Within-host bacterial growth dynamics with both mutation and horizontal gene transfer

Abstract

The evolution and emergence of antibiotic resistance is a major public health concern. The understanding of the within-host microbial dynamics combining mutational processes, horizontal gene transfer and resource consumption, is one of the keys to solving this problem. We analyze a generic model to rigorously describe interactions dynamics of four bacterial strains: one fully sensitive to the drug, one with mutational resistance only, one with plasmidic resistance only, and one with both resistances. By defining thresholds numbers (i.e. each strain’s effective reproduction and each strain’s transition threshold numbers), we first express conditions for the existence of non-trivial stationary states. We find that these thresholds mainly depend on bacteria quantitative traits such as nutrient consumption ability, growth conversion factor, death rate, mutation (forward or reverse), and segregational loss of plasmid probabilities (for plasmid-bearing strains). Next, concerning the order in the set of strain’s effective reproduction thresholds numbers, we show that the qualitative dynamics of the model range from the extinction of all strains, coexistence of sensitive and mutational resistance strains, to the coexistence of all strains at equilibrium. Finally, we go through some applications of our general analysis depending on whether bacteria strains interact without or with drug action (either cytostatic or cytotoxic).

Appendices

Proof of Theorem 3.1

The right-hand side of System (2.2) is continuous and locally lipschitz on ${\mathbb {R}}^5$. Using a classic existence theorem, we then find $T>0$ and a unique solution $E(t)\omega _0= \left( B(t),N_s(t),N_m(t),N_p(t),N_{m.p}(t)\right)$ of (2.2) from $[0,T) \rightarrow {\mathbb {R}}^5$ and passing through the initial data $\omega _0$ at $t=0$. Let us now check the positivity and boundedness of the solution E on [0, T).

Since $E(\cdot )\omega _0$ starts in the positive orthant ${\mathbb {R}}^5_+$, by continuity, it must cross at least one of the five borders $\{B=0\}$, $\{N_j=0\}$ (with $j\in {\mathcal {J}}$) to become negative. Without loss of generality, let us assume that E reaches the border $\{N_s=0\}$. This means we can find $t_1 \in (0,t)$ such that $N_j(t)>0$ for all $t \in (0,t_1)$, $j \in {\mathcal {J}}$ and $N_s(t_1)=0$, $B(t_1) \ge 0$, $N_j(t_1) \ge 0$ for $j \in {\mathcal {J}}$. Then, the $\dot{N}_s$-equation of (2.2) yields $\dot{N}_s(t_1)= \theta \tau _p\beta _p B(t_1)N_p(t_1) +\varepsilon _m\tau _m\beta _m B(t_1)N_m(t_1) \ge 0$, from where the orbit $E(\cdot )\omega _0$ cannot cross ${\mathbb {R}}^5_+$ through the border $\{N_s=0\}$. Similarly, we prove that at any borders $\{B=0\}$, $\{N_j=0\}$ (with $j \in {\mathcal {J}}$), either the resulting vector field stays on the border or points inside ${\mathbb {R}}^5_+$. Consequently, $E([0,T))\omega _0 \subset {\mathbb {R}}^5_+$.

Recalling that $N=\sum _{j \in {\mathcal {J}}} N_j$ and adding up the $\dot{N}$- and $\dot{B}$-equations, it comes

$$\begin{aligned} \frac{{\mathrm{d}}}{{\mathrm{d}}t} \left( \tau _{\text {max}}B + N \right) \le \tau _{\text {max}}\Lambda - \min \left( d,d_{\text {min}} \right) \left( \tau _{\text {max}}B + N \right) ; \end{aligned}$$

from where one deduces estimate (3.2). So, the aforementioned local solution of System (2.2) is a global solution i.e. defined for all $t\in {\mathbb {R}}_+$. Which ends the proof of Theorem 3.1.

Proof of Theorem 3.3

If we consider a small perturbation of the bacteria-free steady state $E^0$, the initial phase of the invasion can be described by the linearized system at $E^0$. Since the linearized equations for bacteria populations do not include the one for the nutrient, we then have

$$\begin{aligned} \begin{aligned} \frac{{\mathrm{d}}}{{\mathrm{d}}t} (u,v)^T= J[E^0] (u,v)^T, \end{aligned} \end{aligned}$$

(C.1)

with $J[E^0]= \left( \begin{array}{cc} B_0G-D &{}\quad B_0L_p\\ 0 &{}\quad B_0(G_p-L_p)-D_p \end{array} \right) .$

We claim that

Claim B.1

For small mutation rates $\varepsilon _j$, the principal eigenvalue $r(B_0G-D )$ and $r( B_0(G_p-L_p)-D_p)$, of matrices $(B_0G-D )$ and $( B_0(G_p-L_p)-D_p)$ writes

$$\begin{aligned} \begin{aligned} r(B_0G-D )&= \frac{1}{2} \left\{ B_0\tau _s\beta _s(1-\varepsilon _s)-d_s + B_0\tau _m\beta _m(1-\varepsilon _m) -d_m + \right. \\&\left. \left[ \left( B_0\tau _s\beta _s(1-\varepsilon _s)-d_s - B_0\tau _m\beta _m(1-\varepsilon _m)+d_m \right) ^2 +4\varepsilon _m\varepsilon _s\tau _m\tau _s\beta _m\beta _sB_0^2 \right] ^{1/2} \right\} \\&= \frac{1}{2} \left\{ d_s ({\mathcal {R}}_s-1) + d_m ({\mathcal {R}}_m-1) + \left| d_s ({\mathcal {R}}_s-1) - d_m ({\mathcal {R}}_m-1) \right| \right\} + {\mathcal {O}}(\varepsilon _m\varepsilon _s), \end{aligned} \end{aligned}$$

i.e.

$$\begin{aligned} r(B_0G-D )= \left\{ \begin{aligned}&B_0\tau _s\beta _s(1-\varepsilon _s)-d_s + {\mathcal {O}}(\varepsilon _m\varepsilon _s), \quad \text { if } d_s({\mathcal {R}}_s -1) > d_m({\mathcal {R}}_m-1),\\&B_0\tau _m\beta _m(1-\varepsilon _m)-d_m + {\mathcal {O}}(\varepsilon _m\varepsilon _s), \quad \text { if } d_s({\mathcal {R}}_s -1) < d_m({\mathcal {R}}_m-1). \end{aligned} \right. \end{aligned}$$

In the same way, we have

$$\begin{aligned} \begin{aligned}&r(( B_0(G_p-L_p)-D_p))= \\&\left\{ \begin{aligned}&B_0\tau _p\beta _p(1-\theta )(1- \varepsilon _p)-d_p + {\mathcal {O}}(\varepsilon _p\varepsilon _{m.p}), \quad \text { if } d_p ({\mathcal {R}}_p-1)> d_{m.p}({\mathcal {R}}_{m.p}-1),\\&B_0\tau _{m.p}\beta _{m.p}(1-\theta )(1-\varepsilon _{m.p})-d_{m.p} + {\mathcal {O}}(\varepsilon _p\varepsilon _{m.p}), \quad \text { if } d_p ({\mathcal {R}}_p-1)< d_{m.p}({\mathcal {R}}_{m.p}-1). \end{aligned} \right. \end{aligned} \end{aligned}$$

Denoting by $\sigma (J[E^0])$ the spectrum of $J[E^0]$, we recall that the stability modulus (Li and Wang 1998) or spectral bound (Engel and Nagel 2001) of $J[E^0]$ is $s_0(J[E^0])=\{\max \text {Re}(z): z\in \sigma (J[E^0]) \}$ and $J[E^0]$ is said to be locally asymptotically stable (l.a.s.) if $s_0(J[E^0])<0$. Following Claim B.1 it comes

$$\begin{aligned} \begin{aligned}&\sigma (J[E^0])= \sigma (B_0G-D ) \cup \sigma ( B_0(G_p-L_p)-D_p) \simeq \\&\left\{ B_0\tau _s\beta _s(1-\varepsilon _s)-d_s, B_0\tau _m\beta _m(1-\varepsilon _m)-d_m, B_0\tau _p\beta _p(1-\varepsilon _p)(1-\theta )-d_p,\right. \\&\left. B_0\tau _{m.p}\beta _{m.p}(1-\theta )(1-\varepsilon _{m.p})-d_{m.p} \right\} , \end{aligned} \end{aligned}$$

where the last approximation holds for small mutation rates.

Note that, when mutation rates are small enough, we obtain $s_0(J[E^0])<0$ if and only if ${\mathcal {R}}^*<1$, i.e. $E^0$ is l.a.s if ${\mathcal {R}}^*<1$ and unstable if ${\mathcal {R}}^*>1$.

We now check the global stability of $E^0$ when ${\mathcal {T}}^* <1$. The $\dot{B}$-equation of (2.2) gives $\dot{B} \le \Lambda - dB$. Further, $B_0$ is a globally attractive stationary state of the upper equation $\dot{w} = \Lambda - dw$, i.e. $w(t)\rightarrow B_0$ as $t\rightarrow \infty$. Which gives $w(t)\le B_0$ for sufficiently large time t, from where $B(t)\le B_0$ for sufficiently large time t. Combining this last inequality with the total bacteria dynamics described by (2.1), we find $\dot{N}\le \sum _{j \in {\mathcal {J}}}\left( \beta _j\tau _j B_0 -d_j\right) N\le c_0({\mathcal {T}}^*-1)N$, with $c_0>0$ a positive constant. Therefore $N(t) \le N(0) e^{c_0({\mathcal {T}}^*-1)t} \rightarrow 0$ as as $t\rightarrow \infty$. This ends the proof of the global stability of $E^0$ when ${\mathcal {T}}^* <1$.

Item (iii) of the theorem remains to be checked. To do so we will apply results in Hale and Waltman (1989). Let us first notice that $E^0$ is an unstable stationary state with respect to the semiflow E. To complete the proof, it is sufficient to show that $W^s(\{E^0\})\cap X_0=\emptyset$, where $W^s(\{E^0\})= \left\{ w \in \Omega : \lim _{t\rightarrow \infty } E(t)w= E^0\right\}$ is the stable set of $\{E^0\}$. To prove this assertion, let us argue by contradiction by assuming that there exists $w \in W^s(\{E^0\})\cap X_0$. We set $E(t)w=(B(t),N_s(t),N_m(t),N_p(t),N_{m.p}(t))$. The $\dot{N}$-equation defined by (2.1) gives

$$\begin{aligned} \begin{aligned} \dot{N}(t)&= \sum _{j \in {\mathcal {J}}}d_j \left( {\mathcal {T}}_j \frac{B(t)}{B_0} -1\right) N_j(t)\\ \ge&d_{\text {min}} \left( \frac{B(t)}{B_0} \min _j {\mathcal {T}}_j -1\right) N(t), \text { for all time } t. \end{aligned} \end{aligned}$$

Since $\min _j {\mathcal {T}}_j>1$ and it is assumed that $B(t) \rightarrow B_0$ as $t\rightarrow \infty$, we find that the function $t \mapsto N(t)=N_s(t)+N_m(t)+N_p(t)+N_{m.p}(t)$ is not decreasing for t large enough. Hence there exists $t_0\ge 0$ such that $N(t) \ge N(t_0)$ for all $t\ge t_0$. Since $N(t_0)>0$, this prevents the component $(N_s,N_m,N_p,N_{m.p})$ from converging to (0, 0, 0, 0) as $t \rightarrow \infty$. A contradiction with $E(t)\omega \rightarrow E^0$. This completes the proof of Theorem 3.3.

Proof of Theorem 4.1

The stationary state $E^*_{s-m}= \left( B^*,u^*,0 \right)$. Here, it is useful to considered the abstract formulation of the model given by (2.3). Setting $v=0$, the $\dot{u}$-equation of (2.3) gives $\left[ BG-D\right] u=0$ i.e. $D^{-1}Gu= u/B$. Since $D^{-1}G= \left[ \begin{array}{cc} {\mathcal {R}}_s/B_0 &{}\quad {\mathcal {K}}_{m\rightarrow s}/B_0\\ {\mathcal {K}}_{s\rightarrow m}/B_0 &{}\quad {\mathcal {R}}_m/B_0 \end{array} \right]$ is a positive and irreducible matrix, from the so-called Perron-Frobenius theorem, $1/B= r(D^{-1}G)$ and $u=c\phi$, where $\phi >0$ is the eigenvector of $D^{-1}G$ corresponding to $r(D^{-1}G)$ and normalized such that $\Vert \phi \Vert _1=1$ and $c>0$ is a positive constant. Notice that $\phi >0$ means all components of the vector $\phi$ are positive. More precisely, we have

$$\begin{aligned} \begin{aligned}&B^{-1}=r(D^{-1}G)= \frac{1}{2B_0} \left\{ {\mathcal {R}}_s +{\mathcal {R}}_m + \left[ \left( {\mathcal {R}}_s -{\mathcal {R}}_m \right) ^2 +4 {\mathcal {K}}_{m\rightarrow s} {\mathcal {K}}_{s\rightarrow m} \right] ^{1/2} \right\} ,\\&\phi = \frac{\phi _0}{\Vert \phi _0\Vert _1}, \end{aligned} \end{aligned}$$

with $\phi _0= \left( {\mathcal {R}}_s -{\mathcal {R}}_m + \left[ \left( {\mathcal {R}}_s -{\mathcal {R}}_m \right) ^2 +4 {\mathcal {K}}_{m\rightarrow s} {\mathcal {K}}_{s\rightarrow m} \right] ^{1/2} ,2 {\mathcal {K}}_{s\rightarrow m}\right) ^T$.

Further, from the $\dot{B}$-equation we find $\Lambda -dB= c B \left\langle (\beta _s,\beta _m),\phi \right\rangle$, i.e.

$$\begin{aligned} c = \frac{d\left( B_0r(D^{-1}G)-1\right) }{\left\langle (\beta _s,\beta _m),\phi \right\rangle }>0 \Longleftrightarrow B_0r(D^{-1}G)>1. \end{aligned}$$

Approximation of ${\mathcal {P}} \left( E^*_{s-m}\right)$ for small mutation rates. Now, let us assume that mutation rates $\varepsilon _j$ are small enough. Without loss of generality, we express parameters $\varepsilon _j$ as functions of the same quantity, let us say $\eta$, with $\eta \ll 1$. We have

$$\begin{aligned} {\mathcal {P}} \left( E^*_{s-m}\right) = \frac{2 {\mathcal {K}}_{s\rightarrow m}}{{\mathcal {R}}_s -{\mathcal {R}}_m + \left[ \left( {\mathcal {R}}_s -{\mathcal {R}}_m \right) ^2 +4 {\mathcal {K}}_{m\rightarrow s} {\mathcal {K}}_{s\rightarrow m} \right] ^{1/2} +2 {\mathcal {K}}_{s\rightarrow m}}. \end{aligned}$$

(C.2)

By setting $\zeta _\eta ={\mathcal {R}}_s -{\mathcal {R}}_m + \left[ \left( {\mathcal {R}}_s -{\mathcal {R}}_m \right) ^2 +4 {\mathcal {K}}_{m\rightarrow s} {\mathcal {K}}_{s\rightarrow m} \right] ^{1/2}$, it comes

$$\begin{aligned} \zeta _\eta = \left( {\mathcal {T}}_s -{\mathcal {T}}_m\right) (1-\eta )+ |{\mathcal {T}}_s -{\mathcal {T}}_m| \left[ 1- \eta + \frac{1}{2} \left( \left( \frac{{\mathcal {T}}_s +{\mathcal {T}}_m}{{\mathcal {T}}_s -{\mathcal {T}}_m} \right) ^2 -1\right) \eta ^2 \right] +{\mathcal {O}}(\eta ^3). \end{aligned}$$

(C.3)

Next, we find a simple approximation of the frequency ${\mathcal {P}} \left( E^*_{s-m}\right)$ for cases ${\mathcal {R}}_s>{\mathcal {R}}_m$ and ${\mathcal {R}}_s<{\mathcal {R}}_m$, i.e. ${\mathcal {T}}_s>{\mathcal {T}}_m$ and ${\mathcal {T}}_s<{\mathcal {T}}_m$ for small mutations $\varepsilon _j$’s.

Case ${\mathcal {T}}_s>{\mathcal {T}}_m$. From estimates (C.2) and (C.3) it comes

$$\begin{aligned} \left\{ \begin{aligned}&{\mathcal {P}} \left( E^*_{s-m}\right) = \frac{d_s}{d_m} \frac{{\mathcal {T}}_s}{{\mathcal {T}}_s- {\mathcal {T}}_m} \eta + {\mathcal {O}}(\eta ^2),\\&\text {with} \quad {\mathcal {R}}_s= {\mathcal {T}}_s (1-\varepsilon _s)>1, \quad \text {and } {\mathcal {T}}_s > {\mathcal {T}}_m. \end{aligned} \right. \end{aligned}$$

Case ${\mathcal {T}}_s<{\mathcal {T}}_m$. From estimates (C.2) and (C.3) it comes

$$\begin{aligned} \left\{ \begin{aligned}&{\mathcal {P}} \left( E^*_{s-m}\right) = 1- \frac{d_m \left( {\mathcal {T}}_m -{\mathcal {T}}_s \right) }{4B_0\tau _s\beta _s} \left( \left( \frac{{\mathcal {T}}_s +{\mathcal {T}}_m}{{\mathcal {T}}_s -{\mathcal {T}}_m} \right) ^2 -1\right) \eta + {\mathcal {O}}(\eta ^2),\\&\text {with} \quad {\mathcal {R}}_m= {\mathcal {T}}_m (1-\varepsilon _m)>1, \quad \text {and } {\mathcal {T}}_m > {\mathcal {T}}_s. \end{aligned} \right. \end{aligned}$$

The stationary state $E^*= \left( B^*,N_s^*,N_m^*,N_p^*,N^*_{m.p} \right)$. By setting $u=(N_s,N_m)^T$, $v=(N_p,N_{m.p})^T$ and taking $\dot{u}= \dot{v}=0$ in (2.3), we have

$$\begin{aligned} \left\{ \begin{aligned}&B\left( G u+ L_pv\right) =Du+ H(N)[N_p+N_{m.p}] \left( N_s, N_m \right) ^T,\\&B(G_p-L_p)v= D_pv- H(N)[N_p+N_{m.p}] \left( N_s, N_m \right) ^T , \end{aligned} \right. \end{aligned}$$

i.e.

$$\begin{aligned} B L (u,v)+\alpha F(u,v) = (u,v), \end{aligned}$$

(C.4)

with

$$\begin{aligned} \begin{aligned}&L= {\mathcal {D}}^{-1} {\mathcal {G}}= \left[ \begin{array}{cc} D^{-1}G &{}\quad D^{-1}L_p\\ 0 &{}\quad D_p^{-1}(G_p-L_p) \end{array} \right] , \\&{\mathcal {G}}= \left[ \begin{array}{cc} G &{}\quad L_p\\ 0 &{}\quad G_p-L_p \end{array} \right] ; \quad {\mathcal {D}}= \left[ \begin{array}{cc} D &{}\quad 0\\ 0 &{}\quad D_p \end{array} \right] ,\\&F(u,v)= \frac{N_p+ N_{m.p}}{a_H+b_HN} \left[ \begin{array}{c} -D^{-1}u \\ D_p^{-1}u \end{array} \right] . \end{aligned} \end{aligned}$$

Note that the spectrum of L is $\sigma (L)= \sigma \left( D^{-1}G\right) \cup \sigma \left( D_p^{-1}(G_p-L_p)\right)$ and the spectral radius of matrices $D^{-1}G$ and $D_p^{-1}(G_p-L_p)$ are given by

$$\begin{aligned} \begin{aligned}&r(D^{-1}G)= \frac{1}{2B_0} \left\{ {\mathcal {R}}_s +{\mathcal {R}}_m + \left[ \left( {\mathcal {R}}_s -{\mathcal {R}}_m \right) ^2 +4 {\mathcal {K}}_{m\rightarrow s} {\mathcal {K}}_{s\rightarrow m} \right] ^{1/2} \right\} ,\\&r(D_p^{-1}(G_p-L_p))= \frac{1}{2B_0} \left\{ {\mathcal {R}}_p +{\mathcal {R}}_{m.p} + \left[ \left( {\mathcal {R}}_p -{\mathcal {R}}_{m.p} \right) ^2 +4 {\mathcal {K}}_{m.p\rightarrow p} {\mathcal {K}}_{p\rightarrow m.p} \right] ^{1/2} \right\} . \end{aligned} \end{aligned}$$

We now introduced a parametric representation of the stationary state $E^*$ with respect to the small parameter $\alpha$. Using the Lyapunov–Schmidt expansion (see Cushing 1998 for more details), the expanded variables are

$$\begin{aligned} \begin{aligned}&u= u^0+ \alpha u^1+ \cdots ,\\&v= v^0+ \alpha v^1+ \cdots ,\\&B= b^0+ \alpha b^1+ \cdots ,\\&F(u,v)= \frac{N_p^0+ N_{m.p}^0}{a_H+b_HN^0} \left[ \begin{array}{c} -D^{-1}u^0 \\ D_p^{-1}u^0 \end{array} \right] + \cdots , \end{aligned} \end{aligned}$$

(C.5)

with $u^0=(N_s^0,N_m^0)$, $v^0=(N_p^0,N_{m.p}^0)$ and $N^0=N_s^0+N_m^0+N_p^0+N_{m.p}^0.$

Evaluating the substitution of expansions (C.5) into the eigenvalue equation (C.4) at ${\mathcal {O}}(\alpha ^0)$ produces $b_0 L (u^0,v^0)= (u^0,v^0)$, i.e.

$$\begin{aligned} \left\{ \begin{aligned}&b^0D^{-1}Gu^0 + b^0D^{-1}L_pv^0=u^0,\\&b^0D_p^{-1}(G_p-L_p)v^0=v^0. \end{aligned}\right. \end{aligned}$$

(C.6)

Since we are interested in $v_0>0$, System (C.6) leads to $b^0 D_p^{-1}(G_p-L_p)v^0=v^0$. The irreducibility of the matrix $D_p^{-1}(G_p-L_p)$ gives that $(1/b_0,v^0)$ is the principal eigenpair of $D_p^{-1}(G_p-L_p)$. That is $1/b^0= r(D_p^{-1}(G_p-L_p))$ and $v^0= c_0 \frac{\varphi _0}{\Vert \varphi _0\Vert _1}$, wherein $c_0$ is a positive constant and

$\varphi _0= \left( {\mathcal {R}}_p -{\mathcal {R}}_{m.p} + \left[ \left( {\mathcal {R}}_p -{\mathcal {R}}_{m.p} \right) ^2 +4 {\mathcal {K}}_{m.p\rightarrow p} {\mathcal {K}}_{p\rightarrow m.p} \right] ^{1/2} ,2 {\mathcal {K}}_{p\rightarrow m.p}\right) ^T.$

Again, System (C.6) gives

$$\begin{aligned} (D^{-1}G -1/b^0{\mathbb {I}})u^0 = -D^{-1}L_pv^0. \end{aligned}$$

(C.7)

Since $D^{-1}L_pv^0 = \text {diag} \left( {\mathcal {K}}_{p\rightarrow s}, {\mathcal {K}}_{m.p\rightarrow m}\right) v^0$ which is positive, equation (C.7) can be solved for $u_0>0$ iff $1/b^0> r\left( D^{-1}G\right)$ and so $u^0 = -(D^{-1}G -1/b^0{\mathbb {I}})^{-1} D^{-1}L_pv^0.$

From the $\dot{B}$-equation of the model, the term of order ${\mathcal {O}}(\alpha ^0)$ leads to

$$\begin{aligned} c_0= & {} \frac{\Vert \varphi _0\Vert _1 d\left[ B_0/b^0-1 \right] }{ \left\langle \beta , \left( -(D^{-1}G -1/b^0{\mathbb {I}})^{-1} D^{-1}L_p\varphi _0, \varphi _0\right) \right\rangle }>0 \\&\Longleftrightarrow B_0/b^0=B_0r(D_p^{-1}(G_p-L_p)) >1. \end{aligned}$$

Consequently, it comes $b_0>0$ and $(u^0,v^0)>0$ are such that

$$\begin{aligned} \left\{ \begin{aligned}&1/b^0=r(D_p^{-1}(G_p-L_p)),\\&v^0= \frac{ d\left[ B_0/b^0-1 \right] }{ \left\langle \beta , \left( -(D^{-1}G -1/b^0{\mathbb {I}})^{-1} D^{-1}L_p\varphi _0, \varphi _0\right) \right\rangle } \varphi _0,\\&u^0 = -(D^{-1}G -1/b^0{\mathbb {I}})^{-1} D^{-1}L_pv^0, \end{aligned}\right. \end{aligned}$$

conditioned by

$$\begin{aligned} r(D_p^{-1}(G_p-L_p))> r\left( D^{-1}G\right) \quad \text { and } \quad B_0r(D_p^{-1}(G_p-L_p)) >1. \end{aligned}$$

(C.8)

Again, evaluating the substitution of expansions (C.5) into the eigenvalue equation (C.4) at ${\mathcal {O}}(\alpha )$ produces

$$\begin{aligned} \left( b^0L-{\mathbb {I}}\right) (u^1,v^1) =- \frac{b^1}{b^0} (u^0,v^0) - \frac{N_p^0+ N_{m.p}^0}{a_H+b_HN^0} \left[ \begin{array}{c} -D^{-1}u^0 \\ D_p^{-1}u^0 \end{array} \right] . \end{aligned}$$

that is

As $1/b^0$ is a characteristic value of $D_p^{-1}(G_p-L_p)$, $(b^0D_p^{-1}(G_p-L_p)-{\mathbb {I}})$ is a singular matrix. Thus, for (C.9b) to have a solution, the right-hand side of (C.9b) must be orthogonal to the null space of the adjoint $(b^0D_p^{-1}(G_p-L_p)-{\mathbb {I}})^{T}$ of $(b^0D_p^{-1}(G_p-L_p)-{\mathbb {I}})$. The null space of $(b^0D_p^{-1}(G_p-L_p)-{\mathbb {I}})^{T}$ is spanned by $\omega _0$, where $\omega _0^T$ is the eigenvector of $(D_p^{-1}(G_p-L_p))^T$ corresponding to the eigenvalue $1/b^0$ and normalized such that $\Vert \omega _0\Vert _1=1$. The Fredholm condition for the solvability of (C.9b) is $\omega _0 \cdot \left( \frac{b_1}{b_0} v^0 + \frac{N_p^0+ N_{m.p}^0}{a_H+b_HN^0} D_p^{-1}u^0 \right) = 0$. Which gives

$$\begin{aligned} b^1= -b^0 \frac{N_p^0+ N_{m.p}^0}{a_H+b_HN^0} \frac{ \omega _0\cdot D_p^{-1}u^0 }{\omega _0 \cdot v^0}, \end{aligned}$$

and so

$$\begin{aligned} \left\{ \begin{aligned}&(N_s,N_m)= u^0+ \alpha u^1+ {\mathcal {O}}(\alpha ^2),\\&(N_p,N_{m.p})= v^0+ \alpha v^1+ {\mathcal {O}}(\alpha ^2),\\&B= b^0+ \alpha b^1+ {\mathcal {O}}(\alpha ^2). \end{aligned}\right. \end{aligned}$$

Approximation of $E^*$ for small mutation and HGT flux rates. Here we derive a simple approximation of the stationary state $E^*$ when mutation and HGT flux rates $\varepsilon _j$ and $\alpha$ are small. Without loss of generality, we express parameters $\varepsilon _j$ and $\alpha$ as functions of the same quantity, let us say $\eta$, with $\eta \ll 1$. First, we have

$$\begin{aligned} \begin{aligned} 1/b^0&= r(D_p^{-1}(G_p-L_p))= B_0^{-1} \max \left( {\mathcal {R}}_p, {\mathcal {R}}_{m.p} \right) + {\mathcal {O}}(\eta ^2)\\&= (1-\theta )(1-\eta ) B_0^{-1} \max \left( {\mathcal {T}}_p, {\mathcal {T}}_{m.p} \right) + {\mathcal {O}}(\eta ^2). \end{aligned} \end{aligned}$$

Next, we consider two cases ${\mathcal {R}}_p> {\mathcal {R}}_{m.p}$ and ${\mathcal {R}}_p< {\mathcal {R}}_{m.p}$, i.e. ${\mathcal {T}}_p> {\mathcal {T}}_{m.p}$ and ${\mathcal {T}}_p< {\mathcal {T}}_{m.p}$ for small mutations $\varepsilon _j$’s.

Case ${\mathcal {T}}_p> {\mathcal {T}}_{m.p}$. Recall that condition (C.8) for the existence of the stationary state $E^*$ simply rewrites

$$\begin{aligned} {\mathcal {R}}_p> \max \left( {\mathcal {R}}_s, {\mathcal {R}}_m \right) \quad \text { and } \quad {\mathcal {R}}_p >1, \end{aligned}$$

which, for small $\eta$, rewrites

$$\begin{aligned} (1-\theta ){\mathcal {T}}_p> \max \left( {\mathcal {T}}_s, {\mathcal {T}}_m \right) \quad \text { and } \quad (1-\theta ){\mathcal {T}}_p >1. \end{aligned}$$

We have $\varphi _0= \left( ({\mathcal {T}}_p -{\mathcal {T}}_{m.p})(1-\theta )(1-\eta ) , \eta B_0\tau _p\beta _p/d_{m.p}\right) ^T + {\mathcal {O}}(\eta ^2)$ and $\omega _0= \left( ({\mathcal {T}}_p -{\mathcal {T}}_{m.p})(1-\theta )(1-\eta ) , \eta B_0\tau _{m.p}\beta _{m.p}/d_p \right) + {\mathcal {O}}(\eta ^2)$. Which gives the following approximation of the stationary state $E^*$:

$$\begin{aligned} \begin{aligned}&B^*= \frac{B_0 }{(1-\theta ){\mathcal {T}}_p} \left( 1+ \eta \right) + {\mathcal {O}}(\eta ^2),\\&(N_p^*,N_{m.p}^*)= \frac{c_0}{\Delta ^\eta _{p-m.p} + \eta \frac{B_0\tau _p\beta _p}{d_{m.p}}} \left( \Delta ^\eta _{p-m.p} , \eta \frac{B_0\tau _p\beta _p}{d_{m.p}} \right) ^T + {\mathcal {O}}(\eta ^2),\\&(N_s^*,N_m^*)= \frac{B_0c_0 \theta }{\Delta ^\eta _{p-s} \left( \Delta ^\eta _{p-m.p} + \eta \frac{B_0\tau _p\beta _p}{d_{m.p}}\right) } \\&\times \left( \Delta ^\eta _{p-m.p} \frac{B_0\tau _p\beta _p}{d_s}, \eta \frac{\Delta ^\eta _{p-m.p} \frac{B_0\tau _s\beta _s}{d_m} \frac{B_0\tau _p\beta _p}{d_s} + \Delta ^\eta _{p-s} \frac{B_0\tau _{m.p}\beta _{m.p}}{d_m} \frac{B_0\tau _p\beta _p}{d_{m.p}} }{ \Delta ^\eta _{p-m} } \right) ^T+ {\mathcal {O}}(\eta ^2) \end{aligned} \end{aligned}$$

(C.10)

with $\Delta ^\eta _{p-m.p}= ({\mathcal {T}}_p -{\mathcal {T}}_{m.p})(1-\theta )(1-\eta )$, $\Delta ^\eta _{p-m}= {\mathcal {T}}_p (1-\theta )(1-\eta )- {\mathcal {T}}_m (1-\eta )$,

$\Delta ^\eta _{p-s}= {\mathcal {T}}_p (1-\theta )(1-\eta )- {\mathcal {T}}_s (1-\eta )$, $c_0= \frac{d \left( {\mathcal {R}}_p -{\mathcal {R}}_{m.p} + {\mathcal {K}}_{p\rightarrow m.p}\right) \left( {\mathcal {R}}_p-1 \right) }{ \beta ^* }>0$ and

$$\begin{aligned} \begin{aligned} \beta ^*&= \beta _p ({\mathcal {R}}_p-{\mathcal {R}}_{m.p}) + \beta _{m.p} {\mathcal {K}}_{p\rightarrow m.p}\\&\quad + \frac{B_0}{ ({\mathcal {R}}_s -{\mathcal {R}}_p) ({\mathcal {R}}_m-{\mathcal {R}}_p)} \left\{ \beta _s ({\mathcal {R}}_p-{\mathcal {R}}_m)({\mathcal {R}}_p -{\mathcal {R}}_{m.p}) {\mathcal {K}}_{p\rightarrow s}\right. \\&\quad \left. +\,\beta _m \left[ ({\mathcal {R}}_p-{\mathcal {R}}_{m.p}){\mathcal {K}}_{s\rightarrow m} {\mathcal {K}}_{p\rightarrow s} + ({\mathcal {R}}_p-{\mathcal {R}}_s){\mathcal {K}}_{m.p\rightarrow m} {\mathcal {K}}_{p\rightarrow m.p}\right] \right\} . \end{aligned} \end{aligned}$$

From where, ${\mathcal {P}}(E^*)= {\mathcal {P}}^*_m+ {\mathcal {P}}^*_p+ {\mathcal {P}}^*_{m.p}$,

with ${\mathcal {P}}^*_m= \eta \frac{B_0 \theta }{\Delta ^\eta _{p-s} } \frac{\Delta ^\eta _{p-m.p} \frac{B_0\tau _s\beta _s}{d_m} \frac{B_0\tau _p \beta _p}{d_s} + \Delta ^\eta _{p-s} \frac{B_0\tau _{m.p}\beta _{m.p}}{d_m} \frac{B_0\tau _p\beta _p}{d_{m.p}} }{ \Delta ^\eta _{p-m} N^\eta }$, ${\mathcal {P}}^*_p= \frac{\Delta ^\eta _{p-m.p}}{N^\eta }$, ${\mathcal {P}}^*_{m.p} = \eta \frac{B_0\tau _p\beta _p}{d_{m.p} N^\eta }$ and $N^\eta =\Delta ^\eta _{p-m.p} + \eta \frac{B_0\tau _p\beta _p}{d_{m.p}} + \frac{B_0 \theta }{\Delta ^\eta _{p-s} } \left( \Delta ^\eta _{p-m.p} \frac{B_0\tau _p\beta _p}{d_s}+ \eta \frac{\Delta ^\eta _{p-m.p} \frac{B_0\tau _s\beta _s}{d_m} \frac{B_0\tau _p\beta _p}{d_s} + \Delta ^\eta _{p-s} \frac{B_0\tau _{m.p}\beta _{m.p}}{d_m} \frac{B_0\tau _p\beta _p}{d_{m.p}} }{ \Delta ^\eta _{p-m} } \right) .$

With the Taylor expansion it comes

$$\begin{aligned} \begin{aligned} {\mathcal {P}}^*_m=&\eta \frac{B_0 \theta }{{\mathcal {T}}_p (1-\theta )- {\mathcal {T}}_s} \frac{ ({\mathcal {T}}_p -{\mathcal {T}}_{m.p})(1-\theta ) \frac{B_0\tau _s \beta _s}{d_m} \frac{B_0\tau _p\beta _p}{d_s} + \left( {\mathcal {T}}_p (1-\theta ) - {\mathcal {T}}_s\right) \frac{B_0\tau _{m.p}\beta _{m.p}}{d_m} \frac{B_0\tau _p\beta _p}{d_{m.p}} }{ \left( {\mathcal {T}}_p (1-\theta ) - {\mathcal {T}}_m \right) ({\mathcal {T}}_p -{\mathcal {T}}_{m.p})(1-\theta ) \left( 1 + \frac{B_0 \theta }{{\mathcal {T}}_p (1-\theta )- {\mathcal {T}}_s} \frac{B_0\tau _p\beta _p}{d_s} \right) }+ {\mathcal {O}}(\eta ^2), \\ {\mathcal {P}}^*_p=&\frac{1}{\left( 1 + \frac{B_0 \theta }{{\mathcal {T}}_p (1-\theta )- {\mathcal {T}}_s} \frac{B_0\tau _p\beta _p}{d_s} \right) } + {\mathcal {O}}(\eta ),\\ {\mathcal {P}}^*_{m.p}&= \eta \frac{B_0\tau _p\beta _p}{d_{m.p} ({\mathcal {T}}_p -{\mathcal {T}}_{m.p})(1-\theta ) \left( 1 + \frac{B_0 \theta }{{\mathcal {T}}_p (1-\theta )- {\mathcal {T}}_s} \frac{B_0\tau _p\beta _p}{d_s} \right) }+ {\mathcal {O}}(\eta ^2). \end{aligned} \end{aligned}$$

Case ${\mathcal {T}}_p< {\mathcal {T}}_{m.p}$. Again, condition (C.8) for the existence of the stationary state $E^*$ becomes

$$\begin{aligned} {\mathcal {R}}_{m.p}> \max \left( {\mathcal {R}}_s, {\mathcal {R}}_m \right) \quad \text { and } \quad {\mathcal {R}}_{m.p} >1, \end{aligned}$$

which, for sufficiently $\eta$, rewrites

$$\begin{aligned} (1-\theta ){\mathcal {T}}_{m.p}> \max \left( {\mathcal {T}}_s, {\mathcal {T}}_m \right) \quad \text { and } \quad (1-\theta ){\mathcal {T}}_{m.p} >1. \end{aligned}$$

Again, by Taylor expansion, we find

$$\begin{aligned} \begin{aligned}&B^*= \frac{B_0 }{(1-\theta ){\mathcal {T}}_{m.p}} \left( 1+ \eta \right) + {\mathcal {O}}(\eta ^2),\\&\text {and}\\&v^0= c_0 \left( 0, 1\right) + {\mathcal {O}}(\eta ^2). \end{aligned} \end{aligned}$$

(C.11)

Since $u^0= -(D^{-1}G -1/b^0{\mathbb {I}})^{-1} D^{-1}L_pv^0,$ we then have

$$\begin{aligned} u^0= \frac{c_0 \theta B_0^2 \tau _p\beta _p }{d_m \left[ {\mathcal {T}}_{m.p}(1-\theta )(1-\eta )-{\mathcal {T}}_m(1-\eta ) \right] } \left( 0,1\right) + {\mathcal {O}}(\eta ^2). \end{aligned}$$

From where, ${\mathcal {P}}(E^*)= {\mathcal {P}}^*_m+ {\mathcal {P}}^*_p+ {\mathcal {P}}^*_{m.p}$, with

$$\begin{aligned} \begin{aligned} {\mathcal {P}}^*_m&= \frac{\theta B_0^2 \tau _p\beta _p}{\theta B_0^2 \tau _p\beta _p +d_m \left[ {\mathcal {T}}_{m.p}(1-\theta )-{\mathcal {T}}_m \right] } + {\mathcal {O}}(\eta ),\\ {\mathcal {P}}^*_p&= {\mathcal {O}}(\eta ),\\ {\mathcal {P}}^*_{m.p}&= \frac{d_m \left[ {\mathcal {T}}_{m.p}(1-\theta )-{\mathcal {T}}_m \right] }{\theta B_0^2 \tau _p\beta _p +d_m \left[ {\mathcal {T}}_{m.p}(1-\theta )-{\mathcal {T}}_m \right] } + {\mathcal {O}}(\eta ). \end{aligned} \end{aligned}$$

Proof of Theorem 4.2

The linearized system at a given stationary state $E^*=(B^*,u^*,v^*)$ writes

$$\begin{aligned} \left( \dot{B},\dot{u},\dot{v}\right) ^T=J[E^*] \left( B,u,v\right) ^T \end{aligned}$$

(D.12)

wherein $J[E^*]$ is defined by the Jacobian matrix associated to (D.12) and is given by

$$\begin{aligned} \begin{aligned}&J[E^*]= \left( \begin{array}{ccc} -\left[ d + \left\langle \beta ,(u^*,v^*)\right\rangle \right] &{}\quad - B^* (\beta _s, \beta _m) &{}\quad - B^* (\beta _p,\beta _{m.p})\\ Gu^*+L_pv^* &{}\quad B^*G-D- M[E^*] &{} -N[E^*] +B^*L_p\\ (G_p-L_p)v^* &{}\quad M[E^*] &{} B^*(G_p-L_p)-D_p +N[E^*] \end{array} \right) ,\\&M[E^*]= \left( \begin{array}{cc} H(N^*)(N^*_p+N^*_{m.p}) + \xi ^*_s &{}\quad \xi ^*_s \\ \xi ^*_m &{}\quad H(N^*)(N^*_p+N^*_{m.p}) + \xi ^*_m \end{array} \right) ,\\&N[E^*]= \left( \begin{array}{cc} H(N^*)N^*_s+ \xi ^*_s &{}H(N^*)N^*_s+ \xi ^*_s \\ H(N^*)N^*_m+ \xi ^*_m &{}\quad H(N^*)N^*_m + \xi ^*_m \end{array} \right) , \end{aligned}\nonumber \\ \end{aligned}$$

(D.13)

with $\xi ^*_s=H'(N^*)N^*_s(N^*_p+N^*_{m.p}),$ and $\xi ^*_m=H'(N^*)N^*_m(N^*_p+N^*_{m.p})$. Without loss of generality, and when necessary, we express parameters $\varepsilon _j$ and $\alpha$ as functions of the same quantity, let us say $\eta$, with $\eta \ll 1$.

Recall that the stability modulus of a matrix M is $s_0(M)=\{\max \text {Re}(z): z\in \sigma (M) \}$ and M is said to be locally asymptotically stable (l.a.s.) if $s_0(M)<0$ (Li and Wang 1998).

Stablity of $E^*_{s-m}$. At point $E^*_{s-m}$, the Jacobian matrix $J[E^*_{s-m}]$ writes

$$\begin{aligned} J[E^*_{s-m}]= \left( \begin{array}{cc} Z[E^*_{s-m}] &{}\quad X[E^*_{s-m}]\\ 0 &{}\quad Y[E^*_{s-m}] \end{array} \right) , \end{aligned}$$

with

$$\begin{aligned} \begin{aligned}&Z[E^*_{s-m}]= \left( \begin{array}{cc} -\left[ d + \left\langle (\beta _s,\beta _m),u^*\right\rangle \right] &{}\quad - B^* (\beta _s,\beta _m) \\ Gu^* &{}\quad B^*G-D \end{array} \right) ,\\&X[E^*_{s-m}]= \left( \begin{array}{c} - B^* (\beta _p,\beta _{m.p})\\ -N[E^*_{s-m}] +B^*L_p \end{array} \right) ,\\&Y[E^*_{s-m}]= B^*(G_p-L_p)-D_p +N[E^*_{s-m}]. \end{aligned} \end{aligned}$$

Then, $E^*_{s-m}$ is unstable if $s_0 \left( Y[E^*_{s-m}]\right) >0$ and a necessary condition for the stability of $E^*_{s-m}$ is that $s_0 \left( Y[E^*_{s-m}]\right) <0$. By setting $h^*_j= \frac{N^*_j}{a_H+b_HN^*}$, with $j=s,m$; note that

$$\begin{aligned} Y[E^*_{s-m}]= \left[ \begin{array}{ll} &{}B^*\tau _p\beta _p(1-\varepsilon _p)(1-\theta ) -d_p +\alpha h^*_s \quad B^* \varepsilon _{m.p}(1-\theta )\tau _{m.p}\beta _{m.p} +\alpha h^*_s \\ &{} B^*\varepsilon _p(1-\theta )\tau _p\beta _p +\alpha h^*_m \quad B^*\tau _{m.p}\beta _{m.p} (1-\varepsilon _{m.p})(1-\theta ) -d_{m.p} +\alpha h^*_m \end{array} \right] \end{aligned}$$

and eigenvalues $z_1$, $z_2$ of $Y[E^*_{s-m}]$ are such that

$$\begin{aligned} \begin{aligned} z_1&= B^*\tau _p\beta _p(1-\varepsilon _p)(1-\theta ) -d_p +\alpha h^*_s + {\mathcal {O}}(\eta ^2)\\&= d_p \left( \frac{B^*}{B_0}{\mathcal {R}}_p +\frac{\alpha h^*_s}{d_p} -1\right) + {\mathcal {O}}(\eta ^2),\\ z_2&= B^*\tau _{m.p}\beta _{m.p} (1-\varepsilon _{m.p})(1-\theta ) -d_{m.p} +\alpha h^*_m + {\mathcal {O}}(\eta ^2)\\&=d_{m.p} \left( \frac{B^*}{B_0}{\mathcal {R}}_{m.p} +\frac{\alpha h^*_m}{d_{m.p}} -1\right) +{\mathcal {O}}(\eta ^2). \end{aligned} \end{aligned}$$

From where, for sufficiently small mutation and flux of HGT rates, we have

$$\begin{aligned} s_0(Y[E^*_{s-m}])<0 \Longleftrightarrow \left\{ \begin{aligned}&\frac{{\mathcal {R}}_p}{\max \left( {\mathcal {R}}_s,{\mathcal {R}}_m\right) } +\alpha \frac{ h^*_s}{d_p}<1,\\&\frac{{\mathcal {R}}_{m.p}}{\max \left( {\mathcal {R}}_s,{\mathcal {R}}_m\right) } + \alpha \frac{ h^*_m}{d_{m.p}} <1. \end{aligned} \right. \end{aligned}$$

Next, it remains to check the stability of the block matrix $Z[E^*_{s-m}]$:

$$\begin{aligned} Z[E^*_{s-m}]= \left( \begin{array}{ccc} -\left[ d + \beta _sN^*_s+ \beta _m N^*_m\right] &{}\quad - B^* \beta _s &{}\quad -B^*\beta _m \\ \tau _s\beta _s(1-\varepsilon _s)N^*_s +\varepsilon _m\tau _m\beta _mN^*_m &{}\quad B^*\tau _s\beta _s(1-\varepsilon _s) -d_s &{}\quad B^*\varepsilon _m\tau _m\beta _m\\ \varepsilon _s\tau _s\beta _s N^*_s+ \tau _m\beta _m(1-\varepsilon _m)N^*_m &{}\quad B^*\varepsilon _s\tau _s\beta _s &{}\quad B^*\tau _m\beta _m(1-\varepsilon _m) -d_m \end{array} \right) . \end{aligned}$$

Again, recalling that $(N^*_s,N^*_m)= \frac{d\left( B_0r(D^{-1}G)-1\right) }{\left\langle (\beta _s,\beta _m),\phi _0\right\rangle } \phi _0$, the expansion of the matrix $Z[E^*_{s-m}]$ takes the form $Z[E^*_{s-m}]= Z^0+ \eta Z^\eta$, wherein

$$\begin{aligned} Z^0= \left( \begin{array}{ccc} -\left[ d + \beta _s n_s^0 \right] &{}\quad - b^0 \beta _s &{}\quad -b^0\beta _m \\ \tau _s\beta _sn^0_s &{}\quad b^0\tau _s\beta _s -d_s &{}\quad 0\\ 0 &{}\quad 0 &{}\quad b^0\tau _m\beta _m -d_m \end{array} \right) . \end{aligned}$$

with $n^0_s= \frac{c^0}{2} \left( {\mathcal {T}}_s- {\mathcal {T}}_m +|{\mathcal {T}}_s-{\mathcal {T}}_m | \right)$, $\frac{1}{b^0}= \frac{1}{2B_0} \left( {\mathcal {T}}_s+ {\mathcal {T}}_m +|{\mathcal {T}}_s-{\mathcal {T}}_m | \right)$, and where $c^0$ is a positive constant (which does not depends on $\eta$). We first assume that ${\mathcal {T}}_s > {\mathcal {T}}_m$. Then, eigenvalues $z^0_k$ ($k=1,2,3$) of $Z^0$ are such that $z^0_1= b^0\tau _m\beta _m -d_m$, and $z^0_2$, $z^0_3$ are solution of

$$\begin{aligned} z^2+ \left[ d + \beta _s n_s^0+ d_s- b^0\tau _s\beta _s \right] z +b^0\tau _s\beta _s^2n^0_s+ (d + \beta _s n_s^0)( d_s- b^0\tau _s\beta _s)=0. \end{aligned}$$

Since $d_s- b^0\tau _s\beta _s= d_s \left( 1- \frac{2}{{\mathcal {T}}_s+ {\mathcal {T}}_m +|{\mathcal {T}}_s-{\mathcal {T}}_m |} {\mathcal {T}}_s \right) \ge 0$, coefficients of the previous polynomial are positive and so ${\mathcal {R}}_e(z^0_2)<0$, ${\mathcal {R}}_e(z^0_3)<0$. Moreover, $z^0_1= d_m \left( \frac{2}{{\mathcal {T}}_s+ {\mathcal {T}}_m +|{\mathcal {T}}_s-{\mathcal {T}}_m |} {\mathcal {T}}_m -1 \right) \le 0$, from where $s_0(Z^0)<0$, which gives $s_0(Z[E^*_{s-m}])<0$ for $\eta$ sufficiently small (Kato 1976). Consequently, for small $\eta$, $E^*_{s-m}$ is l.a.s if and only if $s_0(Y[E^*_{s-m}])<0$.

Stability of $E^*$. Let

$$\begin{aligned} \left\{ \begin{aligned}&u^*=(x_s^*,x_m^*)+ {\mathcal {O}}(\eta ),\\&v^*=(x_p^*,x_{m.p}^*)+ {\mathcal {O}}(\eta ),\\&B^*= b^*_0+ {\mathcal {O}}(\eta ). \end{aligned} \right. \end{aligned}$$

Then $J[E^*]$ takes the form $J[E^*]= J^*_0 + \eta J^*_\eta$, with $J^*_0=$

$$\begin{aligned} \begin{aligned}&\left( \begin{array}{ccccc} -\left( d + \sum _j \beta _jx^*_j\right) &{} - b^*_0 \beta _s &{} - b^*_0\beta _m &{} - b^*_0 \beta _p &{} - b^*_0 \beta _{m.p}\\ \mu _s x^*_s +\theta \mu _p x^*_p&{} b^*_0\mu _s-d_s &{} 0 &{}b^*_0\theta \mu _p &{}0 \\ \mu _m x^*_m +\theta \mu _{m.p} x^*_{m.p} &{} 0 &{} b^*_0\mu _m -d_m &{} 0 &{} b^*_0\theta \mu _{m.p}\\ (1-\theta ) \mu _p x^*_p &{} 0 &{}0 &{}b^*_0(1-\theta )\mu _p -d_p&{} 0\\ (1-\theta ) \mu _{m.p} x^*_{m.p} &{} 0 &{}0 &{}0 &{} b^*_0(1-\theta )\mu _{m.p}-d_{m.p} \end{array} \right) .\quad \end{aligned} \end{aligned}$$

(D.14)

Recall that, for $\eta$ sufficiently small, the existence of $E^*$ is ensured by

$$(1-\theta )\max \left( {\mathcal {T}}_p,{\mathcal {T}}_{m.p}\right) > \max \left( {\mathcal {T}}_s,{\mathcal {T}}_{m}\right) .$$

If ${\mathcal {T}}_p> {\mathcal {T}}_{m.p}$, from (C.10), we have $b^*_0=\frac{B_0}{(1-\theta ){\mathcal {T}}_p}$, $x^*_m=x^*_{m.p}=0$, and $J^*_0$ rewrites $J^*_0=$

$$\begin{aligned} \begin{aligned}&\left( \begin{array}{ccccc} -\left( d + \sum _j \beta _jx^*_j\right) &{}\quad - b^*_0 \beta _s &{}\quad - b^*_0\beta _m &{}\quad - b^*_0 \beta _p &{}\quad - b^*_0 \beta _{m.p}\\ \mu _s x^*_s +\theta \mu _p x^*_p&{}\quad b^*_0\mu _s-d_s &{}\quad 0 &{}\quad b^*_0\theta \mu _p &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad b^*_0\mu _m -d_m &{}\quad 0 &{}\quad b^*_0\theta \mu _{m.p}\\ (1-\theta ) \mu _p x^*_p &{}\quad 0 &{}\quad 0 &{}\quad 0&{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad b^*_0(1-\theta )\mu _{m.p}-d_{m.p} \end{array} \right) . \end{aligned} \end{aligned}$$

Denoting by $\sigma (\cdot )$ the spectrum of a given matrix, we have $\sigma (J^*_0)=\left\{ b^*_0(1-\theta )\mu _{m.p}-d_{m.p}, b^*_0\mu _m-d_m \right\} \cup \sigma (Y^*_0)$, where

$$\begin{aligned} Y^*_0=\left( \begin{array}{ccc} -\left( d + \sum _j \beta _jx^*_j\right) &{}\quad - b^*_0 \beta _s &{}\quad - b^*_0 \beta _p \\ \mu _s x^*_s +\theta \mu _p x^*_p&{}\quad b^*_0\mu _s-d_s &{}\quad b^*_0\theta \mu _p \\ (1-\theta ) \mu _p x^*_p &{}\quad 0 &{}\quad 0 \end{array} \right) . \end{aligned}$$

Since $b^*_0(1-\theta )\mu _{m.p}-d_{m.p}= d_{m.p} \left( {\mathcal {T}}_{m.p}/{\mathcal {T}}_p-1\right) <0$ and $b^*_0\mu _{m}-d_m= d_{m} \left( {\mathcal {T}}_{m}/((1-\theta ){\mathcal {T}}_p)-1\right) <0$, then $E^*$ is l.a.s. iff $s_0\left( Y^*_0\right) <0$. The characteristic polynomial of $Y^*_0$ writes $\pi (z)=z^3 +\pi _2 z^2 + \pi _1 z +\pi _0$, with

$$\begin{aligned} \begin{aligned}&\pi _2= d + \beta _sx^*_s + \beta _px^*_p+ d_s-b^*_0\mu _s,\\&\pi _1= (d + \beta _sx^*_s + \beta _px^*_p)(d_s-b^*_0\mu _s) +b_0^*\beta _s (\mu _s x^*_s +\theta \mu _p x^*_p) +b_0^*\beta _p(1-\theta ) \mu _p x^*_p,\\&\pi _0= (1-\theta ) \mu _p x^*_p \left( (b_0^*)^2\beta _s\theta \mu _p + (d_s-b^*_0\mu _s)b_0^*\beta _p \right) . \end{aligned} \end{aligned}$$

By the Routh–Hurwitz stability criterion we deduce that $\pi$ is a stable polynomial (i.e. $s_0\left( Y^*_0\right) <0$), and then $E^*$ is l.a.s..

If ${\mathcal {T}}_p< {\mathcal {T}}_{m.p}$, again from (C.11), we have $b^*_0=\frac{B_0}{(1-\theta ){\mathcal {T}}_{m.p}}$, $x^*_s=x^*_{p}=0$, and the characteristic polynomial of $J^*_0$ writes $|J^*_0 -z {\mathbb {I}}|= (b^*_0\mu _s-d_s-z ) (b^*_0(1-\theta )\mu _p -d_p-z) \times$

$$\begin{aligned} \left| \begin{array}{ccccc} -\left( d + \sum _j \beta _jx^*_j\right) -z &{} \quad - b^*_0\beta _m &{}\quad - b^*_0 \beta _{m.p}\\ \mu _m x^*_m +\theta \mu _{m.p} x^*_{m.p} &{}\quad b^*_0\mu _m -d_m -z &{}\quad b^*_0\theta \mu _{m.p}\\ (1-\theta ) \mu _{m.p} x^*_{m.p} &{}\quad 0 &{}\quad -z \end{array} \right| . \end{aligned}$$

Same arguments as previously lead to the local stability of $E^*$.

The model with mutations depending on the abundance of the parental cells

When the occurrence of new mutants depends on the abundance of the parental cells, the model writes

$$\begin{aligned} \left\{ \begin{aligned}&\dot{B}(t)= \Lambda -dB -B\sum _{j \in {\mathcal {J}}}\beta _jN_j ,\\&\dot{N}_s(t)=\tau _s\beta _s BN_s -(d_s+\varepsilon _s)N_s +\varepsilon _mN_m + \theta \tau _p\beta _p BN_p - H(N) N_s[N_p +N_{m.p}] ,\\&\dot{N}_m(t)= \tau _m\beta _m BN_m - (d_m+\varepsilon _m)N_m + \varepsilon _s N_s +\theta \tau _{m.p}\beta _{m.p} BN_{m.p} - H(N) N_m[N_{m.p}+N_p],\quad \\&\dot{N}_p(t)= \tau _p\beta _p(1-\theta ) BN_p - (d_p+\varepsilon _p)N_p +\varepsilon _{m.p} N_{m.p} +H(N) N_s[N_p +N_{m.p}] ,\\&\dot{N}_{m.p}(t)= \tau _{m.p}\beta _{m.p}(1-\theta ) BN_{m.p} - (d_{m.p}+\varepsilon _{m.p}) N_{m.p} + \varepsilon _p N_p + H(N) N_m[N_{m.p}+N_p], \end{aligned} \right. \end{aligned}$$

(D.15)

wherein state variables and model parameters are the same as for Model (2.2).