Case of Majority Preference (i.e., β=a)
In this section, we set β=a to analyze the case in which the majority preference is present and the minority aversion is absent. Substitution of Eq. (3) in Eq. (1) yields
where 〈⋅〉 represents the average over the population, i.e., the average of a state-dependent variable with weight x
i
(1≤i≤n). Equation (4) is a replicator equation [14] in which \(s_{i}x_{i}^{a-1}\) and \(\langle sx^{a-1}\rangle =\sum_{\ell=1}^{n} s_{\ell}x_{\ell}^{a}\) play the role of the fitness for state i and the average fitness in the population, respectively. The dynamics given by Eq. (4) has n trivial equilibria corresponding to the consensus, i.e., the monopoly of a single state, and an interior equilibrium given by
$$ x_i^*=\frac{s_i^{\frac{1}{1-a}}}{\sum_{\ell=1}^n s_{\ell}^{\frac {1}{1-a}}} \quad (1\le i\le n). $$
(5)
V(x)≡−〈sx
a−1〉, where x=(x
1,…,x
n
), is a Lyapunov function of the dynamics given by Eq. (4) because
V has a unique global extremum at x
∗, which is minimum for a<1 and maximum for a>1 (Appendix A). Therefore, the coexistence equilibrium given by Eq. (5) is globally stable for a<1 and unstable for a>1.
Equation (4) also admits a unique equilibrium for each subset of the n states. When n=3, for example, the equilibrium in which states 1 and 2, but not 3, coexist is given by Eq. (5) for i=1 and 2, with the denominator replaced by \(s_{1}^{\frac{1}{1-a}} + s_{2}^{\frac{1}{1-a}}\), and x
3=0. In general, there are \(\binom{n}{n^{\prime}}\) equilibria containing n′ states. If 2≤n′≤n−1, these equilibria are unstable. For a<1, the instability immediately follows from the fact that x
∗ is the unique global minimum of V(x). For a>1, any equilibrium containing n′ (2≤n′≤n−1) states is unstable because it realizes the global maximum of the same Lyapunov function restricted to the simplex spanned by the n′ states.
When a=1, we obtain \(V(\boldsymbol{x})\equiv-\langle s\rangle = \sum_{i=1}^{n} s_{i} x_{i}\). Therefore, if s
i
>s
j
(j≠i), the consensus of state i is eventually reached. If s
i
=s
i′>s
j
(j≠i,i′), for example, the n−2 states corresponding to s
j
(j≠i,i′) are eventually eliminated. The dynamics then stops such that states i and i′ coexist. If all the three s
i
values are equal, any population is neutrally stable.
Figure 1 represents the dynamics in the two regimes with n=3, which we obtained by numerically integrating Eq. (4). For a<1, a trajectory starting from anywhere in the interior of the phase space, i.e., x that satisfies x
1+x
2+x
3=1, x
1, x
2, x
3>0, asymptotically approaches the coexistence equilibrium (Fig. 1a). It should be noted that a point in the triangle in Fig. 1a corresponds to a configuration of the population, i.e., x. For example, corner e
i
(i=1, 2, or 3) represents the consensus (i.e., x
i
=1 and x
j
=0 (j≠i)), and the normalized Euclidean distance from the point to the edge e
2–e
3 of the triangle is equal to the x
1 value. For a>1, a trajectory starting from the interior of the triangle converges to one of the n consensus equilibria, depending on the initial condition (Fig. 1b).
In Fig. 2, a bifurcation diagram in which we plot \(x_{1}^{*}\) against a is shown for s
1=0.40, s
2=0.35, and s
3=0.25. As a approaches unity from below, the stable coexistence equilibrium approaches the unstable consensus equilibrium corresponding to the largest s
i
value (e
1 in Fig. 1). At a=1, the two equilibria collide, and an unstable coexistence equilibrium simultaneously bifurcates from the consensus equilibrium corresponding to the smallest s
i
value (e
3 in Fig. 1).
Case of Minority Aversion (i.e., β=0)
In this section, we set β=0 to analyze the case in which the majority preference is absent and the minority aversion is present. Substitution of Eq. (3) in Eq. (1) yields
In contrast to the case of the majority preference (Sect. 3.1), the simplex spanned by n′ (2≤n′≤n−1) states is not invariant under the dynamics given by Eq. (7). Therefore, a state that once gets extinct may reappear. In this section, we numerically analyze Eq. (7) for n=3 and n=4. For general n, we analytically examine the special case in which s
i
is independent of i in Sect. 3.3.
For n=3, the dynamics for various values of a is shown in Fig. 3. We set s
1=0.36, s
2=0.33, and s
3=0.31. When a is small (a<1), there is a unique globally stable coexistence equilibrium in the interior (Fig. 3a). The three consensus equilibria e
1, e
2, and e
3 are unstable. At a=1, e
1, e
2, and e
3 change the stability such that they are stable beyond a=1 (Appendix B). Simultaneously, a saddle point bifurcates from each consensus equilibrium. The bifurcation occurs simultaneously for the three equilibria at a=1 irrespective of the values of s
1, s
2, and s
3. Slightly beyond a=1, the three consensus equilibria and the interior coexistence equilibrium are multistable (see Fig. 3b for the results at a=1.3). As a increases, the attractive basin of the coexistence equilibrium becomes small, and that of each consensus equilibrium becomes large (see Fig. 3c for the results at a=1.4). At a=a
c1≈1.43, the coexistence equilibrium that is stable for a<a
c1 and the unstable interior equilibrium that bifurcates from e
i
at a=1, where i corresponds to the largest s
i
value (i=1 in the present example), collide. This is a saddle-node bifurcation.
Numerically obtained a
c1 values are shown in Fig. 4a for different values of s
1, s
2, and s
3. A point in the triangle in the figure specifies the values of s
1, s
2, and s
3 under the constraint s
1+s
2+s
3=1, s
i
>0 (1≤i≤3). It seems that a
c1 is the largest when s
i
=1/3 (1≤i≤3). Figure 4 suggests that heterogeneity in s
i
makes a
c1 smaller and hence makes the stable coexistence of the three states difficult. When s
i
≈1 and s
j
≈0 (j≠i), we obtain a
c1≈1.
When a is slightly larger than a
c1, there are two saddle points in the interior. In this situation, one of the three consensus equilibria, which depends on the initial condition, is eventually reached (Fig. 3d). However, the manner with which the triangular phase space is divided into the three attractive basins is qualitatively different from that in the case of the majority preference (Fig. 1b). In particular, in the present case of the minority aversion, even if x
1 is initially equal to 0, the consensus of state 1 (i.e., e
1) can be reached. This behavior never occurs in the case of the majority preference and less likely for a larger a value in the case of the minority aversion (Fig. 3e).
The sizes of the attractive basins of the different equilibria are plotted against a in Fig. 5a. Up to our numerical efforts with various initial conditions, we did not find limit cycles. A discrete jump in the basin size of the coexistence equilibrium is observed at a
c1≈1.43, reminiscent of the saddle-node bifurcation. Interestingly, the attractive basin of the consensus equilibrium e
1 is the largest just beyond a
c1.
As a increases further, the second saddle-node bifurcation occurs at a=a
c2≈2.81, where an unstable node and a saddle point coappear (Fig. 3f). Logically, the sizes of the attractive basins could be discontinuous at a=a
c2 because some initial conditions with small x
1 might be attracted to e
1 when a is slightly smaller than a
c2 and to e
2 or e
3 when a is slightly larger than a
c2. However, up to our numerical efforts, we did not observe the discontinuity, as implied by Fig. 5a.
Numerically obtained a
c2 values are shown in Fig. 4b for different values of s
1, s
2, and s
3. Heterogeneity in s
i
makes a
c2 larger. In addition, a
c2 is equal to a
c1 when s
1=s
2=s
3=1/3. In this symmetric case, the three saddle points simultaneously collide with the stable star node at a=1. Beyond a=1, the equilibrium that is the stable star node when a<1 loses its stability to become an unstable star node. The three saddle points move away from the unstable star node as a increases. This transition can be interpreted as three simultaneously occurring transcritical bifurcations.
The unstable node that emerges at a=a
c2 approaches \(x_{i}^{*}=1/3\) (1≤i≤3) in the limit a→∞, as shown in Appendix C. The three saddle points approach (x
1,x
2,x
3)=(1/2,1/2,0),(1/2,0,1/2), and (0,1/2,1/2), as shown in Fig. 3g. This is a trivial consequence of the proof given in Appendix C. Therefore, the heterogeneity in s
i
does not play the role in the limit a→∞ such that the phase space is symmetrically divided into the three attractive basins corresponding to e
1, e
2, and e
3.
For n=4, the relationship between a and the sizes of the attractive basins of the different equilibria is shown in Fig. 5b. The results are qualitatively the same as those for n=3 (Fig. 5a).
Symmetric Case
In the previous sections, we separately considered the effect of the majority preference (Sect. 3.1) and the minority aversion (Sect. 3.2). In this section, we examine the extended AS model when both effects can be combined. To gain analytical insight into the model, we focus on the symmetric case s
i
=s (1≤i≤n). Although normalization \(\sum_{i=1}^{n} s_{i}=1\) leads to s
i
=1/n, we set s=1 in this section to simplify the notation; s just controls the time scale of the dynamics. Then, Eqs. (1) and (3) are reduced to
$$ \frac{dx_i}{dt}=x_i^\beta\sum _{j=1,\, j\neq i}^n x_j(1-x_j)^{a-\beta }-x_i(1-x_i)^{a-\beta} \sum_{j=1,\, j\neq i}^n x_j^\beta. $$
(8)
Equation (8) implies that, regardless of the parameter values, there exist n trivial consensus equilibria and symmetric coexistence equilibria of n′ (2≤n′≤n) states given by \(x_{i}^{*}=1/n^{\prime}\), where i varies over the n′ surviving states arbitrarily selected from the n states.
Owing to the conservation law \(\sum_{i=1}^{n}x_{i}=1\), the dynamics are (n−1)-dimensional. The eigenvalues of the Jacobian matrix of the dynamics at the coexistence equilibrium containing the n states are (n−1)-fold and given by \((\frac{1}{n} )^{\beta}(1-\frac{1}{n} )^{a-\beta -1} [(n-2)\beta+a-n+1 ]\), as shown in Appendix D. Therefore, the coexistence equilibrium is stable if and only if
$$ (n-2)\beta+a-n+1<0. $$
(9)
Similarly, we show in Appendix B that the consensus equilibria are stable if and only if
Coexistence equilibria of n′ (2≤n′≤n−1) states are always unstable (Appendix D).
Figure 6 is the phase diagram of the model in which the stable equilibria for given parameter values are indicated. The thin solid and dashed lines separating two phases are given by Eqs. (9) and (10), respectively. A multistable parameter region exists when n≥3; Eq. (9) is reduced to a<1 when n=2. When n≥3, the multistability occurs except in the case of the pure majority preference (i.e., β=a). The multistable parameter region enlarges as n increases.