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Some mathematical models from population genetics 4: Spatial models Alison Etheridge University of Oxford joint work with Jochen Blath (TU Berlin), Mark Meredith (Oxford) Nick Barton (Edinburgh), Frantz Depaulis (Paris VI) New York, Sept. 07 – p.1 Lessons learned so far The key message of our balancing selection example is that if we want to study the genealogy of a sample from a structured population, then fluctuations in background frequencies matter. A central question then is how should we model fluctuations of spatially distributed populations. New York, Sept. 07 – p.2 Large population, long timescales, measured in units of size N. Write Zn for the population size after n generations. 0 N If Φ (1) = 1 + a, then E[ZN ] = (1 + a) Z0, so for non-trivial limit a assume a = N . If Z0 converges, so does ZbNtc . N N≥1 N N≥1 Limit process: dXt = aXtdt + pγXtdBt: Reminder: Feller’s rescaling Galton Watson process, offspring generating function Φ(s). Assume Φ00(1) < . 1 New York, Sept. 07 – p.3 0 N If Φ (1) = 1 + a, then E[ZN ] = (1 + a) Z0, so for non-trivial limit a assume a = N . If Z0 converges, so does ZbNtc . N N≥1 N N≥1 Limit process: dXt = aXtdt + pγXtdBt: Reminder: Feller’s rescaling Galton Watson process, offspring generating function Φ(s). Assume Φ00(1) < . 1 Large population, long timescales, measured in units of size N. Write Zn for the population size after n generations. New York, Sept. 07 – p.3 If Z0 converges, so does ZbNtc . N N≥1 N N≥1 Limit process: dXt = aXtdt + pγXtdBt: Reminder: Feller’s rescaling Galton Watson process, offspring generating function Φ(s). Assume Φ00(1) < . 1 Large population, long timescales, measured in units of size N. Write Zn for the population size after n generations. 0 N If Φ (1) = 1 + a, then E[ZN ] = (1 + a) Z0, so for non-trivial limit a assume a = N . New York, Sept. 07 – p.3 Reminder: Feller’s rescaling Galton Watson process, offspring generating function Φ(s). Assume Φ00(1) < . 1 Large population, long timescales, measured in units of size N. Write Zn for the population size after n generations. 0 N If Φ (1) = 1 + a, then E[ZN ] = (1 + a) Z0, so for non-trivial limit a assume a = N . If Z0 converges, so does ZbNtc . N N≥1 N N≥1 Limit process: dXt = aXtdt + pγXtdBt: New York, Sept. 07 – p.3 Galton-Watson branching process branching Brownian motion/ branching random walk. Offspring born where parent died. 1 Feller rescaling: individual represented by atom of mass N , time in units of size N. Spatially distributed populations Populations dispersed in Rd or Zd, New York, Sept. 07 – p.4 1 Feller rescaling: individual represented by atom of mass N , time in units of size N. Spatially distributed populations Populations dispersed in Rd or Zd, Galton-Watson branching process branching Brownian motion/ branching random walk. Offspring born where parent died. New York, Sept. 07 – p.4 Spatially distributed populations Populations dispersed in Rd or Zd, Galton-Watson branching process branching Brownian motion/ branching random walk. Offspring born where parent died. 1 Feller rescaling: individual represented by atom of mass N , time in units of size N. New York, Sept. 07 – p.4 Super-random walk: dX (t) = m (X (t) X (t)) dt+aX (t)dt+ γX (t)dW (t); i Zd; i ij j − i i i i 2 Xj p where W (t); t 0 Zd is a collection of independent Brownian mo- f i ≥ gi2 tions and Xi(t) is the size of the population in deme i at time t. The limiting processes The Dawson-Watanabe superprocess: For positive, twice differentiable test functions φ, t t φ, Xt φ, X0 D∆φ, Xs ds aφ, Xs ds h i − h i − Z0 h i − Z0 h i is a martingale with quadratic variation t γφ2; X ds: 0 h si R New York, Sept. 07 – p.5 The limiting processes The Dawson-Watanabe superprocess: For positive, twice differentiable test functions φ, t t φ, Xt φ, X0 D∆φ, Xs ds aφ, Xs ds h i − h i − Z0 h i − Z0 h i is a martingale with quadratic variation t γφ2; X ds: 0 h si Super-random walk: R dX (t) = m (X (t) X (t)) dt+aX (t)dt+ γX (t)dW (t); i Zd; i ij j − i i i i 2 Xj p where W (t); t 0 Zd is a collection of independent Brownian mo- f i ≥ gi2 tions and Xi(t) is the size of the population in deme i at time t. New York, Sept. 07 – p.5 Clumping and extinction. E [ φ, X ] = eat T φ, X ; h ti h t 0i Take a = 0. t 2 var ( φ, Xt ) = γTt−s (Tsφ) ; X0 ds: h i Z h i 0 In one and two dimensions grows without bound. New York, Sept. 07 – p.6 As we saw in the first lecture, in Zd can specify the population size locally The Classical Stepping Stone Model. Populations should be regulated by local rules. Individuals living in locally crowded regions will have a lower reproduc- tive success than those living in sparsely populated regions. Controlling the population Exogenously specify total population size Fleming-Viot superprocess. New York, Sept. 07 – p.7 Populations should be regulated by local rules. Individuals living in locally crowded regions will have a lower reproduc- tive success than those living in sparsely populated regions. Controlling the population Exogenously specify total population size Fleming-Viot superprocess. As we saw in the first lecture, in Zd can specify the population size locally The Classical Stepping Stone Model. New York, Sept. 07 – p.7 Individuals living in locally crowded regions will have a lower reproduc- tive success than those living in sparsely populated regions. Controlling the population Exogenously specify total population size Fleming-Viot superprocess. As we saw in the first lecture, in Zd can specify the population size locally The Classical Stepping Stone Model. Populations should be regulated by local rules. New York, Sept. 07 – p.7 Controlling the population Exogenously specify total population size Fleming-Viot superprocess. As we saw in the first lecture, in Zd can specify the population size locally The Classical Stepping Stone Model. Populations should be regulated by local rules. Individuals living in locally crowded regions will have a lower reproduc- tive success than those living in sparsely populated regions. New York, Sept. 07 – p.7 For infinite initial measures, to prevent immediate catastrophe, 1 h(r)rd−1dr < . 0 1 R The stepping-stone version of the Bolker-Pacala model: In the super-random walk setting the corresponding model is dXt(i) = mij (Xt(j) Xt(i)) dt + α 0M λij Xt(j)1 Xt(i)dt − − Xj Xj @ A (i) + γXt(i)dBt : p Note that moment equations not closed. Locally regulated populations a(s; x) = α M h(x; y); X (dy) : − h s i For simplicity h(x; y) = h( x y ). k − k New York, Sept. 07 – p.8 The stepping-stone version of the Bolker-Pacala model: In the super-random walk setting the corresponding model is dXt(i) = mij (Xt(j) Xt(i)) dt + α 0M λij Xt(j)1 Xt(i)dt − − Xj Xj @ A (i) + γXt(i)dBt : p Note that moment equations not closed. Locally regulated populations a(s; x) = α M h(x; y); X (dy) : − h s i For simplicity h(x; y) = h( x y ). For infinite initial measures, to k − k prevent immediate catastrophe, 1 h(r)rd−1dr < . 0 1 R New York, Sept. 07 – p.8 Note that moment equations not closed. Locally regulated populations a(s; x) = α M h(x; y); X (dy) : − h s i For simplicity h(x; y) = h( x y ). For infinite initial measures, to k − k prevent immediate catastrophe, 1 h(r)rd−1dr < . 0 1 R The stepping-stone version of the Bolker-Pacala model: In the super-random walk setting the corresponding model is dXt(i) = mij (Xt(j) Xt(i)) dt + α 0M λij Xt(j)1 Xt(i)dt − − Xj Xj @ A (i) + γXt(i)dBt : p New York, Sept. 07 – p.8 Locally regulated populations a(s; x) = α M h(x; y); X (dy) : − h s i For simplicity h(x; y) = h( x y ). For infinite initial measures, to k − k prevent immediate catastrophe, 1 h(r)rd−1dr < . 0 1 R The stepping-stone version of the Bolker-Pacala model: In the super-random walk setting the corresponding model is dXt(i) = mij (Xt(j) Xt(i)) dt + α 0M λij Xt(j)1 Xt(i)dt − − Xj Xj @ A (i) + γXt(i)dBt : Note that moment equations not closed. p New York, Sept. 07 – p.8 Survival and Extinction Theorem For each fixed interaction kernel h and γ; K > 0 there exists α0 = α0(K; γ; h) such that for α > α0, the superprocess version of the Bolker-Pacala model with parameters (h; K/α; α; γ) started from any finite initial measure dies out in finite time. If h also satisfies h(r)rd−1dr < , then when started from any tempered initial 1 measureR (with p > d) the process with these parameters suffers local extinction. New York, Sept. 07 – p.9 Let α > 0 be fixed. If r2−δh(r) is unbounded for some δ > 0, then for each fixed • γ > 0, there is an M0 > 0 such that for M < M0 the superprocess version of the Bolker-Pacala model with parameters (h; M; α; γ) started from any finite initial measure dies out in finite time.
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