Using genetic data to estimate diffusion rates in heterogeneous landscapes

Abstract

Having a precise knowledge of the dispersal ability of a population in a heterogeneous environment is of critical importance in agroecology and conservation biology as it can provide management tools to limit the effects of pests or to increase the survival of endangered species. In this paper, we propose a mechanistic-statistical method to estimate space-dependent diffusion parameters of spatially-explicit models based on stochastic differential equations, using genetic data. Dividing the total population into subpopulations corresponding to different habitat patches with known allele frequencies, the expected proportions of individuals from each subpopulation at each position is computed by solving a system of reaction–diffusion equations. Modelling the capture and genotyping of the individuals with a statistical approach, we derive a numerically tractable formula for the likelihood function associated with the diffusion parameters. In a simulated environment made of three types of regions, each associated with a different diffusion coefficient, we successfully estimate the diffusion parameters with a maximum-likelihood approach. Although higher genetic differentiation among subpopulations leads to more accurate estimations, once a certain level of differentiation has been reached, the finite size of the genotyped population becomes the limiting factor for accurate estimation.

Appendices

Appendix 1: gradual release of the pre-dispersal populations

The Eq. (2.2) describes a simultaneous release of all the individuals at $t=0.$ To account for a possible gradual release of the individuals, the Eq. (2.2) can be replaced by:

$$\begin{aligned} \frac{\partial u}{\partial t}=\varDelta (D(x) \, u) -\frac{u}{\nu }+u_0(x)\, f(t), \ t>0, \, x\in \varOmega , \end{aligned}$$

(7.1)

where the term $u_0(x)\, f(t)$ describes the release of the individuals; $u_0(x)$ still corresponds to the pre-dispersal density and the function f(t) is the release rate. It can be described by any nonnegative function or distribution with integral 1 and with support in [0, T], T corresponding to the end of the release period. In this framework, the density of dispersers coming from habitat $\varOmega ^h$ satisfies the equation:

$$\begin{aligned} \frac{\partial u^h}{\partial t}=\varDelta (D(x) \, u^h) -\frac{u^h}{\nu }+u_0^h(x)\, f(t), \ t>0, \, x\in \varOmega , \end{aligned}$$

(7.2)

where $u_0^h$ is still given by (2.7).

Appendix B: precise shape of the diffusion terms

In our numerical computations, we took

$$\begin{aligned} \phi (x)=\mu _{2\, R}(\Vert x\Vert ) \hbox { and }\psi (x)=\psi (x_1,x_2)=\mu _{R}\left( x_1- q\right) , \end{aligned}$$

for the function $\mu$ defined by (see Fig. 7):

$$\begin{aligned} \mu _R(r)=\exp \left( \frac{-r^4}{(r^2-R^2)^2}\right) \hbox { for }r \in (-R,R) \hbox { and }\mu _R(r)=0 \hbox { otherwize}. \end{aligned}$$

Fig. 7

The function $\mu _R(r)$, for $R=0.05$ and $r\in (-0.1, 0.1)$

Appendix C: computation of the $F_{ST}$

The index $F_{ST}$ is used as a measure of genetic differentiation among the subpopulations. It was computed as follows: we set

$$\begin{aligned} J_S=\frac{1}{\varLambda }\sum \limits _{\lambda =1}^{\varLambda } \sum \limits _{a=1}^{A} \sum \limits _{h=1}^{H} \frac{1}{H}\left( p_{h \lambda a}\right) ^2 \hbox { and } J_T=\frac{1}{\varLambda }\sum \limits _{\lambda =1}^{\varLambda }\sum \limits _{a=1}^{A} \left( \frac{1}{H} \sum \limits _{h=1}^{H} p_{h \lambda a}\right) ^2, \end{aligned}$$

where $\varLambda$ is the number of loci, A,  the number of alleles per locus whose frequency is measured and H the number of subpopulations. Here, $J_S$ and $J_T$ denote the mean homozygosity across subpopulations and the homozygosity of the total population, respectively. Then, we can write

$$\begin{aligned} F_{ST}= \frac{J_S-J_T}{1-J_T}. \end{aligned}$$

(9.1)

This formula corresponds to Nei’s $G_{ST}$ for a single locus (Nei 1973), with numerator and denominator averaged over the $\varLambda$ loci. In our computations, all the subpopulations had the same size; in other situations, the weight 1 / H in the above formulas for $J_S$ and $J_T$ should be replaced by the relative sizes of the subpopulations.

Appendix D: numerical computation of the cumulated population densities

In order to compute the cumulated densities $w_\infty (x)$ and $w_\infty ^h(x),$ we used the time-dependent partial differential equation solver Comsol Multiphysics$^{\copyright }$ applied to the evolution equations (10.2) and (10.4) below at large time ($t=20$), with default parameter values (finite element method with second order basis elements) and a triangular mesh adapted to the geometry of our landscape and made of 5296 elements.

We defined the cumulated population density at intermediate times t and position x by:

$$\begin{aligned} w_t(x)=\int _0^{t}u(s,x)\, ds, \ \hbox { for all }t>0, \ x\in \varOmega . \end{aligned}$$

(10.1)

Integrating (2.2) between 0 and $t>0$ we note that $w_t(x)$ satisfies the following equation:

$$\begin{aligned} \frac{\partial w_t}{\partial t}=\varDelta (D(x) \, w_t) -\frac{w_t}{\nu }+u_0(x), \ t>0, \, x\in \varOmega , \end{aligned}$$

(10.2)

and $w_0(x)=0.$

Similarly, the cumulated population density of individuals coming from $\varOmega ^h$ is:

$$\begin{aligned} w_t^h(x)=\int _0^{t}u^h(s,x)\, ds, \ \hbox { for all }t>0, \ x\in \varOmega . \end{aligned}$$

(10.3)

This function satisfies:

$$\begin{aligned} \frac{\partial w_t^h}{\partial t}=\varDelta (D(x) \, w_t^h) -\frac{w_t^h}{\nu }+u_0^h(x), \ t>0, \, x\in \varOmega , \end{aligned}$$

(10.4)

and $w_0^h(x)=0.$

Appendix E: using abundance data in the inference of the diffusion rates

As already mentioned, an important feature of our approach is that the likelihood does not depend on the capture rates $\beta _\tau$. As the expected number of individuals captured in a trap $\theta _\tau$ is proportional to $\alpha \, \beta _\tau$, the absolute number of individuals captured in $\theta _\tau$ cannot be used directly to infer the diffusion parameters if the capture rates are not known. However, if the capture rate was the same ($=\beta$) for all traps, we could include the information on the absolute number of captured individuals $\mathbf {I}=\{I_1, \ldots ,I_J\}$ by computing the likelihood

$$\begin{aligned} \begin{array}{ll} \mathcal {L}(D,\alpha \, \beta ) &{} =\mathbb {P}(\mathbf {\mathcal {G}},\mathbf {I}|D, \alpha \, \beta ) \\ &{} = \mathbb {P}(\mathbf {\mathcal {G}}|\mathbf {I},D, \alpha \, \beta ) \mathbb {P}(\mathbf {I}|D, \alpha \, \beta ), \end{array} \end{aligned}$$

where $\mathbf {\mathcal {G}}$ is the genotype information. In our framework, the genotype information does not depend on the number of captured individuals in each trap, as we assumed a constant number of genotyped individuals per trap, G. Besides, we have shown that the quantity $\mathbb {P}(\mathbf {\mathcal {G}}|D, \alpha \, \beta )$ was independent of $\alpha \, \beta$. Using the assumptions of Sect. 2 on the capture process, the quantity $\mathbb {P}(\mathbf {I}|D, \alpha \, \beta )$ can be computed explicitly:

$$\begin{aligned} \mathbb {P}(\mathbf {I}|D, \alpha \, \beta )=\prod \limits _{\tau =1}^{J}\exp (-C_\tau )\frac{C_\tau ^{I_\tau }}{(I_\tau !)}. \end{aligned}$$

Finally, one can infer the diffusion parameters, together with the product $\alpha \, \beta$ by maximising the likelihood:

$$\begin{aligned} \mathcal {L}(D,\alpha \, \beta )=2^{k} \prod \limits _{\tau =1, \ldots , J} \prod \limits _{i=1, \ldots , G} \exp (-C_\tau )\frac{C_\tau ^{I_\tau }}{(I_\tau !)} \sum \limits _{h=1}^{H}\left[ \frac{C^h_\tau }{C_\tau }\prod \limits _{\lambda =1}^{\varLambda }p_{h \lambda a^1} \, p_{h \lambda a^2}\right] , \end{aligned}$$

where k is the total number of heterozygous loci in the genotyped population. For the computation of $C_\tau$ and $C^h_\tau ,$ the pre-dispersal density $\alpha$ can be fixed arbitrarily to 1.

Appendix F: modelling sharp transitions between regions with different diffusion rates

For the sake of simplicity, we assumed in this paper that the coefficient D was a smooth function of the position x, leading to a scalar equation (2.2), with a unique classical solution. Sharp transitions could be modelled by replacing the Eq. (2.2) by a system of N equations, where N is the number of patches $\varOmega _i$ where the diffusion coefficient takes a constant value $D_i$ and $u_i$ is the population density in the patch $\varOmega _i$:

$$\begin{aligned}\left\{ \begin{array}{l} \displaystyle \frac{\partial u_i}{\partial t}=D_i \, \varDelta u_i - \frac{u_i}{\nu }, \ x \in \varOmega _i, \\ \displaystyle u_i=u_j, \ x \in \partial \varOmega _i\cap \partial \varOmega _j, \\ \displaystyle D_i \, \nabla u_i \cdot \mathbf {n}_i=-D_j \, \nabla u_j \cdot \mathbf {n}_j, \ x \in \partial \varOmega _i\cap \partial \varOmega _j, \end{array}\right. \end{aligned}$$

where $\partial \varOmega _i$ denotes the boundary of $\varOmega _i$ and $\mathbf {n}_i$ the outward unit normal to the boundary. The first boundary condition corresponds to the continuity of the population density in $\varOmega =\bigcup \nolimits _{i}\varOmega _i$. The second boundary condition guaranties the conservation of mass in the absence of mortality ($\nu =+\infty$) and with reflecting boundary conditions on $\partial \varOmega$.