The goal of this part is to validate the model of Section 2 and the theoretical results of Sections 3 and 4. In that respect, we compare these results with Individual-Based simulations of stochastic Wright-Fisher Model presented in Figure 4. In particular, we compare expected population sizes, expected mean fitnesses and characteristic times. In the context-independent case, we also study the dependence of the cha-racteristic time with respect to the model parameters.
5.1 Context-independent models
We assume that the ditribution of the migrants p? is a Dirac at m? = −0.18. We take the migration rate d equal to 90 (such that d = −500m?), and the mean of the mutation kernel µJ to 0.1. We focus on two cases: Dirac mutation kernel and Gaussian mutation kernel, with variance σ2J = 0.01. To study weak and strong mutation events, we take the mutation rate equals respectively to either 0.1×µJ (= 0.01) or µJ.
Our PDE framework (10) gives accurate results for both the prediction ofhmi(t) and N(t) (respectively (16) and (17)) at small times (Figures 7 and 8). Then, the accuracy of the predictions of our PDE framework at layer times depends on the
0 20 40 60 80 100t -0.18
-0.16 -0.14 -0.12 -0.10 -0.08 -0.06 mt
(a)U = 0.01
0 20 40 60 80 100t
200 400 600 800 Nt
(b)U = 0.01
5 10 15 20 t
-0.20 -0.15 -0.10 -0.05 0.00 m t
(c)U = 0.1
0 5 10 15 20 t
0 200 400 600 800 Nt
(d)U = 0.1
Figure 7: Individual-Based simulations vs PDE theory. We assume here a Dirac distribution of migrant fitnesses (p? = δm?) and a Dirac mutation kernel J = δµJ. (a,c): dynamics of the expected mean fitness hmi(t) in the sink. (b,d):
dynamics of the expected population size N(t) in the sink. The dark dashed lines represent the solution given by our PDE theory (respectively (16) and (17)). The light dotted lines represent the mean values obtained by Individual-Based simu-lations (model in Figure 4). The shaded regions correspond to higher and lower values obtained by Individual-Based simulations. The parameters have been chosen asµJ = 0.1,m? =−0.18and d = 90.
value of the mutation rate. For low mutation rates (upper panels in Figures 7 and 8), the mean fitness hmi(t) tends to be overestimated by our theory: this is probably a consequence of the “large population size assumption”(3). As mutations are rare, they often do not even occur in the Individual-Based Model framework, whereas the PDE framework assumes that they always occur, but with a low rate. For higher mutation rates (lower panels in Figures 7 and 8) the PDE theory gives satisfactory results, even at large times.
Futhermore, the dynamics in the sink has three phases, with an inversion of the convexity ofN(t): for a short time, the quantity is rapidly increasing, thanks to the migration events, then becomes stable (approximately equal to−d/m?), because of the selection, and finally is rapidly increasing again, with an exponential growth, due to mutations (see Figures 7d and 8d).
Now we analyse the dependence of t0 with respect to the model parameters, namely the migration rate d, migrant fitness m? and the mutation rate U (Figure 9). The PDE theory and the Individual-Based simulations give consistent results regarding the dependence of t0 with respect to U (Figure 9a) and m? (Figure 9b).
0 10 20 30 40 t -0.20
-0.15 -0.10 -0.05 mt
(a)U = 0.01
0 10 20 30 40 t
0 100 200 300 400 500 600 700Nt
(b)U = 0.01
5 10 15 20 t
-0.20 -0.15 -0.10 -0.05 m t
(c)U = 0.1
0 5 10 15 20 t
200 400 600 800 1000Nt
(d)U = 0.1
Figure 8: Individual-Based simulations vs PDE theory. We assume here a Dirac distribution of migrant fitnesses (p? =δm?) and a Gaussian mutation kernel J(m) = √1
2πσ2J exp
−(m−µ2σ2J)2 J
. (a,c): dynamics of the expected mean fitnesshmi(t) in the sink. (b,d): dynamics of the expected population size N(t) in the sink. The dark dashed lines represent the solution given by our PDE theory (respectively (16) and (17)). The light dotted lines represent the mean values obtained by Individual-Based simulations (model in Figure 4). The shaded regions correspond to higher and lower values obtained by Individual-Based simulations. The parameters have been chosen as µJ = 0.1, σJ2 = 0.01,m? =−0.18and d = 90.
0 1 2 3 4 5 U 0
5 10 15 20 25 30t0
(a)
-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 m* 10 20 30 40 50 60t0
(b)
0 50 100 150 200 d
16 18 20 22 24 t0
(c)
Figure 9: Dynamics of the characteristic time. We assume here a Dirac dis-tribution of migrant fitnesses (p? =δm?) and a Dirac mutation kernel J =δµJ with µJ = 0.1. (a) with respect to d (U = µJ, m? = −0.18). (b) with respect to U (d = 90, m? = −0.18). (c) with respect to m? (d = 90, U = µJ). The plain lines represent the mean value obtained with Individual-Based simulations, and the dashed lines are the analytic value of t0, given by (25).
However, our PDE theory predicts (when N0 = 0) that t0 is independent of d, which is apparently not the case in the Individual-Based simulations (Figure 9c).
The function t0(d) obtained with Individual-Based simulations seems to converge towards the value predicted by the PDE theory for larged. We conjecture that this is due to the fact that large d lead to higher population sizes, and therefore the approximation (3) becomes more realistic.
5.2 Context-dependent case
We compare here the results given by our PDE approach theory (13) with the results given by Individual-Based simulations when the dynamics in the sink follows a Fisher’s Geometric Model. We make two different assumptions on the distribution of m?.
The parameters have been choosen to see correctly the different phases, described in the previous section.
Dirac distribution of the migrant fitnessp? =δm?: The corresponding results are represented in Figure 10.
We first assume that there is no mutation and so U = 0 (Figures 10a and 10b).
We take the migration rated= 25, the migrant fitnessm? =−0.05, the phenotypic dimension n= 6 and the optimal sink fitness mmax= 0.1. As we have no mutation events, we can see that the expected mean fitness is constant, and the population size grows, until stabilizing. Our PDE theory (13) accurately describes the mean dynamics obtained with Individual-Based simulations (Figures 10a and 10b).
Second, we assume a positive mutation rate (U > 0) in the sink (Figures 10c and 10d). We have choosen the migration rate d = 250, the migrant fitness m? = −0.025, the phenotypic dimensionn= 3, the mutation variance λ= 0.002/3 and the optimal fitness mmax = 0.1. Now mutation occurs and produces diversity.
This leads to an increase in hmi(t). Thus, as in the context-dependent case, the population size dynamics has three phases: first N(t) is rapidly growing, then is
“stabilizing”, and finally grows exponentially as a source. Again the PDE theory is very accurate (Figures 10c and 10d).
Migrant fitness given by Fisher’s Geometric Model at mutation-selection balance: The corresponding results are represented in Figure 11. Here, the migra-tion rated is equal to 12 500 (choosen large enough to see correctly the dynamics).
The parameters are the fitness in the sink of the source optimum mD =−0.5, the phenotypic dimensionn = 24, the mutation variance λ= 0.005/12 and the optimal fitness in the sink mmax = 0.5.
We firt assume that there is no mutation in the sink: U = 0 (Figures 11a and 11b). We can see that the analytic line has the same trajectory – three phases – than in the context-dependent cases. For a short time our PDE theory accurately describes the mean dynamics obtained with Individual-Based simulations, but the error increases after a certain time (Figures 11a and 11b).
Second, we assume that the population can undergo mutation in the sink i.e.
U > 0 (Figures 11c and 11d). The mean fitness increases, and the population size
50 100 150 200t
-0.10 -0.05 0.05 0.10mt
(a)
50 100 150 200t
200 400 600 Nt
(b)
50 100 150 200 250 300t
-0.030 -0.025 -0.020 -0.015 -0.010 -0.005 mt
(c)
0 50 100 150 200 250 300t
2000 4000 6000 8000 10000 12000 14000 Nt
(d)
Figure 10: Individual-Based simulations vs PDE theory. We assume here a Dirac distribution of migrant fitnesses (p? = δm?) and that the dynamics in the sink follows a Fisher’s Geometric Model. (a,c): dynamics of the expected mean fitness hmi(t) in the sink. (b,d): dynamics of the expected population size N(t) in the sink. The dark dashed lines represent the solution given by our PDE theory (13). The light dotted lines represent the mean values obtained by Individual-Based simulations (model in Figure 4). The shaded regions correspond to higher and lower values obtained by Individual-Based simulations.
has three phases. Unlike the previous case, we have a good enough approximation if mutation events occur in the sink. For example, thanks to Figure 11c, the analytic value of the characteristic time is 50.5, and the Individual-Based Model value is approximately equal to 52: our PDE approach is very accurate.