Homogenization of a Directed Dispersal Model for Animal Movement in a Heterogeneous Environment

Abstract

The dispersal patterns of animals moving through heterogeneous environments have important ecological and epidemiological consequences. In this work, we apply the method of homogenization to analyze an advection–diffusion (AD) model of directed movement in a one-dimensional environment in which the scale of the heterogeneity is small relative to the spatial scale of interest. We show that the large (slow) scale behavior is described by a constant-coefficient diffusion equation under certain assumptions about the fast-scale advection velocity, and we determine a formula for the slow-scale diffusion coefficient in terms of the fast-scale parameters. We extend the homogenization result to predict invasion speeds for an advection–diffusion–reaction (ADR) model with directed dispersal. For periodic environments, the homogenization approximation of the solution of the AD model compares favorably with numerical simulations. Invasion speed approximations for the ADR model also compare favorably with numerical simulations when the spatial period is sufficiently small.

Appendix: Homogenized Diffusion Coefficient for Periodic, Zero Mean Velocity

We derive the simple formula (73) for the diffusion coefficient (28) in the case that the advection velocity u is periodic (29) with zero mean (30). These two conditions imply that I and its reciprocal, $I^{-1}$ are also periodic with period $2\ell$,

$$\begin{aligned} I(y+2\ell )=I(y). \end{aligned}$$

(53)

For convenience, we set the lower limits of integration to $y_0=-\ell$ in the definitions of I, $\gamma$, and $\alpha$ (5, 14, 24).

From the definition of $\alpha$ (24),

$$\begin{aligned} \alpha (y)=\int _{-\ell }^yI(s)^{-1}\,\hbox {d}s. \end{aligned}$$

(54)

Since we are interested in the limit as $y\rightarrow \infty$ in (28), we may assume that $y>0$ in what follows. In this case, $\alpha$ can be expressed as the sum

$$\begin{aligned} \alpha (y)=\int _{(2m-1)\ell }^{y}I(s)^{-1}\,\hbox {d}s+\sum _{i=0}^{m-1}\int _{(2i-1)\ell }^{(2i+1)\ell }I(s)^{-1}\,\hbox {d}s, \end{aligned}$$

(55)

where m is chosen so that

$$\begin{aligned} (2m-1)\ell \le y < (2m+1)\ell . \end{aligned}$$

(56)

Since $I^{-1}$ is periodic with period $2\ell$, each of last m terms on the right-hand side of (55) are equal to $\int _{-\ell }^{\ell } I^{-1}(s)\,\hbox {d}s$. Consequently,

$$\begin{aligned} \alpha (y)=m\int _{-\ell }^{\ell } I(s)^{-1}\,\hbox {d}s+\int _{(2m-1)\ell }^{y}I(s)^{-1}\,\hbox {d}s. \end{aligned}$$

(57)

Now, we turn our attention to the integral in the denominator of (28). Similar to above, we express the integral as the sum

$$\begin{aligned} \int _{-\ell }^{y}I(s)\alpha (s)\,\hbox {d}s=\int _{(2n-1)\ell }^{y}I(s)\alpha (s)\,\hbox {d}s+\sum _{i=0}^{n-1}\int _{(2i-1)\ell }^{(2i+1)\ell }I(s)\alpha (s)\,\hbox {d}s, \end{aligned}$$

(58)

where n is chosen so that

$$\begin{aligned} (2n-1)\ell \le y < (2n+1)\ell . \end{aligned}$$

(59)

We will develop bounds for the last n terms on the right-hand side of (58). The limits of integration of these integrals restrict the relevant values of s, so that $\alpha (s)$ needs only to be considered for $(2i-1)\ell \le s<(2i+1)\ell$. Thus, according to (57) and (56), these integrals can be rewritten as,

$$\begin{aligned} \int _{(2i-1)\ell }^{(2i+1)\ell }I(s)\alpha (s)\,\hbox {d}s= & {} i\left( \int _{-\ell }^{\ell }I(s)^{-1}\,\hbox {d}s\right) \left( \int _{-\ell }^{\ell }I(s)\,\hbox {d}s\right) \nonumber \\&+\int _{(2i-1)\ell }^{2(i+1)\ell }I(s)\left( \int _{(2i-1)\ell }^sI(w)^{-1}\,dw\right) \,\hbox {d}s. \end{aligned}$$

(60)

Because I and $I^{-1}$ are positive, the double-integral on the right-hand side of Eq. (60) must also be positive. Furthermore, the assumption (6) implies that $I(s)\le I_M$ and $I^{-1}(s)\le (I_m)^{-1}$, for all $s\in \mathbb {R}^1$, where $I_M$ and $I_m$ are fixed positive real numbers. Consequently,

$$\begin{aligned} 0<\int _{(2i-1)\ell }^{2(i+1)\ell }I(s)\left( \int _{(2i-1)\ell }^sI(w)^{-1}\,dw\right) \,\hbox {d}s\le (2\ell )^2\frac{I_M}{I_m}. \end{aligned}$$

(61)

Next, we develop bounds for the first term on the right-hand side of Eq. (58). First, note that since $\alpha (s)$ is a positive, increasing function and $y\ge (2n-1)\ell$,

$$\begin{aligned} 0\le \int _{(2n-1)\ell }^y\alpha (s)\,\hbox {d}s\le \alpha (y)[y-(2n-1)\ell ]. \end{aligned}$$

(62)

Inequality (59) also implies that

$$\begin{aligned} y-(2n-1)\ell \le 2\ell . \end{aligned}$$

(63)

The two preceding inequalities, along with (6) result in the following bound:

$$\begin{aligned} 0\le \int _{(2n-1)\ell }^yI(s)\alpha (s)\,\hbox {d}s\le 2\ell I_M\alpha (y) \end{aligned}$$

(64)

Using inequalities (61) and (64) and equations (58) and (60), we arrive at the upper bound

$$\begin{aligned} \int _{-\ell }^{y}I(s)\alpha (s)\,\hbox {d}s\le & {} \left( \frac{(n-1)n}{2}\right) \left( \int _{-\ell }^{\ell }I(s)^{-1}\,\hbox {d}s\right) \left( \int _{-\ell }^{\ell }I(s)\,\hbox {d}s\right) \nonumber \\&+\,2\ell I_M\alpha (y) + (2\ell )^2\frac{I_M}{I_m}, \end{aligned}$$

(65)

and the lower bound

$$\begin{aligned} \int _{-\ell }^{y}I(s)\alpha (s)\,\hbox {d}s \ge \left( \frac{(n-1)n}{2}\right) \left( \int _{-\ell }^{\ell }I(s)^{-1}\,\hbox {d}s\right) \left( \int _{-\ell }^{\ell }I(s)\,\hbox {d}s\right) . \end{aligned}$$

(66)

Inequality (59) implies that

$$\begin{aligned} \frac{y-2\ell }{2\ell }<n\le \frac{y}{2\ell }. \end{aligned}$$

(67)

Consequently,

$$\begin{aligned} \left( \frac{(n-1)n}{2}\right) =\frac{1}{2}\left( \frac{y}{2\ell }\right) ^2+O(y^1), \end{aligned}$$

(68)

as $y\rightarrow \infty$. Furthermore, the last two terms on the right-hand side of the upper bound (65) satisfy

$$\begin{aligned} 2\ell _M\alpha (y)=O(y^1), \end{aligned}$$

(69)

and

$$\begin{aligned} \left( 2\ell \right) ^2\frac{I_M}{I_m}=O(y^0). \end{aligned}$$

(70)

Finally, we evaluate the limit

$$\begin{aligned} \lim _{y\rightarrow \infty }\frac{\int _{\ell }^{y}I(s)\alpha (s)\,\hbox {d}s}{y^2}, \end{aligned}$$

(71)

which arises in the formula for the diffusion coefficient (28). Note that if the right-hand side of the upper bound (65) is divided by $y^2$, and the limit taken as $y\rightarrow \infty$, we obtain

$$\begin{aligned} \frac{1}{2}\left( \frac{\int _{-\ell }^{\ell }I(s)^{-1}\,\hbox {d}s}{2\ell }\right) \left( \frac{\int _{-\ell }^{\ell }I(s)\,\hbox {d}s}{2\ell }\right) , \end{aligned}$$

(72)

due to (68–70). The same limit is obtained if the right-hand side of the lower bound (66) is divided by $y^2$. Thus, by the Squeeze Theorem, the limit (71) is given by the expression (72). Consequently, from this limit and Eq.  (28), we can conclude that the slow-scale diffusion coefficient for a periodic, zero-mean velocity is simply

$$\begin{aligned} D=\frac{\eta }{I_\mathrm{av}I^{-1}_\mathrm{av}}, \end{aligned}$$

(73)

where $I_{av}$ is the average value of I,

$$\begin{aligned} I_{av}=\frac{\int _{-\ell }^{\ell }I(s)\,\hbox {d}s}{2\ell }, \end{aligned}$$

(74)

and $I^{-1}_\mathrm{av}$ is the average value of $I^{-1}$,

$$\begin{aligned} I^{-1}_{av}=\frac{\int _{-\ell }^{\ell }I(s)^{-1}\,\hbox {d}s}{2\ell }. \end{aligned}$$

(75)