Abstract
We study fixation probabilities for the Moran stochastic process for the evolution of a population with three or more types of individuals and frequency-dependent fitnesses. Contrary to the case of populations with two types of individuals, in which fixation probabilities may be calculated by an exact formula, here we must solve a large system of linear equations. We first show that this system always has a unique solution. Other results are upper and lower bounds for the fixation probabilities obtained by coupling the Moran process with three strategies with birth–death processes with only two strategies. We also apply our bounds to the problem of evolution of cooperation in a population with three types of individuals already studied in a deterministic setting by Núñez Rodríguez and Neves (J Math Biol 73:1665–1690, 2016). We argue that cooperators will be fixated in the population with probability arbitrarily close to 1 for a large region of initial conditions and large enough population sizes.
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Acknowledgements
This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. AGMN thanks Fábio Chalub for many fruitful discussions. The authors acknowledge the anonymous referees and the Associate Editor for many corrections improving the clarity of this work and for suggestions, including combining Lemma 1 in Benaim and Weibull (2003), Theorem 7 with our Theorem 6, strengthening a lot the results we had in a previous version of this paper.
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EMF had a Capes scholarship. AGMN was partially supported by Fundação de Amparo à Pesquisa de Minas Gerais (FAPEMIG, Brazil).
Appendices
Some results on birth–death processes
This appendix collects some results which we did not want to insert in the main text of the paper, because they have to do only with birth–death processes. Despite that, these results were used in the proof of the theorems in Sect. 5, all of them referring to Moran processes with three strategies. The more important results here are Theorems 11 and 12, which are cited in the proofs of Appendix B. The propositions which precede them are necessary for their proofs.
The first result here deals with comparing fixation probabilities for two birth–death processes in which the birth to death ratio is larger in one process than in the other. We observe that this result might be proved by a coupling argument similar to the one shown in Theorem 2. We opt here for a direct proof using the exact expressions (4) for the fixation probabilities.
Proposition 2
Consider two birth–death processes with the same set of states \(S=\{0,1,2, \dots , N\}\). Let \(r_i\equiv a_i/b_i\) be the birth to death ratio in the first process and \(s_i\equiv a'_i/b'_i\) be the ratio in the second process. Let also \(\pi _i\) and \(\pi '_i\) denote the respective fixation probabilities in state N. If \(r_i >s_i\) for \(i=1,2, \dots , N-1\), then \(\pi _i>\pi '_i\) for all \(i \in S \setminus \{0,N\}\).
Proof
We start by rewriting expression (4) for \(\pi _i\):
A similar expression may be written for \(\pi '_i\) just by writing \(s_k\) in place of \(r_k\). Let
be the denominator minus 1 in the last expression and \(d'_i\) be the same expression with \(r_k\) replaced by \(s_k\). We will prove that \(d'_i-d_i>0\), which of course implies \(\pi _i>\pi '_i\).
Using the fact that \(s_k^{-1}>r_k^{-1}>0\) for all k, both numerator and denominator in the first term in the last expression are clearly positive. To see that the second term is positive, too, notice that its denominator is the same as the first term, and its numerator, with some patience, may be rewritten as
now manifestly positive. \(\square \)
Proposition 3
Consider a birth–death process with state space \(S=\{0,1,2, \dots , N\}\). If \(r_i\) is the ratio of birth to death probabilities and \(i^{*} \in \{1, 2, \dots , N-1\}\) is some fixed state, then the probability \(\pi _{i \backslash i^{*}}\) that the process starts at \(i>i^{*}\) and fixates at state N without ever passing by state \(i^{*}\) is
Proof
Our result (44) may be obtained from (4), noticing that the boundary condition \(\pi _0=0\) is replaced by \(\pi _{i^{*} \backslash i^{*}}=0\) and the set of states \(\{0,1,2, \dots , N\}\) is replaced by \(\{i^{*}, i^{*}+1, \dots , N\}\). \(\square \)
The following result, Proposition 4, is just a straightforward adaptation of the result in Proposition 2 to the fixation probability \(\pi _{i \backslash i^{*}}\) defined in Proposition 3. As the proof is a mere repetition, we do not write it here.
Proposition 4
Consider two birth–death processes with the same set of states \(S=\{0,1,2, \dots , N\}\). Let \(r_i\equiv a_i/b_i\) and \(s_i\equiv a'_i/b'_i\) be the respective birth to death ratios. Suppose that there exists \(i^*\) such that \(r_i >s_i\) for \(i=i^*+1,i^*+2, \dots , N-1\). If \(i>i^*\) and \(\pi _{i \backslash i^{*}}\) and \(\pi '_{i \backslash i^{*}}\) denote the fixation probabilities in state N with the additional condition that the process never passes by state \(i^*\), then \(\pi _{i \backslash i^{*}}>\pi '_{i \backslash i^{*}}\) for all \(i \in \{i^*+1,i^*+2, \dots , N-1\}\).
The next result is the key ingredient in the proof of the important Theorem 13, which generalizes Theorem 6 in the main text.
Theorem 11
Suppose that for large enough values of N we have a family of birth–death processes with birth to death ratios \(r_i^{(N)}\) and fixation probabilities \(\pi _i^{(N)}\), \(i=1, 2, \dots , N-1\). Let \(x\in (0,1)\) and \(\varPi _N(x) \equiv \pi _{[Nx]}^{(N)}\). If there exist \(s>1\) and \(x^* \in (0,1)\) such that for N large enough and \(i>N x^*\) we have \(r_i^{(N)}>s\), then
for \(x>x^*\).
Proof
Let \(\pi '_i\) be the fixation probability of a birth–death process with frequency independent fitness \(s_i=s\) and let \(\pi '_{i\setminus i^*}\) be as in Proposition 4. Summing the geometric progressions appearing in (44) when \(r_i\) is replaced by s, we get
Suppose \(x>x^*\) and N large enough so that \([Nx]>[Nx^*]\). Of course, \(\pi _{[Nx]}^{(N)} \ge \pi _{[Nx]\setminus [Nx^*]}^{(N)}\). As, by Proposition 4, we have \(\pi _{[Nx]\setminus [Nx^*]}^{(N)} > \pi '_{[Nx]\setminus [Nx^*]}\), then
Our conclusion follows because, if \(s>1\), the last expression tends to 1 when \(N \rightarrow \infty \). \(\square \)
The next result here is quite analogous to Theorem 11 in its proof, but it comes with a surprise: although we will be able to prove that, under the stated hypotheses, \(\pi _{[Nx]\setminus [Nx^*]}^{(N)}\) tends to 0 as \(N \rightarrow \infty \), we cannot conclude that \(\pi _{[Nx]}^{(N)}\) tends to 0, too.
Theorem 12
Suppose that for large enough values of N we have a family of birth–death processes with birth to death ratios \(r_i^{(N)}\) and fixation probabilities \(\pi _i^{(N)}\), \(i=1, 2, \dots , N-1\). Suppose also that there exist \(0<s<1\) and \(x^* \in (0,1)\) such that for all N and \(i>N x^*\) we have \(r_i^{(N)}<s\). If \(x>x^*\) and \(\varPi _{N\backslash x^{*}}(x) \equiv \pi _{[Nx]\backslash [Nx^{*}]}^{(N)}\), then
The proof of the above result is analogous to the proof of Theorem 11 and is left to the interested reader. We comment instead on why we cannot arrive at a result completely analogous to Theorem 11.
The first reason is that inequality \(\pi _{[Nx]}^{(N)} \ge \pi _{[Nx]\setminus [Nx^*]}^{(N)}\) used in proving Theorem 11 is still valid and we cannot in general conclude that a quantity larger than or equal to something tending to 0 tends to 0, too.
More than that, we know that in a birth–death process for two strategies we can fulfill the hypotheses of Theorem 12 and still have \(\lim _{N \rightarrow \infty } \pi _{[Nx]}^{(N)} =1\). This is proved for Moran processes with two strategies in de Souza et al. (2019), Theorem 5, if certain conditions on the pay-off matrix are valid. The conditions are \(m_{11}<m_{21}\), \(m_{12}>m_{22}\), i.e. neither of the two strategies is a Nash equilibrium, and
In the interesting situation in which the hypotheses of Theorem 12 are fulfilled and we also have \(\lim _{N \rightarrow \infty } \pi _{[Nx]}^{(N)} =1\), we have for large N and \(x>x^*\) both \(\varPi _N(x)\) close to 1 and \(\varPi _{N\backslash x^{*}}(x)\) close to 0. This means that although fixation at state N is very probable, most probably the chain will pass at least once (thus, it will probably pass many times) by \(x^*\) before fixation occurs. An application of the above phenomenon, in which a source strategy in the replicator dynamics fixates with high probability in the Moran process, is given by McLoone et al. (2018).
Proofs of the theorems in the main text
1.1 Proof of Theorem 2
For each \((i,j) \in \varLambda _N\), consider a first partition of the interval [0, 1] illustrated at the left-hand part of Fig. 6 and defined as follows. Let \(q_k(i,j)\), \(k= 1, 2, \dots , 7\) be the accumulated sums of the transition probabilities at (i, j) ordered as \(p_{ij}^{AB}\), \(p_{ij}^{AC}\), \(p_{ij}^{BC}\), \(p_{ij}^{const}\), \(p_{ij}^{CB}\), \(p_{ij}^{CA}\) and \(p_{ij}^{BA}\). For definiteness, \(q_1(i,j)=p_{ij}^{AB}\), \(q_2(i,j)= q_1(i,j)+p_{ij}^{AC}\), \(q_3(i,j)= q_2(i,j)+p_{ij}^{BC}\), and so on. Of course, \(q_7(i,j)=1\).
For \(k=1, 2, \dots , 7\), let \(I_k(i,j)\) be disjoint intervals with lengths \(q_k(i,j)\):
and
Notice that the sum of the lengths of \(I_1(i,j)\) and \(I_2(i,j)\) is \(Z_{ij}^{+}\) and that the sum of the lengths of \(I_6(i,j)\) and \(I_7(i,j)\) is equal to \(Z_{ij}^{-}\).
For each \(i \in S\), we construct a second partition of [0, 1] illustrated at the right-hand side of Fig. 6, similar to the first partition and related to the comparison chain:
and
Conditions (25) imply
The following vectors are the possible displacements in the state of the target chain: \(d_1=(1,-1)\), \(d_2=(1,0)\), \(d_3=(0,1)\), \(d_4= (0,0)\), \(d_5=(0,-1)\), \(d_6=(-1,0)\) and \(d_7=(-1,1)\).
Suppose that at time 0 the states of the target and comparison chains are respectively \(X_0=(i_0,j_0)\) and \(Y_0=i_0\). The coupling of the target and comparison chains is accomplished by a sequence of independent uniformly distributed random variables \(U_1, U_2, U_3, \dots \in [0,1]\) which will determine the state of both chains at all subsequent times.
The state of both chains at time 1 will be obtained by displacements calculated as functions of \(U_1\), then at time 2 by displacements calculated as functions of \(U_2\), and so on. The way these displacements are calculated is as follows.
We declare that if the state of the target chain at time \(\ell -1\) is \((i_{\ell -1},j_{\ell -1})\), then the \(\ell \)-th displacement of the target chain will be \(d_k\) if \(U_{\ell } \in I_k(i_{\ell -1},j_{\ell -1})\), \(k=1,2, \dots , 7\), \(\ell =1,2, \dots \). For the comparison chain, we declare that if its state at time \(\ell -1\) is \(i'_{\ell -1}\), \(\ell =1,2, \dots \), then the displacement of the state of the comparison chain will be 1, 0, or \(-1\), respectively, if \(U_{\ell }\) is in \(J_1(i'_{\ell -1})\), \(J_2(i'_{\ell -1})\) or \(J_3(i'_{\ell -1})\). Notice that the construction up to now is such that the probabilities of the possible displacements of both chains are all correctly distributed according to the chains’ transition probabilities.
A fundamental observation is that, due to (46), whenever \(U_1\) is such that there is a birth in the comparison chain, then the number of A individuals in the target chain will increase. And also, whenever \(U_1\) is such that the number of A individuals decreases in the target chain, then there is a death in the comparison chain. As \(i'_0=i_0\), it follows that \(i_1 \ge i'_1\).
We will prove by induction that \(i_k \ge i'_k \; \forall k \in {\mathbb {N}}\). Suppose that \(i_k \ge i'_k\) for a certain \(k \in {\mathbb {N}}\). By the same reasoning used in proving that \(i_1 \ge i'_1\), we see that if \(i_k = i'_k\), then \(i_{k+1} \ge i'_{k+1}\). If \(i_{k} \ge i'_k+2\), then the conclusion \(i_{k+1} \ge i'_{k+1}\) also holds, because \(i_{k+1} \ge i_k-1\) and \(i'_{k+1} \le i'_k+1\).
The only case in which it remains to prove that \(i_{k+1} \ge i'_{k+1}\) is when \(i_{k} = i'_k+1\). In this case, we use condition (26), which we had not used, yet. This condition proves that if \(U_k\) is such that a birth occurs in the comparison chain, then the number of A individuals in the target chain will not decrease and we will still have \(i_{k+1} \ge i'_{k+1}\).
We have thus realized simultaneously the target and comparison chains according to their respective transition matrices in a way that the initial states are respectively \((i_0,j_0)\) and \(i_0\) and whenever there is fixation at state N for the comparison chain, then there will be fixation of strategy A in the target chain. Thus \(\alpha _{i_0,j_0} \ge \pi _{i_0}\). As \(i_0\) is arbitrary, the theorem is proved. \(\square \)
1.2 Proof of Theorem 4
Before proving the theorem, we state and prove the following result, used in the proof of Theorem 4 and other results ahead.
Proposition 5
Let \(Z_{ij}^{\pm }\) be defined as in (24). Then, for each fixed value of i, the minimum of \(Z_{ij}^+\) and the maximum of \(Z_{ij}^-\) for \(j \in \{0,1,\dots , N-i\}\) are attained at the same value of j. Also, the maximum of \(Z_{ij}^+\) and the minimum of \(Z_{ij}^-\) for \(j \in \{0,1,\dots , N-i\}\) are attained at the same value of j.
Proof
Just notice that \( Z_{ij}^{+} = i\frac{f_{ij}}{S_{ij}}\frac{N-i}{N}\) and \( Z_{ij}^{-} = \frac{j g_{ij}+(N-i-j)h_{ij}}{S_{ij}}\frac{i}{N}\) may be rewritten as \((1-i\frac{f_{ij}}{S_{ij}})\frac{i}{N}\). For fixed i the value of j minimizing \(i \frac{f_{ij}}{S_{ij}}\) will maximize \(1-i\frac{f_{ij}}{S_{ij}}\). \(\square \)
We can now prove Theorem 4.
Let \(a_i\) be the probability of increasing the number of A individuals from i to \(i+1\) in a population with only A and B individuals. We also define \(b_i\) as the probability of decreasing the number of A individuals from i to \(i-1\) in a population with only A and B individuals. As in (24), let \(Z_{ij}^{+}\) and \(Z_{ij}^{-}\) be respectively the probabilities of increasing and decreasing the number of A individuals from i to \(i\pm 1\) in a population with A, B and C individuals and frequency independent fitnesses. We have
As \(g>h\), then \(if + (N-i)g= if + jg + (N-i-j)g \ge if + jg + (N-i-j)h\) for all \((i,j)\in \varLambda _N\). It follows that
for all \((i,j)\in \varLambda _N\).
By Proposition 5, \(b_i= Z_{i,N-i}^{-}\ge Z_{ij}^{-}\) for all \((i,j)\in \varLambda _N\). We have thus proved that conditions (25) in Theorem 2 are fulfilled. The lower bound in (31) will result if we prove that (26) is true.
In fact, it can be seen, after some tedious manipulations, that
As all terms around the curly brackets in the above expression are obviously non-negative, as well as the denominator \(N S_{ij}S_{i-1,N-i+1}\), then condition (26) is satisfied and the lower bound in (31) proved.
All the remaining bounds can be proved in an analogous way, either using Theorem 2 or Theorem 3. \(\square \)
1.3 Proof of Theorem 5
We will show that the birth–death process (35) satisfies the hypotheses of Theorem 2 for N large enough. The proof that the process defined by (36) satisfies the hypotheses of Theorem 3 for large enough N is analogous.
By (35), we automatically have for each i that \(a_{i}^{low} \le Z_{ij}^{+}\) and \(b_{i}^{low} \ge Z_{ij}^{-}\) for all j such that \((i,j) \in \varLambda _N\). To complete the proof, we need to show that \(a_{i-1}^{low} \le 1 - Z_{ij}^{-}\) if N is large enough.
To see that, we write
The first term \(1-Z_{ij}^--Z_{ij}^+\) is the probability at state (i, j) that the number of A individuals remains constant. It can be written as the sum \(p_{ij}^{AA}+(p_{ij}^{BC}+p_{ij}^{CB}+p_{ij}^{BB}+p_{ij}^{CC})\), in which \(p_{ij}^{AA}\) vanishes only if \(i=0\) and the sum of the remaining four terms vanishes only if \(i=N\).
Writing \(x=i/N\) and \(y=j/N\) and using a reasoning similar to the one exemplified in (19), we get
where
comes from \(p_{ij}^{AA}\) and
comes from the sum \(p_{ij}^{BC}+p_{ij}^{CB}+p_{ij}^{BB}+p_{ij}^{CC}\).
Observe that both \(C_1\) and \(C_2\) are continuous functions with values in [0, 1] in the compact triangle \(\varLambda \) defined in (13). Moreover \(C_1(1,0)=1\) and \(C_2(0,y)=1\), so that there exist \(x_1,x_2 \in [0,1]\), \(x_1<x_2\) such that \(C_2(x,y) \ge 1/2\) if \((x,y)\in \varLambda \) with \(x \le x_1\) and \(C_1(x,y) \ge 1/2\) if \((x,y)\in \varLambda \) with \(x \ge x_2\). As neither \(C_1\) nor \(C_2\) vanishes for the points \((x,y) \in \varLambda \) with \(x \in [x_1,x_2]\), then their sum has a positive minimum value \(\mu \) in this set. Of course the minimum value of \(C_1+C_2\) in \(\varLambda \) cannot be smaller than the smallest between \(\mu \) and 1/2, being then positive and independent of N. This proves that \(1-Z_{ij}^--Z_{ij}^+\) is bounded away from 0 for large enough N.
Using the same ideas,
with
Then the second summand in the right-hand side of (48) becomes
We have thus shown that one of the terms in the right-hand side of (48) is positive and O(1), and the other is \(O(\frac{1}{N})\). This proves that \(1-Z_{ij}^--Z_{i-1,j}^+>0\) for all \((i,j)\in \varLambda _N\) for large enough N and the proof is completed. \(\square \)
1.4 Proof of Theorem 6 and related results
The reader will see that the proof of Theorem 6 is a consequence of this more general result:
Theorem 13
Consider a Moran process with three strategies. Suppose there exist \(s>1\), \(N^* \in {\mathbb {N}}\) and \(x^* \in [0,1)\) such that if \(N \ge N^*\) and \(\frac{i}{N}>x^*\), then
holds \(\forall j \in \{0, 1,\dots , N-i\}\). Then
for all \((x,y) \in \varLambda \) with \(x>x^*\).
Proof
Let \(x \in (0,1)\) and consider the lower bound comparison birth–death process defined in Theorem 5 by (35). By Proposition 5, we know that the maximum over j of \(Z_{[Nx],j}^{-}\) and the minimum over j of \(Z_{[Nx],j}^{+}\) occur at the same value \({{\overline{j}}}(x) \in \{0, 1, \dots , [Nx]\}\). In other words,
Suppose now that \(x>x^*\) and take \(N \ge N^*\) and also large enough so that \([Nx]/N>x^{*}\). Then \(r^{low}_{[Nx]}\) is strictly greater than s for all \(x>x^{*}\). By Theorem 11 in Appendix A, we conclude that \(\lim _{N \rightarrow \infty }\pi ^{low}_{[Nx]}=1\) for all \(x>x^{*}\). As, by Theorem 5, \(\pi ^{low}_{[Nx]}\le \alpha _{[Nx],[Ny]}\), the theorem is proved. \(\square \)
We can now proceed with proving Theorem 6.
Proof
We will show that there exist \(x^*\), s and \(N^*\) as in the hypotheses of Theorem 13. The result will then follow as a consequence of that theorem.
In fact, if strategy A is a strict Nash equilibrium, then \(F(1,0)>G(1,0)\) and \(F(1,0)>H(1,0)\), see (18), and, by continuity, we have a neighborhood of (1, 0) in \(\varLambda \) in which the deterministic fitness F is strictly larger than both G and H.
Let \(x_1\) be the greatest lower bound of the values \(x \in [0,1]\) such that \(F(x,y)>G(x,y)\) and \(F(x,y)>H(x,y)\) hold simultaneously for all y such that \((x,y) \in \varLambda \).
In analogy with what we did in (49), we may rewrite \(Z^{+}_{ij}/Z^{-}_{ij}\) as an asymptotic term R(x, y), where \(x=i/N\) and \(y=j/N\), plus corrections that tend to 0 as \(N \rightarrow \infty \). We obtain
which is continuous in \(\varLambda \setminus (1,0)\).
Choose \(x^*\in (x_1,1)\) and define \(\varLambda ^* = \{(x,y) \in \varLambda \, ; x^* \le x < 1\}\). If we define \(R^* = \inf _{(x,y) \in \varLambda ^*}R(x,y)\), we claim that \(R^*>1\).
In fact, although R is not defined at (1, 0), both F/G and F/H are continuous at this point. So, we define \(S(x,y) = \min \left\{ \frac{F(x,y)}{G(x,y)},\frac{F(x,y)}{H(x,y)}\right\} \), which is continuous in the compact set \(\overline{\varLambda ^*} =\varLambda ^* \cup \{(1,0)\}\). Let \(s^*\) be the minimum value of S on \(\overline{\varLambda ^*}\). As \(F(x,y)>G(x,y)\) and \(F(x,y)>H(x,y)\) in \(\overline{\varLambda ^*}\), then \(s^*>1\). Moreover, \(R(x,y) \ge S(x,y)\) for \((x,y) \in \varLambda ^*\). Thus \(R^* \ge s^*\), proving our claim that \(R^*>1\).
We will now estimate the difference between \(Z^{+}_{ij}/Z^{-}_{ij}\) and \(R(\frac{i}{N}, \frac{j}{N})\). Using (24), we have
Using also the definition (50) of R, we get
where
If \(M_1 \equiv \inf _{N \in {\mathbb {N}}}\min _{(i,j) \in \varLambda _N} \{g_{ij},h_{ij}, G(\frac{i}{N},\frac{j}{N}),H(\frac{i}{N},\frac{j}{N})\}\), then the denominator in (52) is bounded below by \((1-\frac{i}{N})^2 M_1^2\). By the continuity in \(\varLambda \) of G and H and by formulas analogous to (19) for G and H, we know that \(M_1\) is finite and positive.
We can also find an upper bound for the \(\phi _{i,j,N}\) in the numerator. Let \(M_2 \equiv \max _{(x,y) \in \varLambda } \{F(x,y), G(x,y),H(x,y)\}\). By (19) and analogous expressions, there also exists a constant \(c>0\) such that for all \(N \in {\mathbb {N}}\),
Using these bounds, we get \(|\phi _{i,j,N}|<(1-\frac{i}{N})\frac{2c M_2}{N}\).
Putting together the bounds for numerator and denominator in (52), we can see that there exists a constant K such that
Let now \(s=\frac{1}{2}(R^*+1)\) and \(N^*\) be the smallest integer not smaller than \(\frac{K}{R^*-s}\). Then, for \(N>N^*\) and \(\frac{i}{N}>x^*\) we have \(\frac{Z^+_{ij}}{Z^-_{ij}} \ge s\) for all \(j \in \{0,1, \dots , N-i\}\). By Theorem 13 we conclude that \(\lim _{N \rightarrow \infty }A_N(x,y) =1\) for all \((x,y) \in \varLambda \) with \(x>x^*\). \(\square \)
For proving Theorem 9, we start by stating a preparatory result analogous to Theorem 13:
Theorem 14
Consider a Moran process with three strategies. Suppose there exist \(s \in (0,1)\), \(N^* \in {\mathbb {N}}\) and \(x^* \in [0,1)\) such that if \(N \ge N^*\) and \(\frac{i}{N}>x^*\), then
holds \(\forall j \in \{0, 1,\dots , N-i\}\). Then
for all \((x,y) \in \varLambda \) with \(x>x^*\).
We do not write a complete proof of Theorem 14, because it is analogous to the proof of Theorem 13, but we explain the important differences. First of all, instead of using a lower bound comparison birth–death process, we take an upper bound (36). The conclusion is a consequence of Theorem 12 in Appendix A.
For Theorem 10, we only sketch the proof, because it is again similar to preceding ones.
Proof
We use the upper bound (36) for \(\alpha _{ij}\) in Theorem 5. In order to prove this theorem, we should show that for fixed \(x<x^*\) we have \(\pi ^{up}_{[Nx]}\rightarrow 0\) when \(N \rightarrow \infty \). Hypotheses \(F(x,y) < G(x,y)\) and \(F(x,y) < H(x,y)\) if \((x,y) \in \varLambda \) and \(x<x^*\) make sure that \(r_i^{up} \equiv \frac{a_i^{up}}{b_i^{up}}\le s<1\) if N is large enough and \(i<Nx^*\).
The result that, for \(x<x^*\), \(\pi ^{up}_{[Nx]}\rightarrow 0\) when \(N \rightarrow \infty \) may be attained in two equivalent ways. One is to develop for \(i<i^*\) a result analogous to Proposition 3 for the probability of fixation at state 0 of the comparison chain without attaining state \(i^*\). The other way is proving by Theorem 11 that the fixation probability at state N of the dual process, see (5), tends to 1 as \(N \rightarrow \infty \). \(\square \)
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Ferreira, E.M., Neves, A.G.M. Fixation probabilities for the Moran process with three or more strategies: general and coupling results. J. Math. Biol. 81, 277–314 (2020). https://doi.org/10.1007/s00285-020-01510-0
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DOI: https://doi.org/10.1007/s00285-020-01510-0