An ecocultural model predicts Neanderthal extinction through competition with modern humans

We formulate the question of replacement of Neanderthals by modern humans in terms of the instability of an equilibrium involving the size−culture profile of the former to an invading propagule of the latter that may be smaller in numbers but more advanced culturally or in underlying learning ability. Equilibria of the model are obtained by setting Eqs. 1 and 2 to zero. Local stability can be determined from the Jacobian. We introduce the shorthand notation ${\hat{M}}_i=M_i\left({\hat{z}}_i\right)$ to denote the carrying capacity of population i at equilibrium, which equals either $K_i$ or $K_i+D_i$ for the step function (Eq. 3). Here, we summarize the kinds of equilibria and their local stability conditions for general parameter values when the competition coefficients are constants. Later, we consider the case where $b_{ij}$ is a function of $z_j-z_i$.

First, an internal (i.e., coexistence) equilibrium ${\hat{N}}_1>0$, ${\hat{N}}_2>0$, ${\hat{z}}_1\left(\ne z_1^{\ast }\right)>0$, ${\hat{z}}_2\left(\ne z_2^{\ast }\right)>0$ (given that it exists) is locally stable if as shown in Local Stability with Constant Competition Coefficients. In the Lotka−Volterra model, this condition ensures global stability of the (unique) coexistence equilibrium. Second, an edge equilibrium ${\hat{N}}_i={\hat{M}}_i>0$, ${\hat{N}}_j=0$, ${\hat{z}}_i\left(\ne z_i^{\ast }\right)$, ${\hat{z}}_j=0$ (given that it exists) is locally stable if which is also shown in Local Stability with Constant Competition Coefficients. Finally, the corner equilibrium ${\hat{N}}_1=0$, ${\hat{N}}_2=0$, ${\hat{z}}_1=0$, ${\hat{z}}_2=0$ always exists but is necessarily unstable because $r_i>0$ by assumption.

We first investigate the equilibrium properties of the model (Eqs. 1−3) assuming cognitive equality of Neanderthals and modern humans. Suppose that the populations are equivalent with respect to the cognition parameters, ${\gamma }_1={\gamma }_2=\gamma$ and ${\delta }_1={\delta }_2=\delta$, and, with respect to the competition and growth parameters, $b_{12}=b_{21}=b$, $r_1=r_2=r$, and $M_1\left(z_1\right)=M_2\left(z_2\right)=M\left(z_i\right)$. In particular, this assumption entails that both populations have equivalent learning abilities. Under these assumptions, full characterization of the point equilibria and their stability properties is possible.

There are three different kinds of internal equilibria. If ${\hat{z}}_1\le {\hat{z}}_2<z^{\ast }$ (alternatively, ${\hat{z}}_2\le {\hat{z}}_1<z^{\ast }$), Eq. 3 entails ${\hat{M}}_1={\hat{M}}_2=K$. Hence, solving the simultaneous equations ${\hat{N}}_1+b{\hat{N}}_2=K$, $b{\hat{N}}_1+{\hat{N}}_2=K$ yields Then, from Eq. 2, we have ${\hat{z}}_1={\hat{z}}_2=\left(\delta /\gamma \right)K/\left(1+b\right)$, and the consistency requirement ${\hat{z}}_1\le {\hat{z}}_2<z^{\ast }$ entails that this “low” symmetrical internal equilibrium exists only if $\left(\gamma /\delta \right)z^{\ast }>K/\left(1+b\right)$. Inequality 4 entails local stability of this equilibrium if $b<1$ (and instability if $b>1$).

Similarly, a “high” symmetrical internal equilibrium where $z^{\ast }<{\hat{z}}_1\le {\hat{z}}_2$ exists if $\left(\gamma /\delta \right)z^{\ast }<\left(K+D\right)/\left(1+b\right)$, and is locally stable if $b<1$ (unstable if $b>1$). Explicit values of ${\hat{z}}_1={\hat{z}}_2$ can again be obtained using ${\hat{z}}_i=\left({\delta }_i/{\gamma }_i\right){\hat{N}}_i$ from Eq. 2.

The third kind of internal equilibrium is asymmetrical. Suppose, without loss of generality, ${\hat{z}}_1<z^{\ast }<{\hat{z}}_2$, which entails ${\hat{M}}_1=K$, ${\hat{M}}_2=K+D$. Then, Using the consistency requirement and the condition ${\hat{N}}_2>{\hat{N}}_1>0$, this equilibrium exists and is locally stable if In addition, another locally stable asymmetrical internal equilibrium with ${\hat{N}}_1,{\hat{z}}_1$ and ${\hat{N}}_2,{\hat{z}}_2$ interchanged—the symmetrically placed counterpart to Eq. 8—will exist when inequalities 9 and 10 are satisfied. Thus, four stable coexistence equilibria are possible in this special case of our model. Moreover, they may exist and be locally stable at the same time.

The edge equilibria can also be identified. The “low” edge equilibrium, and its counterpart both exist if $\left(\gamma /\delta \right)z^{\ast }>K$. From inequality 5, both are locally stable if $b>1$. Similarly, the “high” edge equilibrium, and its counterpart both exist if $\left(\gamma /\delta \right)z^{\ast }<K+D$, and are locally stable if inequality 10 is reversed.

Thus, the equilibrium structure of our model is quite complex. An edge equilibrium corresponds to the presence of just one of the populations (population 1, say), and is (locally or globally) stable in the absence of the other (i.e., population 2) (13). However, it may or may not be stable when population 2 is introduced. In the replacement problem, we equate populations 1 and 2 with Neanderthals and modern humans, respectively. We show in Numerical Analysis with Equal Parameter Values that the subsequent dynamics could result in coexistence or in competitive exclusion of population 1, either of which can happen when the initial value of $N_2\left(\equiv N_2^0\right)$ is large enough relative to, but still smaller than, the initial value of $N_1\left(\equiv N_1^0\right)$. The more stringent conditions for competitive exclusion are of particular interest, because these are the conditions that predict the phenomenon that we are trying to explain, namely the replacement of a resident Neanderthal population by invading modern humans.

To illustrate the dynamics of competition between the size−culture profiles of Neanderthals and modern humans, we carry out a numerical analysis under the above simplifying assumptions. We assume the step function model with equal parameter values in the two populations. Because the dynamical system in Eqs. 1−3 is four-dimensional, it is difficult to generate a simultaneous 2D pictorial representation of the basins of attraction for all of the equilibria described above. For simplicity, we focus on the case where the system is initially at the low edge equilibrium (Eq. 11), corresponding to our premise that the Neanderthals in isolation would have formed a comparatively small population at a relatively low culture level. Figs. 1 and 2 illustrate the basins of attraction in the $\left(N_2^0,z_2^0\right)$ plane of initial conditions for population 2 ( $N_2^0$ on the horizontal axis, $z_2^0$ on the vertical axis) (33). They show how these basins of attraction depend on deviations of the initial culture levels, $z_2^0$, from the “quasi-equilibrium” values, $\left(\delta /\gamma \right)N_2^0$ (Eq. 2 with ${\stackrel{\dot{}}{z}}_2=0$).

Positive deviations [ $z_2^0>\left(\delta /\gamma \right)N_2^0$] may arise if population 2 is a propagule derived from an equilibrium population of large size at a high culture level (e.g., Eq. 14) but whose culture level has not yet declined to a value commensurate with its reduced size. Using a “serial founder effect” model, it has been argued that such a propagule may be about 0.09–0.18 the size of the parental population (34). Figs. 1 and 2 show that population 1 can be competitively excluded by an initially smaller population 2, provided the latter maintains an advantage in culture level.

Each colored region in Figs. 1 and 2 corresponds to the basin of attraction for a distinct equilibrium. Seven equilibria exist given the parameter values in Fig. 1. The values of the parameters K and D in the numerical analysis do not represent known carrying capacities; rather, they should be regarded as being measured in some unspecified units. The two symmetrical coexistence equilibria (Eqs. 6 and 7) and the two high edge equilibria (Eqs. 13 and 14), are locally stable, but only two of these—Eqs. 6 and 14—are attractors from the initial conditions considered here. The low edge equilibrium 11, representing the state of the system before the introduction of population 2, is unstable. Hence, population 2 will necessarily invade, and the question is whether convergence occurs to equilibrium 6, resulting in coexistence, or to equilibrium 14, resulting in competitive exclusion. The purple and blue regions in Fig. 1B denote the basins of attraction of the low symmetrical coexistence equilibrium 6 and the high edge equilibrium 14, respectively.

To the left of the white broken line in Fig. 1B, population 2 is initially smaller than population 1. Fig. 1B shows that competitive exclusion of population 1 by population 2 is possible even when the latter is initially smaller than the former. For example, convergence to the high edge equilibrium 14 (black disk in the upper right-hand corner) can occur from an initial population ratio of $N_2^0/N_1^0=0.9$ and an initial culture level of $z_2^0=\delta \left(K+D\right)/\gamma$ (right-hand white cross). If population 2 is initially even smaller, e.g., $N_2^0/N_1^0=0.7$ (left-hand white cross in Fig. 1B), then convergence to the low symmetrical internal equilibrium 6 (black disk in the lower left-hand corner) results. Importantly, our ecocultural model—in contrast to the standard Lotka−Volterra model—predicts the simultaneous existence of locally stable internal and edge equilibria, where the former may act as “traps” hindering competitive exclusion. Fig. 1A shows the corresponding trajectories.

Fig. 1 A and B illustrates the default case where the timescales of demographic and cultural change are equal, specifically $r=\gamma =1$. Fig. 1B suggests that $N_2^0$ must be of the same order of magnitude as $N_1^0$ for competitive exclusion to occur in this case. Fig. 1C shows that the minimum $N_2^0/N_1^0$ required for competitive exclusion is sensitive to our assumptions regarding these timescales. Here, we set $r=1$ and vary γ and δ together, subject to the constraint $\gamma /\delta =2.2$. By keeping $\gamma /\delta$ constant, the existence and stability conditions are unaffected and only the basins of attraction may change. Low rates of cultural decay (and innovation) relative to population growth appear to facilitate competitive exclusion by an initially smaller population with an initial cultural advantage. For example, if $\gamma =0.2$ and $r=1$, a ratio as low as $N_2^0/N_1^0=0.1$ may suffice.

Fig. 2 shows the effects of varying the competition coefficient, b. Fig. 2A (Middle) is equivalent to Fig. 1B. For low values of b satisfying inequality 10, the two asymmetrical internal equilibria, 8 and its counterpart, exist and are locally stable. Thus, all nine corner, edge, and internal equilibria exist for the parameter values in Fig. 2A (Left). However, only two attractors are found, and the green region of Fig. 2A (Left) denotes the basin of attraction of the asymmetrical internal equilibrium 8 (where ${\hat{N}}_2>{\hat{N}}_1$); for the parameter values assumed here, ${\hat{N}}_2\approx 49$ and ${\hat{N}}_1\approx 5$, which is close to replacement. As b increases, the asymmetrical coexistence equilibria undergo transcritical bifurcation into the high edge exclusion equilibria, as shown in the bifurcation diagram Fig. 2B.

Eventually, at even larger values of b, as in Fig. 2A (Right), all four coexistence equilibria are eliminated, while the four edge equilibria become locally stable. Here, the attractors are the low edge equilibria 11 and 12, and the high edge equilibrium 14; the basins of attraction are colored in light yellow, light blue, and blue, respectively. The light yellow region corresponds to noninvasion, whereas the light blue and blue regions both correspond to competitive exclusion of population 1 by population 2. The sizes of the populations at equilibrium, ${\hat{N}}_1,{\hat{N}}_2$, also vary with b, as shown in Fig. 2B.

The jury is still out on whether or not Neanderthals and modern humans differed in cognition (35, 36). Aspects of cognition incorporated into our model are the infidelity of social learning, ${\gamma }_i$, and the upgrade innovation rate ${\delta }_i$. Assume again that the two populations are equivalent with respect to the competition and growth parameters $b_{ij},r_i,M_i\left(z_i\right)$. However, we now assume that they differ in the ratio ${\gamma }_i/{\delta }_i$, smaller values of which indicate higher learning ability. Specifically, we take As shown in Effect of a Large Difference in Learning Ability, inequality 15 together with imply that the high edge equilibrium is the sole locally stable equilibrium. Moreover, if cyclical solutions can be ruled out, as the numerical analysis suggests, this equilibrium is globally stable, entailing competitive exclusion of population 1 (Neanderthals) by population 2 (modern humans) from all initial conditions. We can interpret this result to mean that, no matter how small an invading population of modern humans, given this large advantage in learning ability, they could replace a resident population of Neanderthals, no matter how large. The less stringent inequality obtained by setting $b=0$ in 15 ensures a globally stable small population−low culture level equilibrium for population 1 in isolation and a globally stable large population−high culture level equilibrium for population 2 in isolation (13).

Ecological competition coefficients between populations with different culture levels may be functions of this difference, rather than simple constants. We return to our assumption that the two populations are equivalent with respect to the cognition, competition, and growth parameters [ ${\gamma }_1={\gamma }_2$, ${\delta }_1={\delta }_2$, $r_1=r_2$, $M_1\left(z_1\right)=M_2\left(z_2\right)$]. However, we now assume that the competition coefficients, $b_{ij}$, are not constant but depend on the instantaneous culture levels of the two populations. A simple version of this sets where $b_0$ and ε are positive constants, and $z_j-z_i$ is the signed difference in the culture levels of populations i and j ( $i=\mathrm{1,2}$, $j\ne i$). For Eq. 1 to be meaningful, ε must be chosen small enough that $b_{12}$ and $b_{21}$ both remain positive.

In Eq. 16, $b_{ij}>b_{ji}$ if and only if $z_j>z_i$; that is, the population with the higher culture level has a linear competitive edge. Introduction of Eq. 16 into Eq. 1 induces positive feedback, whereby a population (2, say) that enjoys a small initial cultural advantage (i.e., $z_2>z_1$) is able to compound that advantage by first competing more strongly (because $b_{12}>b_{21}$), which then increases its population size ( $N_2$), and in turn increases its culture level via Eq. 2, which in turn further increases its competitive edge. Hence, we expect and find that this feedback enhances the effect of an initial cultural advantage (Fig. 3A).

The feedback model has the same edge and symmetric coexistence equilibria as occur in the model without feedback. The conditions for the existence of these equilibria are unchanged. The conditions for their local stability are given in Local Stability with Feedback. Fig. 3A shows that the minimum $N_2^0/N_1^0$ necessary for competitive exclusion decreases as ε increases. Fig. 3B is a phase diagram illustrating when different solutions exist and are stable for different values of $\left(b_0,\mathit{ϵ}\right)$. The contours demarcating the various regions are determined analytically by the maximum eigenvalues of the Jacobian matrix associated with these equilibria. In general, as ε or $b_0$ increases, feedback causes coexistence solutions to become unstable in favor of exclusion. This includes the classical Lotka−Volterra case ( $\mathit{ϵ}=0$, horizontal axis in Fig. 3B) where coexistence becomes unstable when $b_0>1$.

Somewhat surprising is the existence and stability of a new class of asymmetrical internal equilibria. These equilibria occur in pairs, either both above or below the step ( ${\hat{z}}_1<{\hat{z}}_2<z^{\ast }$ and ${\hat{z}}_2<{\hat{z}}_1<z^{\ast }$, or $z^{\ast }<{\hat{z}}_1<{\hat{z}}_2$ and $z^{\ast }<{\hat{z}}_2<{\hat{z}}_1$). Asymmetrical Internal Equilibria with Feedback derives analytical conditions for their existence and reports a perturbation analysis suggesting that existence implies local stability. However, determination of local stability proves difficult in general, and we check for local stability by numerically obtaining the eigenvalues of the characteristic polynomial. We find that these solutions are, in fact, locally stable whenever they exist (at least for the parameter values assumed in Fig. 3B, blue and red regions).

Assume the ramp function for the carrying capacity of population $i$ (= 1 or 2). This is slightly more general than Eq. 3, and reduces to it when $z_i^{\ast }=z_i^{\ast \ast }$.

For an internal equilibrium, the Jacobian is where ${\hat{M}}_i=M_i\left({\hat{z}}_i\right)$, ${\alpha }_i=D_i/\left(z_i^{\ast \ast }-z_i^{\ast }\right)$ if $z_i^{\ast }<{\hat{z}}_i<z_i^{\ast \ast }$, and ${\alpha }_i=0$ if ${\hat{z}}_i<z_i^{\ast }$ or ${\hat{z}}_i>z_i^{\ast \ast }$. Note that ${\alpha }_i=0$ always holds for the step function 3.

For an internal equilibrium ${\hat{z}}_1,{\hat{z}}_2$ such that ${\alpha }_1={\alpha }_2=0$, the Jacobian is reducible yielding the two eigenvalues $-{\gamma }_1$, $-{\gamma }_2$, and the characteristic quadratic 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 ${\hat{N}}_i={\hat{M}}_i>0$, ${\hat{z}}_1\left(\ne z_1^{\ast },z_1^{\ast \ast }\right)>0$, ${\hat{N}}_2=0$, ${\hat{z}}_2=0$, the Jacobian is The eigenvalues are $r_2\left(1-\left({\hat{M}}_1/{\hat{M}}_2\right)b_{21}\right)$, $-{\gamma }_2$, and the zeros of the characteristic quadratic Hence, the edge equilibrium is locally stable if ${\hat{M}}_1{b}_{21}>{\hat{M}}_2$ and ${\gamma }_1>{\delta }_1{\alpha }_1$, where the latter condition is limiting only for an edge equilibrium situated on the ramp, i.e., $z_1^{\ast }<{\hat{z}}_1<z_1^{\ast \ast }$.

Assume that the two populations have identical competition and growth parameters, $b_{12}=b_{21}=b$, $r_1=r_2=r$, $K_1=K_2=K$, $D_1=D_2=D$, $z_1^{\ast }=z_2^{\ast }=z^{\ast }$, where the carrying capacity is given by the step function 3; however, they differ in their learning abilities, specifically in the ratio ${\gamma }_i/{\delta }_i$, which satisfy In addition, we assume As shown in the next paragraph, these assumptions entail that the high edge equilibrium corresponding to competitive exclusion of population 1, is the sole locally stable point equilibrium.

The low symmetrical internal equilibrium ${\hat{N}}_1=K/\left(1+b\right)$, ${\hat{N}}_2=K/\left(1+b\right)$ does not exist because $\left({\gamma }_2/{\delta }_2\right)z^{\ast }<K/\left(1+b\right)$ by assumption, contrary to the inequality under Eq. 6. The high symmetrical internal equilibrium ${\hat{N}}_1=\left(K+D\right)/\left(1+b\right)$, ${\hat{N}}_2=\left(K+D\right)/\left(1+b\right)$ does not exist because $\left(K+D\right)/\left(1+b\right)<K+D<\left({\gamma }_1/{\delta }_1\right)z^{\ast }$ by assumption, contrary to the inequality under Eq. 7. The asymmetrical internal equilibria ${\hat{N}}_1=\left[K-b\left(K+D\right)\right]/\left(1-b^2\right)$, ${\hat{N}}_2=\left(K+D-bK\right)/\left(1-b^2\right)$ and ${\hat{N}}_1=\left(K+D-bK\right)/\left(1-b^2\right)$, ${\hat{N}}_2=\left[K-b\left(K+D\right)\right]/(1-b^2)$ do not exist because $b>K/\left(K+D\right)$ by assumption, contrary to 10. The low edge equilibrium ${\hat{N}}_1=0$, ${\hat{N}}_2=K$ exists because $K<\left({\gamma }_1/{\delta }_1\right)z^{\ast }$ but is unstable because $b<1$ by assumption. The low edge equilibrium ${\hat{N}}_1=0$, ${\hat{N}}_2=K$ does not exist because $\left({\gamma }_2/{\delta }_2\right)z^{\ast }<K$ by assumption, contrary to the inequality under Eq. 12. The high edge equilibrium ${\hat{N}}_1=K+D$, ${\hat{N}}_2=0$ does not exist because $K+D<\left({\gamma }_1/{\delta }_1\right)z^{\ast }$ by assumption, contrary to the inequality under Eq. 13. Finally, the high edge equilibrium ${\hat{N}}_1=0$, ${\hat{N}}_2=K+D$ exists because $\left({\gamma }_2/{\delta }_2\right)z^{\ast }<K+D$ and is locally stable because $b>K/\left(K+D\right)$, both by assumption.

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 where We now specialize to the case of $\alpha =0$ (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 Hence, where is a cubic function of λ.

For a symmetrical internal equilibrium where $0<{\hat{z}}_1={\hat{z}}_2<z^{\ast }$ or $z^{\ast \ast }<{\hat{z}}_1={\hat{z}}_2$, we can set ${\hat{b}}_{ij}=b_0$, and ${\hat{M}}_1={\hat{M}}_2=M$ where $M=K$ if $0<{\hat{z}}_1={\hat{z}}_2<z^{\ast }$ and $M=K+D$ if $z^{\ast \mathrm{\ast }}<{\hat{z}}_1={\hat{z}}_2$. Provided $b_0\ne 1$, we also have ${\hat{N}}_1={\hat{N}}_2=M/\left(1+b_0\right)$. Hence, the cubic reduces to Both zeros of the quadratic factor are negative or have negative real part if $b_0<1$ and 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 ${\hat{N}}_1={\hat{M}}_1>0$, ${\hat{z}}_1=\left(\delta /\gamma \right){\hat{M}}_1\left(\ne z_1^{\ast },z_1^{\ast \ast }\right)$, ${\hat{N}}_2=0$, ${\hat{z}}_2=0$ is The eigenvalues are $r\left(1-\left[b_0\left(1+\mathit{ϵ}{\hat{z}}_1\right){\hat{M}}_1\right]/K\right)$, $-\gamma$, and the zeros of the characteristic quadratic ${\lambda }^2+\left(r+\gamma \right)\lambda +r\left(\gamma -\delta \alpha \right)$, and there is local stability if $\gamma >\delta \alpha$ and Hence, for $\alpha =0$, both low edge equilibria are locally stable if Similarly, both high edge equilibria are locally stable if

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, Note that $\phi \left(0\right)$ and $f\left(0\right)$ are always of the same sign. To obtain an approximate condition for the local stability of an asymmetrical internal equilibrium where $0<{\hat{z}}_1<z^{\ast }\le z^{\ast \ast }<{\hat{z}}_2$, ${\hat{M}}_1=K$, and ${\hat{M}}_2=K+D$, we assume that ε is small. An internal equilibrium with $\mathit{ϵ}=0$ is locally stable if $b_{12}{b}_{21}<1$ or equivalently if $\phi \left(0\right)>0$, 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 which reduces to In addition, there exists a class of asymmetrical internal equilibria not found when the competition coefficients are constants. Such equilibria are of four kinds: $0<{\hat{z}}_1<{\hat{z}}_2<z^{\ast }$ and ${\hat{M}}_1={\hat{M}}_2=K$, $0<{\hat{z}}_2<{\hat{z}}_1<z^{\ast }$ and ${\hat{M}}_1={\hat{M}}_2=K$, $0<z^{\ast \ast }<{\hat{z}}_1<{\hat{z}}_2$ and ${\hat{M}}_1={\hat{M}}_2=K+D$, or $0<z^{\ast \ast }<{\hat{z}}_2<{\hat{z}}_1$ and ${\hat{M}}_1={\hat{M}}_2=K+D$. Assuming ${\hat{z}}_1<{\hat{z}}_2$ and setting ${\hat{M}}_1={\hat{M}}_2=M$ without loss of generality, we can obtain explicit solutions for ${\hat{N}}_1$ and ${\hat{N}}_2$ as follows.

At equilibrium, we have Transforming variables and solving yields Hence 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 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 ${\hat{N}}_1={\hat{N}}_2=M/\left(1+b_0\right)$. Similarly, replacing the second inequality by an equality yields the edge equilibrium ${\hat{N}}_1=0$, ${\hat{N}}_2=M$.

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 ${\hat{N}}_1$ and ${\hat{N}}_2$, we obtain, after some algebra, 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, $b_0^2-1+\left(2b_0\delta \mathit{ϵ}M/\gamma \right)\equiv \xi$ and $1-b_0-\left(b_0\delta \mathit{ϵ}M/\gamma \right)\equiv \eta$, respectively, to be positive. Hence, existence entails $\phi \left(0\right)>0$.

Next we note that when $\xi =0$, the cubic reduces to One eigenvalue is zero, and the other two are negative. Hence, when we perturb ξ so that $f\left(0\right)$ 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 $\eta =0$, the cubic reduces to and the same argument holds when we perturb η.