Step 2: the mutant density reaches a value C cross (Stochastic Euler

scheme)

Recall that the trait of the successful mutant is R^K+σKh whereh∈ {1, . . . , A}. Due to the regularity assumptions (iv) in Assumption 2, we have the following estimates:

b(R^K+σ_Kh) = b(R^K) +b^′(R^K)σKh+O((σKh)²) (II.7.17) d(R^K+σKh) = d(R^K) +d^′(R^K)σKh+O((σKh)²)

r(R^K+σ_Kh) = r(R^K) +r^′(R^K)σ_Kh+O((σ_Kh)²)

c(R^K+σ_Kh, R^K) = c(R^K, R^K) +∂₁c(R^K, R^K)σ_Kh+O((σ_Kh)²) c(R^K, R^K+σ_Kh) = c(R^K, R^K) +∂₂c(R^K, R^K)σ_Kh+O((σ_Kh)²)

c(R^K+σ_Kh, R^K+σ_Kh) = c(R^K, R^K)+(∂₁c(R^K, R^K)+∂₂c(R^K, R^K))σ_Kh+O((σ_Kh)²). The deterministic system: Although we cannot use a law of large numbers, to understand the behavior of the stochastic system it is useful to look at the properties of the corresponding deterministic Lotka-Volterra system. The limiting system when K → ∞, with σ_K =0, takes the simple form

dm⁰_t

dt = m⁰_t(r(R^K) −c(R^K, R^K)(m⁰_t+m^k_t¹)), (II.7.18) dm^k_t¹

dt = m^k_t¹(r(R^K) −c(R^K, R^K)(m⁰_t+m^k_t¹)). (II.7.19) The corresponding vector field is depicted in Figure II.3. This system has an invariant mani-fold made of fixed points given by the roots of the equation

m⁰+m^k1 =r(R^K)/c(R^K, R^K) =z(R¯ ^K), (II.7.20) with m⁰, m^k1 ≥ 0. This manifold connects the fixed points of the monomorphic equations, (z¯(R^K),0) and (0,z¯(R^K)). Note that z¯(R^K) has the interpretation of the total mass of the population in equilibrium. A simple computation shows that the Hessian matrix on the invariant manifold is given by

H(m⁰, m^k1) = −c(R^K, R^K) (m⁰ m⁰

m^k1 m^k1). (II.7.21) The corresponding eigenvectors are (1,−1) with eigenvalue 0, and (m⁰,¯z(R^K) −m⁰) with eigenvalue −c(R^K, R^K)z¯(R^K).

It follows that the perturbed system dm⁰_t

dt =m⁰_t(r(R^K) −c(R^K, R^K)m⁰_t−c(R^K, R^K+σ_Kh)m^k_t¹), (II.7.22) dm^k_t¹

dt =m^k_t¹(r(R^K+σ_Kh)−c(R^K+σ_Kh, R^K)m⁰_t−c(R^K+σ_Kh, R^K+σ_Kh)m^k_t¹), (II.7.23) has an invariant manifold connecting its fixed points(z¯(R^K),0)and(0,z¯(R^K+σ_Kh)), where z(R¯ ^K+σKh) =r(R^K+σKh)/c(R^K+σKh, R^K+σKh)in aσ_K-neighborhood of the unperturbed invariant manifold (see Figure II.3). Thus the perturbed deterministic system will move quickly towards a small neighborhood of this invariant manifold and then move slowly with speed O(σK) along it. Since the invariant manifold is close to the curve m⁰+m^k1 =z(R¯ ^K),

58 II.7. THE SECOND PHASE OF AN INVASION

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

m⁰ mk1

unperturbed system σK=0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

m⁰ mk1

perturbed system σK=0.01

Figure II.3: Right: Vector field of the unperturbed system (σK = 0), Left: Vector field of the perturbed system (σK=0.01). Parameters are given in Table II.1

Table II.1: Parameters of the Figures II.3

b(R^K) =2 d(R^K) =1 c(R^K, R^K) =1 c(R^K, R^K+σKh) =1−2σK

b(R^K+σKh) =2+σK d(R^K+σKh) =1−σK c(R^K+σKh, R^K) =1−2σK c(R^K+σKh, R^K+σKh) =1−σK

it is reasonable to choose as variables M_t =m⁰_t +m^k_t¹. The motion of the system will then be close to the curve φ(m˜ ^k_t¹) defined by the condition that the derivative ofMt vanishes for M_t=φ˜(m^k_t¹).

Since dM_t

dt =M_t(r(R^K) −c(R^K, R^K)M_t) (II.7.24)

− [(∂1c(R^K, R^K) +∂2c(R^K, R^K))Mt−r^′(R^K)]σ_Khm^k_t¹+O(σ_K² ).

Setting the right hand side to zero yields the leading orders inσ_K

φ˜(m^k_t¹) =z(R^K) +σ_Khm^k_t¹(^r_r^′₍⁽_R^RK^K)⁾−^∂1c(RK,R_c(^K_R⁾⁺K,R^∂2Kc()^RK,RK)) +O(σ_K² ). (II.7.25) We expect that the stochastic system also evolves along this curve. I.e., we will show that m^k1 increases while the total mass stays close to the curve defined in (II.7.25).

Define the function

φ(y) ≡z(R^K) +σ_Khy(^r_r^′₍⁽_R^RK^K)⁾−^∂1c(RK,R_c(^K_R⁾⁺K,R^∂2Kc()^RK,RK)), (II.7.26) and the stopping time

θ_near^K _φ(i

2)≡inf{t≥θ_{mut. size}^K _i(/2)∶ ∣⟨ν˜_t,1⟩ −φ(i(/2))∣ < (M/3)σ_K}. (II.7.27) The dependence ofφwith respect to the mutant density allows us to decompose the increase of the mutant density into successive steps during which the total mass does not move more thanM σ_K.

Lemma II.7.4. Fix >0. Suppose that the assumptions of Theorem II.7.1 hold. Then, there exists a constant M >0 (independent of, K, andi) such that and for all 2≤i≤2⁻¹C_cross , (a) Soon afterθ^K_{mut. sizei(/2)}, the total population size is close to φ(i₂):

II.7. THE SECOND PHASE OF AN INVASION 59

Klim→∞ σ_K⁻¹P[θ_near^K _φ(i

2)> (θ^K_{mut. sizei(/2)}+S_K) ∧θ₂^K_{succ. mut.}∧θ_diversity^K

∧inf{t≥θ^K_{mut. sizei(/2)}∶ ∃k≥1∶M^k(˜νt) = ⌈(i±¹₂)(/2)K⌉} ] =0.

(b) A change of order for the mutant density takes more thano(σ_K⁻¹) time:

Klim→∞σ⁻_K¹P[inf{t≥θ_{mut. size}^K _i(/2)∶ ∃k≥1∶M^k(ν˜_t) = ⌈(i±¹₂)(/2)K⌉}

< (θ_{mut. size}^K _i(/2)+S_K) ∧θ_near^K _φ(i

2)∧θ₂^K_{succ. mut.}∧θ^K_diversity] =0.

(c) At the time when the mutant density has changed of order the total population size is still close to φ(i₂):

Klim→∞σ_K⁻¹P[inf{t≥θ^K_nearφ(i

2)∶∣⟨ν˜_t,1⟩ −φ(i₂)∣ >M σ_K} <θ^K_{2succ. mut.}∧θ^K_diversity

∧inf{t≥θ^K_{mut. sizei(/2)}∶∃k≥1∶M^k(ν˜_t)=⌈(i±1)(/2)K⌉} ] =0.

(d) A change of order for the mutant density takes no more than (iσK)^−1−α/2 time:

Klim→∞σ⁻_K¹P[θ_{mut. size}^K _(i+1)(/2)> (θ_near^K _φ(i

2)+ (iσ_K)^−1−α/2) ∧θ^K_{2succ. mut.}

∧θ^K_diversity∧inf{t≥θ^K_nearφ(i

2)∶ ∣⟨˜νt,1⟩ −φ(i(/2))∣ >M σK} ] =0.

Remark 7. For each >0, Lemma II.7.4 implies that the mutant density reaches the value C_cross with high probability, sinceis independent ofK. Moreover, for all>0,

P[θ^K_{mut. sizeC}

cross > (θ^K_{mut. size}+ ^ln(K)

σ_K^1+α/2) ∧θ^K_{2succ. mut.}∧θ_diversity^K ] =o(σ_K) (II.7.28) and P[ ∣⟨ν˜_θK

mut. sizeCcross

,1⟩ −φ(C_cross )∣ >M σ_K] =o(σ_K). (II.7.29) Proof. We will prove the lemma by induction over i. Base clause: Compare with Lemma II.7.2 and II.7.3 that there exists a constant M>0such that ∣⟨ν˜_θK

mut. size,1⟩ −φ(0)∣is smaller thanM σ_K and that θ^K_{mut. size}<θ^K_{2succ. mut.}∧θ^K_diversity both with probability 1−o(σ_K).

Induction step from i−1to i: Assume that the lemma holds true fori−1, then be prove separately that (a)-(d) are true fori, as long asi<2⁻¹C_cross

Proof. of (a) foriby assuming that the lemma holds for i−1. In the proof we use the following notation

θ˜^K_i ≡θ₂^K_{succ. mut.}∧θ^K_diversity∧inf{t≥θ^K_{mut. sizei(/2)}∶ ∃k≥1∶M^k(˜νt) = ⌈(i±¹₂)₂K⌉}. (II.7.30) Note thatθ˜_i^K differs fromθ˜^K defined in Lemma II.7.3. We will prove(a)provided it happens before θ˜_i^K and we use the estimates of step (b) for ito prove that it indeed happens before θ˜^K_i with high probability.

If the Lemma is true fori−1, we know that (with (d)) P[∣⟨˜ν_θK

mut. sizei(/2),1⟩ −φ((i−1)₂)∣ <M σK] =1−o(σK). (II.7.31) Since φ(x) −φ(y) =O(h(x−y)σ_K), we have with probability1−o(σ_K)either

inf{t≥θ_{mut. size}^K _i(/2)∶ ∣⟨˜νt,1⟩ −φ(i₂)∣ < (M/3)σK} =θ_{mut. size}^K _i(/2), (II.7.32)

60 II.7. THE SECOND PHASE OF AN INVASION which implies (a) for i, or at least

∣⟨ν˜_θK

mut. sizei(/2),1⟩ −φ(i₂)∣ < (M+ ∣h(^r_r^′₍⁽_R^RK^K)⁾−^(∂1c(RK,R_c(^K_R⁾⁺K,R^∂2Kc()^RK,RK)))∣)σ_K. (II.7.33) Similarly as in many previous lemmata we want to coupleK⟨ν˜_t,1⟩with a discrete time Markov chain. Therefore, let

X_tⁱ= ∣K⟨ν˜_t,1⟩ − ⌈φ(i₂)K⌉∣, (II.7.34) and T₀ⁱ = θ_{mut. size}^K _i(/2) and (T_kⁱ)k≥1 be the sequences of the jump times of ⟨ν˜t,1⟩ after θ_{mut. size}^K _i(/2). Then let Y_kⁱ be the associated discrete time process which records the val-ues thatX_tⁱ takes after timeθ_{mut. size}^K _i(/2).

Claim. There exists a constant, C_derivative^b,d,c >0, such that for all ⌈C_derivative^b,d,c σKK⌉ ≤j <

⌈K⌉ and K large enough,

P[Y_nⁱ₊₁ =j+1∣Y_nⁱ =j, Tn+1<θ˜^K_i ] ≤ ¹₂ −σK=∶p^K₊. (II.7.35) Moreover, we can choose

C_derivative^b,d,c =sup

x∈X 1

c(x,x)(4b(x) +A∣^r′(x_r⁾₍^c_x⁽₎^x,x)−∂₁c(x, x) −∂₂c(x, x)∣). (II.7.36) If⟨ν˜_t,1⟩K> ⌈φ(i(/2))K⌉ at timet=T_nⁱ, then ⟨ν˜_Ti

n,1⟩K= ⌈φ(i(/2))K⌉ +Y_nⁱ and, condition-ally onFT_nⁱ, the left hand side of (II.7.35) is equal to the probability that the next event is a birth. Namely,

∑k≥0b(h_k,1(ν˜_Ti

n))M^k(ν˜_Ti n)

∑k≥0(b(hk,1(˜ν_Tnⁱ)) +d(hk,1(ν˜_Tnⁱ)) + ∫N×Xc(hk,1(ν˜_Tnⁱ), ξ)d˜ν_Tnⁱ(ξ))M^k(˜ν_Tnⁱ) (II.7.37)

≤ (b(R^K)∑k≥0M^k(ν˜_Ti

n) +σ_Khb^′(R^K)M^k1(ν˜_Ti

n) +C_L^b2Aσ_K⌈³_α⌉σ_KK+O(σ_K²K))

×(∑k≥0(b(R^K) +d(R^K) + ∑k≥0c(R^K,R^K)

K M^k(ν˜_Ti

n))M^k(˜ν_Ti n)

+σKhM^k1(˜ν_Tnⁱ)(b^′(R^K)+d^′(R^K) +^{∂1c(RK,RK)+}_K^∂2c(RK,RK)(M⁰(˜ν_Tnⁱ)+M^k1(˜ν_Tnⁱ)))

− (C_L^b,d,c)2Aσ_K⌈3/α⌉σ_KK−O(σ_K²K) )⁻

1

.

For the inequality we have used the fact that, conditioned onT_n<θ˜_i^K, there at mostσ_K⌈3/α⌉ many unsuccessful mutant individuals which differ at most2Aσ_K from the resident traitR^K. Since ∑k≥0M^k(ν˜_Ti

n) = ⟨ν˜_Ti

n,1⟩K which equals ⌈φ(i(/2))K⌉ +j conditioned on j = Y_nⁱ, the right hand side of the last inequality is smaller or equals

(b(R^K) +σ_Khb^′(R^K)_⌈φ(^M_i(^k1_/2^(ν₎₎^{˜T i}_Kⁿ⁾_⌉+j +O(σ²_K)) (II.7.38)

×(b(R^K) +d(R^K) +c(R^K, R^K)^⌈φ(i(/_K^2))K⌉+j +σ_K ^hM

k1(ν˜_{T i}

n)

⌈φ(i(/2))K⌉+j

× (b^′(R^K)+d^′(R^K) +^{∂1c(RK,RK)+}_K^∂2c(RK,RK)(M⁰(ν˜_Ti

n)+M^k1(˜ν_Ti

n))) −O(σ_K² ))⁻

1

II.7. THE SECOND PHASE OF AN INVASION 61 and by definition ofφthe denominator equals

2b(R^K)σ_K+2σ_Khb^′(R^K)_⌈φ(^M_i(^k1_/2^(ν₎₎^{˜T i}_Kⁿ⁾_⌉+j+c(R^K, R^K)_K^j −O(σ²_K) (II.7.39) +σ_Kh[i₂(^r_z^′₍⁽_R^RK^K)⁾+∂₁c(R^K, R^K) +∂₂c(R^K, R^K)) +_⌈φ(^M_i(^k1_/2^(ν₎₎^{˜T i}_Kⁿ⁾_⌉+j

× (d^′(R^K)−b^′(R^K) +^{∂1c(RK,RK)+}_K^∂2c(RK,RK)(M⁰(ν˜_Ti

n) +M^k1(ν˜_Ti n)))]. Thus, we obtain that the right hand side of (II.7.37) is bounded from above by

1

2−^c(_3b^RK_(R^,RK^K)⁾jK⁻¹−_4b^σ₍^K_R^hK)[i₂(^r_z^′₍⁽_R^RK^K)⁾−∂₁c(R^K, R^K) −∂₂c(R^K, R^K)) (II.7.40) +_⌈φ(^M_i(^k1_/2^(ν₎₎^{˜T i}_Kⁿ⁾_⌉+j( −r^′(R^K) +^{∂1c(RK,RK)+}_K^∂2c(RK,RK)(M⁰(ν˜_Ti

n) +M^k1(ν˜_Ti

n)))] +O(σ²_K). In the case where ⟨˜νt,1⟩K< ⌈φ(i(/2))K⌉ at timet=T_nⁱ, we obtain the same inequality but with an opposite sign in front of the third term. Since

∣ⁱ₂ ^r_z^′₍⁽_R^RK^K)⁾−^Mk1_K^{(ν˜T in)}_⌈φ((^r′(_/^R₂^K₎₎⁾_K^K_⌉±j− (∂1c(R^K, R^K) +∂₂c(R^K, R^K)) (ⁱ₂ −^Mk1_K^{(ν˜T in)}) ∣ (II.7.41)

< (/2) ∣^r_z^′₍⁽_R^RK^K)⁾−∂₁c(R^K, R^K) −∂₂c(R^K, R^K)∣,

we deduce the claim. Since we choose M such that M ≥ 3C_derivative^b,d,c , we can construct a Markov chainZ_nⁱ such thatZ_nⁱ ≥Y_nⁱ, a.s., for allnsuch thatT_nⁱ <θ˜_i^K∧inf{t≥θ^K_{mut. sizei(/2)}∶

∣⟨ν˜_t,1⟩ −φ(i(/2))∣ < ¹₃M σ_K} and the marginal distribution of Z_n is a Markov chain with Z₀ⁱ =Y₀ⁱ and transition probabilities

P[Z_nⁱ₊₁=j₂∣Z_nⁱ =j₁] =⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

p^K₊ for j1≥1and j2=j1+1, 1−p^K₊ for j₁≥1and j₂=j₁+1,

0 else.

(II.7.42)

Let Cexit =sup_x∈X2A∣^r_r^′₍⁽_x^x₎⁾− ^(∂1c(x,x_c(⁾⁺_x,x^∂2₎^c(x,x))∣. Then,by applying Proposition II.9.5 (b), we obtain, for all a≤ (M+C_exit)σ_KK and K large enough,

Pa[inf{n≥0∶Z_nⁱ ≥2(M+C_exit)σ_KK} <inf{n≥0∶Z_nⁱ ≤ (^M₃ )σ_KK}] (II.7.43)

≤exp(−K^α).

Next defineBⁱ≡inf{n≥0∶Z_nⁱ ≤ ¹₃M σ_KK}. This is the random variable, which counts the number of jumps Zⁱ makes until it is smaller than σ_KK. Note that (T_nⁱ₊₁−T_nⁱ), the times between two jumps of X_tⁱ, are exponential distributed with a parameter (b(R^K) +d(R^K) + c(R^K, R^K)z(R^K))z(R^K)K+O(σ_KK) if T_nⁱ₊₁ is smaller than θ˜^K_i . Thus,

(T_lⁱ₊₁−T_lⁱ) ≼E_lⁱ, (II.7.44) where (E_lⁱ)l≥0 is a sequence of independent exponential random variables with parameter infx∈Xb(x)¯z(x)K. Therefore,

P[θ^K_nearφ(i

2)>θ^K_{mut. sizei(/2)}+S_K∧θ˜^K_i ] (II.7.45)

≤P[ ∑^Bl=ⁱ0E_lⁱ>S_K] +P[θ˜_i^K<θ^K_{mut. sizei(/2)}+S_K∧θ^K_nearφ(i 2)].

62 II.7. THE SECOND PHASE OF AN INVASION Our next goal is to find a number, ni, such that P[Bⁱ >ni] is o(σK). Since the transition probabilities of Zⁱ do not depend on the present state, we have that Z_nⁱ −Z₀ⁱ has the same law as∑ⁿk=1V_kⁱ, where(V_kⁱ)k∈N is a sequence of i.i.d. random variables with

P[V_kⁱ=1] =p^K₊ and P[V_kⁱ= −1] =1−p^K₊ (II.7.46) andE[V_kⁱ] = −2σ_K and∣V_kⁱ∣ =1. Furthermore, we get

P[Bⁱ≤ni] ≥P[inf{j≥0∶Zj−Z0≤ −⌈(³₂M+Cexit)σKK⌉} ≤ni] (II.7.47)

≥P[∑ⁿkⁱ=1V_kⁱ≤ −⌈(³₂M+C_exit)σ_KK⌉]

and by applying the

Hoeffding’s Inequality. (Appendix 2 in [110]) Let Y₁, . . . , Y_n be independent random variables such that, for allj∈N,a_j ≤Y_j−E[Y_j] ≤b_j for some real constantsa_j, b_j. Then, for x>0,

P[ ∑ⁿj=1Y_j−E[Y_j] ≥x] ≤exp(−2x²(∑ⁿj=1(a_j−b_j)²)⁻¹). (II.7.48) we obtain

P[∑ⁿ_k=ⁱ₁V_kⁱ≥ −2σ_Kn_i+ (n_i)^1/2+α/2] ≤2 exp(−(n_i)^α). (II.7.49) With n_i ≡ ⌈K(³₂M +C_exit)⌉, we get −2σ_Kn_i+ (n_i)^1/2+α/2 ≤ −⌈(³₂M +C_exit)σ_KK⌉, since K^−12+α≪σ_K. Applying the exponential Chebyshev’s inequality (with λ=K^α)

P[∑^⌈l=^K0^{(32M+Cexit)⌉}E_lⁱ>SK] ≤exp(−λSK)E[exp(λ∑^⌈l=^K0^{(32M+Cexit)⌉}E_lⁱ)] (II.7.50)

≤exp(−λS_K)⎛

⎝

inf_x∈Xb(x)z¯(x)K inf_x∈Xb(x)z¯(x)K−λ

⎞

⎠

⌈K(³2M+Cexit)⌉+1

≤exp(−λS_K+ (⌈K(³₂M+C_exit)⌉ +1)ln(1+_infx∈Xb(x^λ_{)z¯(x)K−λ}))

≤exp⎛

⎝−λS_K+λ

3

2M+Cexit+1

inf_x∈Xb(x)z¯(x)+O(λ²K⁻¹)⎞

⎠≤exp(−K^α). Hence, the left hand side of (II.7.45) is bounded from above by

exp(−K^α)+2 exp(−(K(³₂M+C_exit))^α)+P[θ˜^K_i < (θ^K_{mut. sizei(/2)}+S_K) ∧θ_near^K _φ(i

2)]. (II.7.51) This proves the lemma if we can show that

P[θ˜_i^K< (θ^K_{mut. sizei(/2)}+S_K) ∧θ_near^K _φ(i

2)] =o(σ_K). (II.7.52) According to Remark 6 and Lemma II.7.3, we have that

P[θ^K_{2succ. mut.}∧θ_diversity^K <θ^K_{mut. sizei(/2)}+S_K] =o(σ_K). (II.7.53) Therefore, the following proof of (b) foriimplies (a) fori.

Proof. of (b) foriby assuming that the lemma holds for i−1. Note that the random elements Bⁱ, Tⁱ, Vⁱ, Wⁱ, Xⁱ, Yⁱ, and Zⁱ are not the ones of the last proof. They will be defined during this proof. In fact, the structure of the proof is similar to the one of (a), except that

II.7. THE SECOND PHASE OF AN INVASION 63 we prove a lower bound for the time of a change of order for the mutant density instead of upper bound for the time of a change of order σ_K of the total mass. We couple M^k_t¹, for t ≥ θ^K_{mut. sizei(/2)}, with a discrete time Markov chain (depending on i). Therefore, let T₀ⁱ =θ^K_{mut. sizei(/2)} and (T_kⁱ)_k≥1 be the sequences of jump times of M^k_t¹ after θ^K_{mut. sizei(/2)}. Furthermore, let(Y_nⁱ)_n≥0be the discrete time process which records the values thatM^k_t¹ takes i.e., Y₀ⁱ=M^k1(ν˜_Ti

0) = ⌈Ki(/2)⌉and Y_nⁱ=M^k1(ν˜_Ti

n).Observe that if θ˜^K_i >θ^K_nearφ(i

2)∧inf{t≥θ_{mut. size}^K _i(/2)∶ ∣⟨ν˜_t,1⟩ −φ(i₂)∣ ≥2(M+Cexit)σ_KK}, (II.7.54) we know from the inequality (II.7.43) that the probability thatθ^K_nearφ(i

2)is larger thaninf{t≥ θ^K_{mut. sizei(/2)}∶ ∣⟨ν˜_t,1⟩ −φ(i(/2))∣ ≥2(M+Cexit)σ_KK} is smaller thanexp(−K^α). Define

θˆ^K_i ≡ inf{t≥θ^K_{mut. sizei(/2)}∶ ∣⟨˜ν_t,1⟩ −φ(i₂)∣ ≥2(M+Cexit)σKK} (II.7.55)

∧θ^K_nearφ(i

2)∧θ₂^K_{succ. mut.}∧θ^K_diversity and

C˜_fitness ≡ inf

x∈X∂₁f(x, x)/b. (II.7.56)

Note that θˆ_i^K ≠ θˆ^K. Then, for all −⌈₄K⌉ ≤ j ≤ ⌈₄K⌉, for K large enough and for small enough, we have that

P[Y_nⁱ₊₁= ⌈i₂K⌉ +j+1∣Y_nⁱ = ⌈i₂K⌉ +j, T_nⁱ₊₁<θˆ_i^K] (II.7.57)

∈ [¹₂+¹₂C˜_fitnessσ_K, ¹₂ +2AC˜_fitnessσ_K],

since the left hand side of (II.7.57) is equal to the expectation of the probability that the next event is a birth without mutation conditioned on FT_nⁱ. Namely,

b(R^K+σKh)(1−uKm(R^K−σKh))

(b(R^K+σKh)+d(R^K+σKh)+∫N×Xc(R^K+σKh, ξ)d˜νTn(ξ)) (II.7.58)

=b(R^K+σ_Kh)[b(R^K+σ_Kh)+d(R^K+σ_Kh)+c(R^K+σ_Kh, R^K)(φ(i₂)−^⌈i2_K^K⌉+j) +c(R^K+σ_Kh, R^K+hσ_K)(^⌈i2_K^K⌉+j)+ξ₁(σ_KC_L^c(⌈_α³⌉+2(M+Cexit)))]⁻¹+O(u_K)

=b(R^K+σ_Kh)[2b(R^K+σ_Kh)−f(R^K+σ_Kh, R^K)+c(R^K+σ_Kh, R^K)(φ(i₂)−_c(^r_R⁽K^R,R^K)K)) +σKh∂2c(R^K, R^K)(^⌈i2_K^K⌉+j)+ξ1(σKC_L^c(⌈_α³⌉+2(M+Cexit)))]⁻¹+O(uK).

for some ξ₁∈ (−1,1). By definition ofφof (II.7.58) is equal to

b(R^K+σ_Kh)[2b(R^K+σ_Kh) −∂₁f(R^K, R^K)σ_Kh+c(R^K+σ_Kh, R^K)σ_Kh (II.7.59)

× (i₂)(^r_r^′₍⁽_R^RK^K)⁾−^∂1_c(^c_R^(RK^K,R^,RKK)⁾) +ξ₁(σ_KC_L^c(⌈_α³⌉ +2(M+C_exit))) ]⁻¹ +O(^σ_K^Kj+σ²_K+u_K)

=b(R^K+σ_Kh)[2b(R^K+σ_Kh) −σ_Kh(1−i₂^c(R_r(^K_R^,RK)^K))∂₁f(R^K, R^K) +ξ₁(σ_KC_L^c(⌈_α³⌉ +2(M+C_exit)))]⁻¹+O(^σ_K^Kj +σ_K² +u_K)

= 1

2+σ_Kh(1−i₂^c(R_r(^K_R^,R_K)^K))^∂1f_b⁽₍^R_R^KK^,R)^K)+σ_Kξ₁^(CLc(⌈

3

α⌉+2(M+Cexit))

b(R^K) +O(^σ_K^Kj +σ_K² +u_K).

64 II.7. THE SECOND PHASE OF AN INVASION Then, becausei<2⁻¹C_cross implies that1−i₂^c(R_r(^K_R^,R_K)^K)>0, we obtain (II.7.57). Thus we can construct a Markov chain Z_nⁱ such that Z_nⁱ ≥Y_nⁱ, a.s., for all n such that T_nⁱ <θˆ^K and such that the marginal distribution ofZ_nⁱ is a Markov chain with transition probabilities

P[Z_nⁱ₊₁=j₂∣Z_nⁱ =j₁] =⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

1

2+2AC˜fitnessσ_K for j2=j1+1,

1

2−2AC˜_fitnessσ_K for j₂=j₁−1,

0 else.

(II.7.60)

We define a continuous time process,Z˜ⁱ, associate to Z_nⁱ. To do this, we define first(T˜_jⁱ)j∈N, the sequence of jump times, byT˜₀ⁱ=0 and

T˜_jⁱ−T˜_jⁱ₋₁=⎧⎪⎪

⎨⎪⎪⎩

T_jⁱ−T_jⁱ₋₁ if T_jⁱ<θ˜^K,

W_jⁱ else, (II.7.61)

whereW_jⁱare exponential random variables with mean(⌈K(i+¹₂)(/2)⌉(b+d+c(4b/c)))⁻¹. We setZ˜_tⁱ=Z_nⁱ ift∈ [T˜_nⁱ,T˜_nⁱ₊₁). Obverse that we obtain by constructionZ˜_tⁱ≥M^k1(ν˜_θK

mut. sizei(/2)+t), for allt such thatθ_{mut. size}^K _i(/2)+t≤θˆ^K_i . Next we want to show that

P[inf{t≥0∶Z˜_tⁱ≥ ⌈K(i+¹₂)(/2)⌉} >S_K] =1−o(σ_K). (II.7.62) Therefore, letB^Z_i =inf{n≥0∶Z_nⁱ = ⌈K(i+¹₂)(/2)⌉}. We can construct(X_jⁱ)j≥1 a sequence of independent, exponential random variables with parameterx^K_i ≡ ⌈K(i+¹₂)(/2)⌉(b+d+c(4b/c)) such that

(T˜_jⁱ₊₁−T˜_jⁱ) ≽X_jⁱ for all1≤j≤B_i^Z. (II.7.63) Our next goal is to find a barrier, ni, such that B_i^Z is smaller than ni only with very small probability. Since the transition probabilities of Zⁱ do not depend on the present state, Zⁱ

B^Z_i −Z₀ is stochastically equivalent to ∑^j_k=1V_kⁱ, where (V_kⁱ)k∈N are i.i.d. random variables taking values±1 with probabilities

P[V_kⁱ=1] = ¹₂ +2AC˜fitnessσ_K and P[V_kⁱ= −1] = ¹₂−2AC˜fitnessσ_K. (II.7.64) Note thatE[V_kⁱ] =4AC˜_fitnessσ_K and ∣V_kⁱ∣ =1. Furthermore, we get

P[B^Z_i ≤ni] =P[∃⌈(/4)K⌉ ≤j≤ni∶ ∑^j_k=1V_kⁱ≥ ⌈(/4)K⌉]. (II.7.65) Hoeffding’s inequality implies that, forj≥ ⌈(/4)K⌉,

P[∑^jk=1V_kⁱ≥4AC˜_fitnessσ_Kj+j^1/2+α/2] ≤2 exp(−j^α). (II.7.66) We taken_i≡K(8AC˜_fitnessσ_K)⁻¹ and get for all⌈(/4)K⌉ ≤j≤n_i,

4AC˜_fitnessσ_Kj+j^1/2+α/2≤ ⌈(/4)K⌉, (II.7.67) since K^−12+α ≪σ_K. Then, the probability that B^Z_i ≤ K(8AC˜_fitnessσ_K)⁻¹ is bounded from above by2 exp(−K^α). Therefore, the left hand side of equation (II.7.62) is larger than

P[∑^Kj=1^{(8AC˜fitnessσK)−1}X_jⁱ>S_K] −2 exp(−K^α), (II.7.68)

II.7. THE SECOND PHASE OF AN INVASION 65 By applying the exponential Chebyshev’s inequality we get, similarly as in (a),

P[∑^Kj=1^{(8AC˜fitnessσK)−1}X_jⁱ≤S_K] (II.7.69)

= P[− ∑^Kj=1^{(8AC˜fitnessσK)−1}X_jⁱ ≥ −S_K]

≤ exp(K^αS_K)E[exp(−K^αX_jⁱ)]^{K(8AC˜fitnessσK)−1}

≤ exp(K^αS_K)exp(K(8AC˜fitnessσ_K)⁻¹ln(_xK^xKi i +K^α))

≤ exp(K^αS_K−K(8AC˜fitnessσ_K)⁻¹CK^−1+α)), for some smallC>0,

≤ exp(−K^α).

This proves that P[inf{t≥0∶Z˜_tⁱ ≥ ⌈K(i+¹₂)(/2)⌉} >S_K] ≥1−3 exp(−K^α), and therefore (b) and (a) for i, provided that the lemma holds fori−1.

Proof. of (c) for iby assuming that the lemma holds for i−1. Note that the random elements Tⁱ, Xⁱ, andYⁱare not the ones of the last proof. As in (a) we coupleK⟨ν˜_t,1⟩ with a discrete time Markov chain. Therefore, let

X_tⁱ= ∣K⟨ν˜_t,1⟩ − ⌈φ(i(/2))K⌉∣ (II.7.70) and T₀ⁱ = θ^K_{mut. sizei(/2)} and (T_kⁱ)k≥1 be the sequences of the jump times of ⟨˜νt,1⟩ after θ^K_{mut. sizei(/2)}. Then, let Y_kⁱ be the associated discrete time process which records the values thatX_tⁱ takes after timeθ^K_{mut. sizei(/2)}.

Claim. There exists a constant C˜_derivative^b,d,c such that for all j< ⌈K⌉ andK large enough, P[Y_nⁱ₊₁=j+1∣Y_nⁱ=j, T_n+1<θ˜_i^K] ≤ 1

2− c

3bjK⁻¹+σ_KC˜_derivative^b,d,c ≡p^K₊(j), (II.7.71) Moreover, we can choose C˜_derivative^b,d,c ≡sup_{x∈X 4b}^A_(x)∣^r_z^′₍⁽_x^x₎⁾−∂1c(x, x) −∂2c(x, x)∣.

From (a) we know that the left hand side of (II.7.71) is smaller or equals 1

2−^c(_3b^R₍^K_R^,RK^K)⁾jK⁻¹+_8b^σ_(R^KK^h) ∣^r_z^′₍⁽_R^RK^K)⁾−∂1c(R^K, R^K) −∂2c(R^K, R^K)∣ +O(σ²_K). (II.7.72) This proves the Claim. Note thatp^K₊(j)depends onj. Since we can chooseM ≥8 ˜C_derivative^{b,d,c 3b}_c, continuing as in Lemma II.6.3 implies that (c) is true for i, provided that the lemma holds for i−1.

Proof. of (d) for i by assuming that the lemma holds for i−1. Again we couple M^k_t¹, for t ≥ θ^K_nearφ(i

2), with a discrete time Markov chain. Let T₀ⁱ = θ_near^K _φ(i

2) and (T_kⁱ)_k≥1 be the sequences of the jump times of M^k_t¹ after θ_near^K _φ(i

2). Then, let (Y_nⁱ)_n≥0 be the discrete time process which records the values thatM^k_t¹, i.e.,

Y₀ⁱ=M^k1(ν˜_Ti

0) ∈ [K(ⁱ₂ −₄) −1, K(ⁱ₂ +₄) +1], (II.7.73) and Y_nⁱ=M^k1(ν˜_Ti

n). Define θˆ^K_i ≡inf{t≥θ_near^K _φ(i

2)∶ ∣⟨ν˜_t,1⟩ −φ(i₂)∣ >M σ_K} ∧θ^K_{2succ. mut.}∧θ_diversity^K . (II.7.74)

66 II.7. THE SECOND PHASE OF AN INVASION Note that this θˆ_i^K differs only a bit from the one defined in (b). From the proof of (b), we know that the density of the mutant trait has the tendency to increase. More precisely, since i≤C_cross (2/), we have, for all−⌈₄K⌉ ≤j≤ ⌈₂K⌉,for K large enough and small enough,

P[Y_nⁱ₊₁= ⌈i₂K⌉ +j+1∣Y_nⁱ= ⌈i₂K⌉ +j, T_nⁱ₊₁<θˆ^K_i ] ≥¹₂ +σ_K^{infx∈X∂1f(x,x)}

2b (II.7.75)

By Continuing in a similar way as in (b) with bounding the random variables in the in the other direction (as in (a)), implies that (d) is true for i, provided that the lemma holds for i−1.

Im Dokument Stochastic individual-based models of adaptive dynamics and applications to cancer immunotherapy (Seite 67-76)