanderthals.

Replacement of one species (or population) by another is ultimately a matter of numbers. One competing species survives while the other is reduced to, or approaches, zero in size. In the classical Lotka−Volterra model of interspecific competition, this process is called competitive exclusion (7). If Neanderthals were indeed outcompeted by modern humans, the question arises: Wherein lay the advantage to the latter species? Many suggestions have been made, including better tools (8), better clothing (9, 10), and better economic organization (11). These hypotheses share the premise that modern humans were culturally more advanced than the coeval Neanderthals.

The purpose of our paper is threefold. First, we extend the Lotka−Volterra-type model of interspecific competition by incorporating the “culture level” of a species as a variable that interacts with population size (12, 13). Here, culture level may be interpreted as the number of cultural traits, toolkit size, toolkit sophistication, etc. Although, as noted above, many anthropological and archaeological discussions invoke interspecific cultural competition, there is, to the best of our knowledge, no mathematical theory of this ecocultural process. A mechanistic resource competition model is difficult to justify at present, because there is a limited understanding of “what the species are competing for… [or] how they compete” (14). Second, we use our interspecific cultural competition model to explore, analytically and numerically, the possibility that a difference in culture level, or in underlying learning ability, may produce competitive exclusion of a comparatively (although not absolutely) large regional (Neanderthal) population by an initially smaller (modern human) one. Third, we assume the competition coefficients of the Lotka−Volterra model to depend explicitly on the difference in the culture levels of the interacting species (rather than to be constants) and ask how this modification affects the invasion and subsequent dynamics.

Dependence of the culture/technology level of a human population on its size has been the focus of many theoretical (15⇓⇓⇓⇓⇓–21) as well as psychological (22⇓–24), archaeological (25, 26), and ethnological (27⇓⇓–30) studies. However, the coupled dynamics of population size and culture level, where both quantities are treated as variables, has received less theoretical attention (12, 13, 31, 32).

Taking refs. 12 and 13 as the point of departure, we extend previous treatments by introducing two such populations in direct competition with each other in the Lotka−Volterra framework. The two populations are described in terms of their size, N i (≥0), their culture level z i (≥0), i (=1, 2), and parameters to be defined below. We ask whether a population can be replaced by an initially smaller one, which has an advantage in culture level or in learning ability. This ecological perspective on the competition between “size−culture profiles” may inform ongoing debate on the replacement of Neanderthals by modern humans.

Theoretical Model Our model is a four-variable system of coupled differential equations, N ˙ i = r i N i ( 1 − N i + b i j N j M i ( z i ) ) ; [1] z ˙ i = − γ i z i + δ i N i , [2]where i = 1,2 , j ≠ i , and the super dot indicates differentiation with respect to time. Eq. 1 entails logistic growth of population i subject to competition from population j, where b i j ( > 0 ) is the competition coefficient. Population i has carrying capacity M i ( z i ) , which is assumed to be a monotone nondecreasing function of the culture level (see below; Eq. 1 with M i ( z i ) constant reduces to the Lotka−Volterra model). Eq. 2 was originally introduced in ref. 12 as a minimal model for the effect of population size on culture level. Parameter δ i ( > 0 ) measures the per capita input rate of innovations that upgrade culture level (i.e., make z i more positive); the total innovation rate is therefore proportional to population size. Parameter γ i ( > 0 ) gives the decay rate of culture level due to the infidelity of social learning (and random drift). Larger values of δ i and smaller values of γ i indicate higher learning ability. The timescales of demographic and cultural change are determined by r i and γ i , respectively. For the carrying capacity of population i, we assume the step function M i ( z i ) = { K i z i < z i ∗ K i + D i z i ≥ z i ∗ , [3]where K i > 0 and D i > 0 . That is, M i ( z i ) undergoes a discontinuous jump when culture level crosses a critical value z i ∗ ( > 0 ). Eq. 3 can be made slightly more general by interpolating a gradual continuous transition between the lower carrying capacity, K i , and the higher one, K i + D i . We call this a “ramp” function, and it is analyzed in SI Text.

Discussion It is likely that Neanderthals went extinct and were replaced by modern humans due to interspecific competition for overlapping resources. Modern humans are believed to have gradually expanded their range into areas inhabited by Neanderthals (and other archaic humans) by a process of iterative propagule formation. This serial founder scenario receives support from a genetic study showing a regular reduction in heterozygosity with distance from a putative origin in Africa (37). The size of each such propagule may have been about 0.09–0.18 that of the parental population (34). These considerations suggest that a modern human group would have been smaller than a Neanderthal one at initial contact. Our paper reports the theoretical conditions under which such a numerical disadvantage could have been more than compensated by an advantage in culture level or learning ability. Note that in ref. 5, the term “ecocultural” refers to the relationship between environmental niches occupied by Neanderthals and modern humans, overlap in which may have led to competition. In deriving these theoretical conditions, we have made two critical assumptions, which we now discuss. First, we have assumed that the culture level of the invading modern human population at first contact would have shown systematic positive deviations from the quasi-equilibrium value. Specifically, we assume that propagule formation does not entail a significant loss of cultural traits due to the founder effect, as it may for genetic variation. The fundamental difference between genetic and cultural transmission in humans is that an individual inherits only a small subset of the genetic variation present in the population into which he/she is born (unless that population is highly inbred), but there is no such intrinsic limitation on the acquisition of cultural traits (38). A caveat that immediately comes to mind here is division of labor. However, division of labor may not be pronounced in hunter−gatherer societies except between the sexes, and a viable propagule that comprises both reproductive males and females should initially lack few of the cultural traits in its parental population, and its culture level may therefore approximate that of the parental population. This situation may be maintained if time between propagule formation and first contact is short. (Eventually, quasi-equilibrium will be reached according to Eq. 2, but, by then, the propagule will also have grown in size.) On the other hand, expert makers of sophisticated artifacts are likely few, so the cultural founder effect (and hence random cultural drift) cannot be completely neglected (39). Second, we have assumed that the timescales of demographic and cultural change may differ, specifically that the potential rate of population growth may be an order of magnitude higher than the potential rate of cultural change. Richerson et al. suggest that this may be true for the Late Pleistocene and most of the Holocene (32). In extant hunter−gatherers, the intrinsic growth rate ( r i in our model) is of the order of 1% per year (40, 41). The realized growth rate during the Paleolithic, however, was likely much lower (42). We know of no direct evidence bearing on the cultural decay rate ( γ i ) or the per capita upgrade innovation rate ( δ i ) in hunter−gatherers. Here we cite two observations relevant to our assumption. One is the so-called “Tasmanian effect” in which a small isolated population loses many valuable skills and technologies. Because the reduction in the culture level of the eponymous Tasmanian hunter−gatherers occurred over a period of 5,000 y during the Holocene (16), we infer that the realized rate of cultural change was low. However, in the framework of our model, it is not clear whether this was because cultural decay was partially balanced by countervailing innovation—i.e., the first and second terms on the right-hand side of Eq. 2 were perhaps large but of similar magnitude—or because the cultural decay rate itself was small. The other observation concerns the variation in Acheulean handaxe dimensions produced (by hominids ancestral to both Neanderthals and modern humans) over a time span of 1.2 million years. Based on a simulation model, Kempe et al. suggest that this is consistent with a copying error rate of 0.17% per generation, which is low compared with the intrinsic population growth rate and supports our assumption (43). The error rate associated with simple social learning has been estimated in laboratory transmission chain experiments (with extant humans as subjects). In one such experiment (43), each participant in a transmission chain was tasked to view and faithfully reproduce, on an iPad screen, the size of a handaxe image made by the previous participant. The estimated error rate was of the order of 3% per act of copying. Kempe et al. suggest possible reasons for this major discrepancy; for example, unlike the Paleolithic knappers, the experimental subjects could not actually handle their artifacts and thereby reduce the copying error rate. These assumptions (minimal cultural founder effect, different timescales of demographic and cultural change) were not invoked in our analysis of a difference in learning ability. We found that replacement of Neanderthals by modern humans was assured, given a large advantage in learning ability to the latter (inequality 15) and provided an auxiliary condition was also met. However, it may be unrealistic to assume an innate cognitive difference of this magnitude between these two closely related species. A smaller difference in learning ability may be sufficient to drive the replacement process, if the assumptions on cultural dynamics discussed immediately above apply even partially. Although our model defined by Eqs. 1−3 is simple, it preserves the qualitatively important features arising from the interaction between population size and culture level that are seen in more detailed models (13). However, two other assumptions also made in the interest of simplicity require comment here. First, we have neglected cultural/technological transfer between the competing populations. Acculturation of European Neanderthals by modern humans has been a hotly debated issue (44, 45), and, in eastern Eurasia, there is strong evidence for archaeological continuity, perhaps due to the reverse acculturation of incoming modern humans by the resident archaic humans (46, 47). Second, we have ignored the demographic and cultural consequences of interbreeding. Introgression of Neanderthal genes is small in scale (1, 2) but may have affected male fertility (48). Clearly, these considerations must be kept in mind. The replacement process predicted by our ecocultural model can be “self-perpetuating.” That is, given moderate competition ( b 12 = b 21 = b ≈ 0.5 ), a modern human population that has replaced a regional Neanderthal population will be large and at a high culture level (e.g., Fig. 2A, Middle). Hence, it will be in a position to generate a new propagule to invade the next area still occupied by Neanderthals. The resulting spatial effects could also have important consequences.

SI Text Local Stability with Constant Competition Coefficients. Assume the ramp function M i ( z i ) = { K i if z i < z i ∗ K i + D i ( z i − z i ∗ ) z i ∗ ∗ − z i ∗ if z i ∗ ≤ z i < z i ∗ ∗ K i + D i if z i ∗ ∗ ≤ z i for the carrying capacity of population i (= 1 or 2). This is slightly more general than Eq. 3, and reduces to it when z i ∗ = z i ∗ ∗ . For an internal equilibrium, the Jacobian is [ − r 1 N ^ 1 M ^ 1 − r 1 N ^ 1 M ^ 1 b 12 r 1 N ^ 1 M ^ 1 α 1 0 − r 2 N ^ 2 M ^ 2 b 21 − r 2 N ^ 2 M ^ 2 0 r 2 N ^ 2 M ^ 2 α 2 δ 1 0 − γ 1 0 0 δ 2 0 − γ 2 ] , where M ^ i = M i ( z ^ i ) , α i = D i / ( z i ∗ ∗ − z i ∗ ) if z i ∗ < z ^ i < z i ∗ ∗ , and α i = 0 if z ^ i < z i ∗ or z ^ i > z i ∗ ∗ . Note that α i = 0 always holds for the step function 3. For an internal equilibrium z ^ 1 , z ^ 2 such that α 1 = α 2 = 0 , the Jacobian is reducible yielding the two eigenvalues − γ 1 , − γ 2 , and the characteristic quadratic g ( λ ) = λ 2 + ( r 1 N ^ 1 M ^ 1 + r 2 N ^ 2 M ^ 2 ) λ + r 1 N ^ 1 M ^ 1 r 2 N ^ 2 M ^ 2 ( 1 − b 12 b 21 ) . The zeros of the quadratic are both negative or have negative real part if and only if b 12 b 21 < 1 . For an edge equilibrium N ^ i = M ^ i > 0 , z ^ 1 ( ≠ z 1 ∗ , z 1 ∗ ∗ ) > 0 , N ^ 2 = 0 , z ^ 2 = 0 , the Jacobian is [ − r 1 − r 1 b 12 r 1 α 1 0 0 r 2 ( 1 − M ^ 1 M ^ 2 b 21 ) 0 0 δ 1 0 − γ 1 0 0 δ 2 0 − γ 2 ] . The eigenvalues are r 2 ( 1 − ( M ^ 1 / M ^ 2 ) b 21 ) , − γ 2 , and the zeros of the characteristic quadratic h ( λ ) = λ 2 + ( r 1 + γ 1 ) λ + r 1 ( γ 1 − δ 1 α 1 ) . Hence, the edge equilibrium is locally stable if M ^ 1 b 21 > M ^ 2 and γ 1 > δ 1 α 1 , where the latter condition is limiting only for an edge equilibrium situated on the ramp, i.e., z 1 ∗ < z ^ 1 < z 1 ∗∗ . Effect of a Large Difference in Learning Ability, γ i / δ i . Assume that the two populations have identical competition and growth parameters, b 12 = b 21 = b , r 1 = r 2 = r , K 1 = K 2 = K , D 1 = D 2 = D , z 1 ∗ = z 2 ∗ = z ∗ , where the carrying capacity is given by the step function 3; however, they differ in their learning abilities, specifically in the ratio γ i / δ i , which satisfy ( γ 2 / δ 2 ) z ∗ < K / ( 1 + b ) < K + D < ( γ 1 / δ 1 ) z ∗ . In addition, we assume K / ( K + D ) < b < 1. As shown in the next paragraph, these assumptions entail that the high edge equilibrium N ^ 1 = 0 , N ^ 2 = K + D , z ^ 1 = 0 , z ^ 2 = ( δ 2 / γ 2 ) ( K + D ) , corresponding to competitive exclusion of population 1, is the sole locally stable point equilibrium. The low symmetrical internal equilibrium N ^ 1 = K / ( 1 + b ) , N ^ 2 = K / ( 1 + b ) does not exist because ( γ 2 / δ 2 ) z ∗ < K / ( 1 + b ) by assumption, contrary to the inequality under Eq. 6. The high symmetrical internal equilibrium N ^ 1 = ( K + D ) / ( 1 + b ) , N ^ 2 = ( K + D ) / ( 1 + b ) does not exist because ( K + D ) / ( 1 + b ) < K + D < ( γ 1 / δ 1 ) z ∗ by assumption, contrary to the inequality under Eq. 7. The asymmetrical internal equilibria N ^ 1 = [ K − b ( K + D ) ] / ( 1 − b 2 ) , N ^ 2 = ( K + D − b K ) / ( 1 − b 2 ) and N ^ 1 = ( K + D − b K ) / ( 1 − b 2 ) , N ^ 2 = [ K − b ( K + D ) ] / ( 1 − b 2 ) do not exist because b > K / ( K + D ) by assumption, contrary to 10. The low edge equilibrium N ^ 1 = 0 , N ^ 2 = K exists because K < ( γ 1 / δ 1 ) z ∗ but is unstable because b < 1 by assumption. The low edge equilibrium N ^ 1 = 0 , N ^ 2 = K does not exist because ( γ 2 / δ 2 ) z ∗ < K by assumption, contrary to the inequality under Eq. 12. The high edge equilibrium N ^ 1 = K + D , N ^ 2 = 0 does not exist because K + D < ( γ 1 / δ 1 ) z ∗ by assumption, contrary to the inequality under Eq. 13. Finally, the high edge equilibrium N ^ 1 = 0 , N ^ 2 = K + D exists because ( γ 2 / δ 2 ) z ∗ < K + D and is locally stable because b > K / ( K + D ) , both by assumption. Local Stability with Feedback. Assume the ramp function model for the carrying capacity, equal parameter values in the two populations, and competition coefficients given by Eq. 16 that depend on the difference in culture levels. The characteristic polynomial is φ ( λ ) = | − r N ^ 1 M ^ 1 − λ − r N ^ 1 M ^ 1 b ^ 12 r N ^ 1 M ^ 1 ( α + b 0 ϵ N ^ 2 ) − r N ^ 1 M ^ 1 b 0 ϵ N ^ 2 − r N ^ 2 M ^ 2 b ^ 21 − r N ^ 2 M ^ 2 − λ − r N ^ 2 M ^ 2 b 0 ϵ N ^ 1 r N ^ 2 M ^ 2 ( α + b 0 ϵ N ^ 1 ) δ 0 − γ − λ 0 0 δ 0 − γ − λ | , where b ^ i j = b 0 ( 1 + ϵ ( z ^ j − z ^ i ) ) . We now specialize to the case of α = 0 (e.g., the step function model). Then, addition of the third column to the fourth, followed by subtraction of the fourth row from the third, yields φ ( λ ) = | − r N ^ 1 M ^ 1 − λ − r N ^ 1 M ^ 1 b ^ 12 r N ^ 2 N ^ 1 M ^ 1 b 0 ϵ 0 − r N ^ 2 M ^ 2 b ^ 21 − r N ^ 2 M ^ 2 − λ r N ^ 2 N ^ 1 M ^ 1 b 0 ϵ 0 δ 0 − γ − λ 0 0 δ 0 − γ − λ | . Hence, φ ( λ ) = ( γ + λ ) f ( λ ) , where f ( λ ) = ( γ + λ ) [ r N ^ 1 M ^ 1 r N ^ 2 M ^ 2 ( 1 − b ^ 12 b ^ 21 ) + ( r N ^ 1 M ^ 1 + r N ^ 2 M ^ 2 ) λ + λ 2 ] − b 0 δ ϵ [ r N ^ 1 M ^ 1 r N ^ 1 N ^ 2 M ^ 2 ( 1 + b ^ 12 ) + r N ^ 2 M ^ 2 r N ^ 1 N ^ 2 M ^ 1 ( 1 + b ^ 21 ) + ( r N ^ 1 N ^ 2 M ^ 1 + r N ^ 1 N ^ 2 M ^ 2 ) λ ] is a cubic function of λ. For a symmetrical internal equilibrium where 0 < z ^ 1 = z ^ 2 < z ∗ or z ∗ ∗ < z ^ 1 = z ^ 2 , we can set b ^ i j = b 0 , and M ^ 1 = M ^ 2 = M where M = K if 0 < z ^ 1 = z ^ 2 < z ∗ and M = K + D if z ∗ ∗ < z ^ 1 = z ^ 2 . Provided b 0 ≠ 1 , we also have N ^ 1 = N ^ 2 = M / ( 1 + b 0 ) . Hence, the cubic reduces to f ( λ ) = ( λ + r ) { λ 2 + [ γ + r ( 1 − b 0 ) 1 + b 0 ] λ + r 1 + b 0 [ γ ( 1 − b 0 ) − 2 b 0 δ ϵ M 1 + b 0 ] } . Both zeros of the quadratic factor are negative or have negative real part if b 0 < 1 and ϵ < γ δ 1 − b 0 2 2 b 0 M , and there is local stability when both inequalities are satisfied. The latter inequality entails that the critical value of ε is a hyperbolic function of b 0 . We deal with asymmetric internal equilibria in Asymmetrical Internal Equilibria with Feedback. The Jacobian for an edge equilibrium of the ramp function model N ^ 1 = M ^ 1 > 0 , z ^ 1 = ( δ / γ ) M ^ 1 ( ≠ z 1 ∗ , z 1 ∗ ∗ ) , N ^ 2 = 0 , z ^ 2 = 0 is [ − r − r b 0 ( 1 − ϵ z ^ 1 ) r α 0 0 r ( 1 − b 0 ( 1 + ϵ z ^ 1 ) M ^ 1 K ) 0 0 δ 0 γ 0 0 δ 0 − γ ] . The eigenvalues are r ( 1 − [ b 0 ( 1 + ϵ z ^ 1 ) M ^ 1 ] / K ) , − γ , and the zeros of the characteristic quadratic λ 2 + ( r + γ ) λ + r ( γ − δ α ) , and there is local stability if γ > δ α and b 0 δ ϵ γ M ^ 1 2 + b 0 M ^ 1 − K > 0. Hence, for α = 0 , both low edge equilibria are locally stable if ϵ > γ δ 1 K ( 1 b 0 − 1 ) . Similarly, both high edge equilibria are locally stable if ϵ > γ δ K ( K + D ) 2 ( 1 b 0 − K + D K ) . Asymmetrical Internal Equilibria with Feedback. As before, we assume the ramp function model for the carrying capacity, equal parameter values in the two populations, and competition coefficients given by Eq. 16 that depend on the difference in culture levels. For an internal equilibrium, we have, in general, φ ( 0 ) = r 2 γ N ^ 1 N ^ 2 M ^ 1 M ^ 2 ( γ ( 1 − b ^ 12 b ^ 21 ) − b 0 δ ϵ ( N ^ 1 ( 1 + b ^ 12 ) + N ^ 2 ( 1 + b ^ 21 ) ) ) . Note that φ ( 0 ) and f ( 0 ) are always of the same sign. To obtain an approximate condition for the local stability of an asymmetrical internal equilibrium where 0 < z ^ 1 < z ∗ ≤ z ∗ ∗ < z ^ 2 , M ^ 1 = K , and M ^ 2 = K + D , we assume that ε is small. An internal equilibrium with ϵ = 0 is locally stable if b 12 b 21 < 1 or equivalently if φ ( 0 ) > 0 , and the latter criterion should continue to apply while ε remains small. Hence, the asymmetrical internal equilibrium is locally stable to first order in ε if γ ( 1 − b 0 2 ) − b 0 δ ϵ ( 1 + b 0 ) [ K − b 0 ( K + D ) 1 − b 0 2 + K + D − b 0 K 1 − b 0 2 ] > 0 , which reduces to ϵ < γ δ 1 − b 0 2 b 0 ( 2 K + D ) . In addition, there exists a class of asymmetrical internal equilibria not found when the competition coefficients are constants. Such equilibria are of four kinds: 0 < z ^ 1 < z ^ 2 < z ∗ and M ^ 1 = M ^ 2 = K , 0 < z ^ 2 < z ^ 1 < z ∗ and M ^ 1 = M ^ 2 = K , 0 < z ∗ ∗ < z ^ 1 < z ^ 2 and M ^ 1 = M ^ 2 = K + D , or 0 < z ∗ ∗ < z ^ 2 < z ^ 1 and M ^ 1 = M ^ 2 = K + D . Assuming z ^ 1 < z ^ 2 and setting M ^ 1 = M ^ 2 = M without loss of generality, we can obtain explicit solutions for N ^ 1 and N ^ 2 as follows. At equilibrium, we have N ^ 1 + b 0 ( 1 + ϵ b 0 γ ( N ^ 2 − N ^ 1 ) ) N ^ 2 = M , N ^ 2 + b 0 ( 1 + ϵ b 0 γ ( N ^ 1 − N ^ 2 ) ) N ^ 1 = M . Transforming variables and solving yields N ^ 1 + N ^ 2 = γ ϵ δ 1 − b 0 b 0 , N ^ 2 − N ^ 1 = γ b 0 ϵ δ b 0 2 − 1 + 2 b 0 ϵ δ M γ . Hence N ^ 1 = γ 2 b 0 ϵ δ [ 1 − b 0 − b 0 2 − 1 + 2 b 0 ϵ δ M γ ] , N ^ 2 = γ 2 b 0 ϵ δ [ 1 − b 0 + b 0 2 − 1 + 2 b 0 ϵ δ M γ ] . Validity of this equilibrium requires that the argument of the square root be positive, the solutions be positive, and the competition coefficients at equilibrium be nonnegative. These three conditions can be written as ϵ > γ δ 1 − b 0 2 2 b 0 M , ϵ < γ δ 1 − b 0 b 0 M , ϵ ≤ γ δ 1 2 b 0 M , respectively, where the second inequality can be satisfied only if b 0 < 1 . When the first inequality is replaced by an equality, the solution reduces to the symmetrical internal equilibrium N ^ 1 = N ^ 2 = M / ( 1 + b 0 ) . Similarly, replacing the second inequality by an equality yields the edge equilibrium N ^ 1 = 0 , N ^ 2 = M . Local stability is difficult to show, in general. Here, we use a perturbation argument to prove that an asymmetrical internal equilibria of this class is locally stable, provided it is located in the parametric neighborhood of either a symmetrical internal equilibrium or an edge equilibrium. Substituting for N ^ 1 and N ^ 2 , we obtain, after some algebra, φ ( 0 ) = r 2 γ 4 b 0 2 δ 2 ϵ 2 M 2 ( b 0 2 − 1 + 2 b 0 δ ϵ M γ ) ( 1 − b 0 − b 0 δ ϵ M γ ) . The first and second inequalities above that are required for existence are exactly the conditions for the first and second factors in parentheses on the right-hand side, b 0 2 − 1 + ( 2 b 0 δ ϵ M / γ ) ≡ ξ and 1 − b 0 − ( b 0 δ ϵ M / γ ) ≡ η , respectively, to be positive. Hence, existence entails φ ( 0 ) > 0 . Next we note that when ξ = 0 , the cubic reduces to f ( λ ) = λ ( r + λ ) ( γ + r ( 1 − b 0 ) 1 + b 0 + λ ) . One eigenvalue is zero, and the other two are negative. Hence, when we perturb ξ so that f ( 0 ) turns slightly positive, the zero eigenvalue will become negative, while the two originally negative eigenvalues will remain negative or become complex conjugates with negative real part. Similarly, when η = 0 , the cubic reduces to f ( λ ) = λ ( r + λ ) ( γ + λ ) , and the same argument holds when we perturb η.

Acknowledgments This research was supported in part by a National Science Foundation Graduate Research Fellowship (to W.G.), the John Templeton Foundation (M.W.F.), and Monbukagakusho Grant 22101004 (to K.A.).

Footnotes Author contributions: W.G., M.W.F., and K.A. designed research, performed research, contributed new reagents/analytic tools, analyzed data, and wrote the paper.

Reviewers: M.E., University of Stockholm; S.G., Brooklyn College; and F.J.W., University of Groningen.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1524861113/-/DCSupplemental.