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Journal of Mathematical Biology

, Volume 77, Issue 1, pp 27–54 | Cite as

Homogenization analysis of invasion dynamics in heterogeneous landscapes with differential bias and motility

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Abstract

Animal movement behaviors vary spatially in response to environmental heterogeneity. An important problem in spatial ecology is to determine how large-scale population growth and dispersal patterns emerge within highly variable landscapes. We apply the method of homogenization to study the large-scale behavior of a reaction–diffusion–advection model of population growth and dispersal. Our model includes small-scale variation in the directed and random components of movement and growth rates, as well as large-scale drift. Using the homogenized model we derive simple approximate formulas for persistence conditions and asymptotic invasion speeds, which are interpreted in terms of residence index. The homogenization results show good agreement with numerical solutions for environments with a high degree of fragmentation, both with and without periodicity at the fast scale. The simplicity of the formulas, and their connection to residence index make them appealing for studying the large-scale effects of a variety of small-scale movement behaviors.

Keywords

Reaction–diffusion–advection Homogenization Spatial heterogeneity Directed movement Invasion speed Residence index 

Mathematics Subject Classification

92D40 

1 Introduction

Environmental heterogeneity leads to variation in individual movement characteristics, survival rates, and reproductive output across multiple spatial scales. An important problem in spatial ecology is to quantify how small-scale variation in these characteristics impacts large-scale, population-level growth and dispersal patterns. Reaction–diffusion (RD) and reaction–diffusion–advection (RDA) equations are common models for the growth and dispersal of populations (Okubo and Levin 2001; Shigesada and Kawasaki 1997; Murray 2002; Skellam 1951; Patlak 1953). By allowing the coefficients of the models to vary in space, one may explore the consequences of a wide variety of movement responses to environmental heterogeneity (Shigesada et al. 1986, 2015; Maciel and Lutscher 2013; Turchin 1998; Lutscher et al. 2006). In this work, we apply the method of homogenization (see, e.g., Pavliotis and Stuart 2008) to study the large (synonomously, slow) scale dynamics of an RDA model in one spatial dimension with bias and motility coefficient functions that vary at small (synonomously, fast) spatial scales. In addition to detailed approximate solutions, we derive simple approximate formulas for asymptotic invasion speeds and persistence conditions, which are closely connected to the concept of residence index. The results apply to a wide variety of movement behaviors and habitat conditions, including those with sharp transitions (fragmented, or patchy habitats) and large scale drift, and are not restricted to periodic environments.

An organism’s size and dispersal characteristics determine the smallest scale at which this heterogeneity induces changes in individual movement behaviors, or rates of reproduction and survival. At or beyond this scale, variation in habitat conditions may exhibit a high degree of regularity, but it is common for conditions to change abruptly at boundaries between patches of different habitat types, for example, between forests and an agricultural fields or at reserve boundaries. Such fragmentation induces discontinuities in movement characteristics and growth parameters when moving across boundaries, and individual decisions at or near the patch interfaces, in particular, may have important impacts on the population densities, including inducing discontinuities (Creegan and Osborne 2005; Robertson and Radford 2009; Turchin 1991; Schultz and Crone 2001; Ovaskainen and Cornell 2003; Maciel and Lutscher 2013; Cantrell and Cosner 1999; Potts et al. 2016).

In modeling dispersal, it is useful to consider animal movement as being composed of directed and random components (Armsworth and Roughgarden 2005), both of which may vary spatially in response to environmental heterogeneity. Directed movement may arise either as an active or passive response to abiotic or biotic factors. Individuals may tend to drift passively in the direction of wind or water currents, which may vary spatially (Okubo and Levin 2001; Lutscher et al. 2006). Alternatively, movement may be actively directed, for example, toward better habitat conditions or to stay within a home range (Moorcroft and Lewis 2006; Armsworth and Bode 1999; Armsworth and Roughgarden 2005; Kawasaki et al. 2012, 2017; Othmer and Stevens 1997). These examples require individuals to sense and respond to non-local information. On the other hand, movement may be actively biased based on purely local information. For example, individuals may bias their movement in the direction of resource gradients (Shigesada et al. 1979; Armsworth and Roughgarden 2005; Cantrell et al. 2006) or toward a more favorable habitat at the interface between two patches (Ovaskainen and Cornell 2003; Maciel and Lutscher 2013). The characteristics of the random (undirected) component of movement may also vary in response to local and non-local factors in passive or active ways. Random movement may be impeded or enhanced by local physical conditions, such as local surface or medium characteristics or the presence of physical obstacles. On the other hand, foraging organisms may increase their turning frequency or decrease their move lengths to better explore higher quality habitats without a resulting movement bias (Turchin 1991), or move frequency may be reduced in better foraging habitats as individuals handle their food items (Kareiva and Odell 1987).

RDA models with spatially varying coefficients of the form
$$\begin{aligned} \partial _t\rho +\partial _x(v(x)\rho )=\partial _x^2(\eta (x)\rho )+f(\rho ,x), \quad x\in {\mathbb {R}}, \end{aligned}$$
(1)
may be used to model the growth and dispersal of populations in heterogeneous environments. The population density at location x at time t is \(\rho (x,t)\). The advection velocity, or bias v(x) represents the directed component of movement, giving the average velocity of individuals at location x. The random component of movement is modeled by ecological or Fokker–Planck diffusion (Okubo and Levin 2001; Garlick et al. 2011; Turchin 1998; Codling et al. 2008; Risken 1996) rather than Fickian diffusion, and \(\eta (x)>0\) gives the diffusion coefficient, or motility, at location x. The reaction term \(f(\rho ,x)\) incorporates the population dynamics. Often it is assumed that the functions v(x) and \(\eta (x)\) and \(f(\rho ,x)\) exhibit sufficient spatial regularity to ensure the existence and uniqueness of solutions for \(x\in {\mathbb {R}}\). In fragmented environments, the coefficient functions do not meet this regularity requirement at patch interfaces. In this case, an RDA equation (1) may be used to model movement and dynamics within each patch, with interface conditions formulated to conserve flux at the boundaries while incorporating the effects of movement decisions made by individuals at the boundaries (Ovaskainen and Cornell 2003; Maciel and Lutscher 2013). If directed movement only occurs at the interfaces, then the RDA model (1) becomes a reaction diffusion (RD) equation within the patches, and the directed component is only apparent through the interface conditions (Ovaskainen and Cornell 2003; Maciel and Lutscher 2013).

The method of homogenization (Pavliotis and Stuart 2008; Holmes 2013) may be used to find asymptotic approximations for problems in which the coefficient functions vary rapidly in space. By appropriately averaging over the fast scale, a simpler model is obtained with variation only appearing at the slow scale. The homogenized model may then be solved more efficiently than the full model, either analytically or numerically. The method is well-developed for problems arising in the physical sciences, including both formal methods for deriving asymptotic approximations and rigorous convergence results, and some of these results are for RDA models (Cioranescu and Donato 2000; Pavliotis and Stuart 2008; Jikov et al. 1994).

Applications of the method of homogenization are less common in spatial ecology. Though the models arising in spatial ecology share much in common with those arising in physical problems, in ecological applications there are important differences that require new approaches. Most existing homogenization results for RDA equations assume that the advection velocity is divergence free (though, see, Goudon and Poupaud 2004; Yurk 2016). However, in ecological settings, individuals may direct their movement away from some locations and towards others, resulting in non-zero divergence. In fact, in one spatial dimension, advection velocity functions with zero divergence are constant and thus cannot model a population in which the directed component of movement varies in response to environmental heterogeneity except, perhaps, at interfaces. Another common assumption is periodicity of the coefficient functions at the fast scale (Pavliotis and Stuart 2008). Although periodic cases can lead to important theoretical insights (Maciel and Lutscher 2013, 2015), and some environments do exhibit quasi-periodicity (Garlick et al. 2011), this assumption is not appropriate in general.

In spatial ecology, habitat fragmentation leads naturally to discontinuous movement characteristics and population densities at patch interfaces (Ovaskainen and Cornell 2003; Maciel and Lutscher 2013). There are a few examples of the method of homogenization applied to these problems. Othmer (1983) applied the method of homogenization to an RD model with discontinuous coefficient functions, but assuming continuous population densities at patch interfaces. Lutscher et al. (2006) made similar continuity assumptions in applying the method to an RDA model of an aquatic population in the presence of large scale drift. More recently, Maciel and Lutscher (2013) have developed a homogenization approach for more general interface conditions and discontinuous densities in one spatial dimension, but the approach is currently limited to RD models with periodic environments. The method has also been applied to other ecological and epidemiological problems with discontinuous densities in one spatial dimension (Powell and Zimmermann 2004; Yurk 2016; Neupane and Powell 2016) and in two spatial dimensions (Garlick et al. 2011, 2013). In these studies the interface conditions are not given explicitly. However, they can be shown to be a special case of those described by Maciel and Lutscher (2013). Only one of these (Yurk 2016) included an RDA model of spatially varying directed movement, but in this case the motility was spatially uniform.

In the present work, we extend the approach of Yurk (2016) to a more general model, which includes spatially varying bias and motility, a logistic reaction term with spatially varying coefficients, and large scale drift. Large scale drift may be explicitly present (e.g., due to currents) or may emerge from the small scale movement characteristics. The method is developed for the case of a fragmented environment with a particular set of interface conditions, which generalize those implicitly studied by Garlick et al. (2011) and Yurk (2016). When the bias is zero, the interface conditions are a special case of those analyzed by Maciel and Lutscher (2013). The present results apply to both periodic and aperiodic settings, and the large-scale, homogenized models are used to develop approximate persistence conditions and invasion speeds, as well as detailed solution approximations. This work is an important step toward a general homogenization approach for RDA models in one spatial dimension that allow spatial variation in both the directed and random components of movement and more general interface conditions.

In the next section (Sect. 2), we present an RDA model that describes population dispersal and growth dynamics at the fast scale, assuming sufficient regularity in the spatial variation of the coefficient functions across \({\mathbb {R}}\). In Sect. 3 we derive the homogenized model with coefficients that are constant or only vary at the slow scale. Next, in Sect. 4, the homogenization results are interpreted in terms of the residence index, and simple formulas are derived for an approximate invasion criterion and approximate asymptotic invasion speeds. In Sect. 5 the model is extended to fragmented environments by defining interface conditions at the boundaries between fragments, and it is demonstrated that the homogenization results extend to this setting without modification for a specific set of interface conditions. In addition to the general case, two specific cases are considered: a patchy environment without periodicity at the fast scale, and a patchy environment with periodicity at the fast scale. In Sect. 6 the homogenization results are compared to full numerical solutions for both of these cases, including detailed solution comparisons and asymptotic invasion speed comparisons. Finally, the results are discussed in Sect. 7 and directions for future work are briefly considered.

2 Model formulation

We consider the growth and dispersal of a population in a strongly heterogeneous environment. Initially we consider an environment that is not fragmented, so that the coefficient functions exhibit sufficient regularity over \({\mathbb {R}}\). Though we are ultimately interested in slow scale population patterns, we formulate the model in terms of fast scale variables to appropriately capture the heterogeneity at that scale. To this end, we define fast scale spatial and temporal variables y and \(\tau \), and let \(\rho (y,\tau )\) be the population density at location y at time \(\tau \). This allows us to “zoom in” on the population at the level of the heterogeneity. Adapting the RDA model (1), we propose the following model of population growth and dispersal at the fast scale:
$$\begin{aligned} \partial _{\tau }\rho +\partial _y[(\varepsilon u(\varepsilon y)+v(y))\rho ]=\partial ^2_y(\eta (y)\rho )+\varepsilon ^2 f(\rho ,y), \quad y\in {\mathbb {R}}. \end{aligned}$$
(2)
Here, \(\varepsilon \) is a small, dimensionless parameter representing the separation between the slow and fast scales, which satisfies \(0<\varepsilon \ll 1\).
The coefficient functions v(y) and \(\eta (y)\) in the model (2) give the spatially varying bias and motility. Large scale drift is introduced through the term \(\varepsilon u(\varepsilon y)\). The input, \(\varepsilon y\), varies slowly, and will later become our slow-scale spatial variable. We assume that the large scale drift is small (\(O(\varepsilon )\)) when viewed from the fast scale. Thus, bias arising in response to local environmental heterogeneity is assumed to be stronger than bias at the slow scale. We assume throughout that the long run average value of \(v/\eta \) is zero. That is,
$$\begin{aligned} \lim _{y\rightarrow \pm \infty }\frac{1}{y}\int _{y_0}^{y} \frac{v(s)}{\eta (s)}\,ds=0 \end{aligned}$$
(3)
for both of the indicated limits. This ensures that variability in the fast scale bias does not give rise to bias at the slow scale. If it does, then it may be possible to determine an effective large scale bias, as demonstrated in Sect. 5.3. The effective large scale drift is then subtracted from the fast scale bias so that the resulting coefficients satisfy the Assumption (3).
The reaction term, \(\varepsilon ^2f(\rho ,y)\), in the model (2) gives the population dynamics. In the analysis that follows we will assume a specific form for f, though the analysis can be extended for other reaction terms. Specifically, the reaction term will assume the following logistic form:
$$\begin{aligned} f(\rho ,y)=(\lambda (y)-\mu (y)\rho )\rho . \end{aligned}$$
(4)
Here \(\lambda (y)\) is the intrinsic growth, and \(\mu (y)\ge 0\) is an intraspecific competition coefficient, both of which are allowed to vary spatially at the fast scale. In terms of the fast scale variables, the reaction rate is assumed to be slow \(O(\varepsilon ^2)\).

3 Multi-scale analysis and homogenization

To capture the population growth and dispersal patterns that emerge from the model (2) at large observational scales, we introduce slow scale temporal and spatial variables t and x, such that \(t=\varepsilon ^2\tau \), \(x=\varepsilon y\). “Zooming out” to the slow scale through the change of variables, the model (2) becomes
$$\begin{aligned} \partial _{t}\rho +\partial _x[(u(x)+\varepsilon ^{-1}v(x/\varepsilon ))\rho ] =\partial ^2_x(\eta (x/\varepsilon )\rho )+f(\rho ,x/\varepsilon ), \end{aligned}$$
(5)
where \(\rho =\rho (x,t)\). We rewrite this equation by introducing the integrating factor
$$\begin{aligned} I(x;\varepsilon )=\exp \left( -\int _{x_0}^x\frac{v(s/\varepsilon )}{\varepsilon \eta (s/\varepsilon )}\,ds\right) . \end{aligned}$$
(6)
Then the slow scale model (5) can be rewritten as
$$\begin{aligned} \partial _{t}\rho +\partial _x(u(x)\rho )=\partial _x\left[ I(x;\varepsilon )^{-1} \partial _x(I(x;\varepsilon )\eta (x/\varepsilon )\rho )\right] +f(\rho ,x/\varepsilon ). \end{aligned}$$
(7)
To analyze this model, we employ the method of multiple scales. This proceeds formally, by assuming the perturbation ansatz
$$\begin{aligned} \rho =\rho _0(x,y,t)+\varepsilon \rho _1(x,y,t) +\varepsilon ^2\rho _2(x,y,t)+\cdots , \end{aligned}$$
(8)
where each term is assumed to be a bounded function of the fast spatial variable y. Treating x and y independently induces the change of derivative, \(\partial _x\rightarrow \varepsilon ^{-1}\partial _y+\partial _x\). We assume that f, \(\eta \), and v vary at the fast scale but not at the slow scale and that u varies only at the slow scale. Then (7) becomes
$$\begin{aligned} \partial _{t}\rho +(\varepsilon ^{-1}\partial _y+\partial _x)(u(x)\rho )= & {} (\varepsilon ^{-1}\partial _y+\partial _x)\left[ I(y)^{-1} (\varepsilon ^{-1}\partial _y+\partial _x)(I(y)\eta (y)\rho )\right] \nonumber \\&+\,f(\rho ,y), \end{aligned}$$
(9)
where \(\rho =\rho (x,y,t)\), and I(y) is the integrating factor (6) after a corresponding change of variables
$$\begin{aligned} I(y)=\exp \left( -\int _{y_0}^y\frac{v(s)}{\eta (s)}\,ds\right) . \end{aligned}$$
(10)

3.1 Homogenization

We apply the method of homogenization, substituting the expansion (8) into the two-scale equation (9), and solving at each order of \(\varepsilon \). The details, which generalize the approach of Yurk (2016), are given in “Appendix A”. The result is an RDA equation with coefficients that are constant or vary only at the slow scale
$$\begin{aligned} \partial _t p +U\partial _x(up)=D\partial _x^2p+(\varLambda -Mp)p, \end{aligned}$$
(11)
where
$$\begin{aligned} \rho _0(x,y,t)=\frac{p(x,t)}{I(y)\eta (y)}. \end{aligned}$$
(12)
Note that (12) separates the scale dependence of the leading order solution between the slow varying numerator p(xt), which satisfies (11), and the fast varying denominator \(I(y)\eta (y)\). The coefficients of the homogenized model (11) are given by
$$\begin{aligned} U= & {} \frac{\displaystyle \lim _{y\rightarrow \infty }\frac{1}{y}\int _{y_0}^y \eta (s)^{-1}\,ds}{\left( \displaystyle \lim _{y\rightarrow \infty } \frac{1}{y}\int _{y_0}^yI(s)^{-1}\eta (s)^{-1}\,ds\right) \left( \displaystyle \lim _{y\rightarrow \infty }\frac{1}{y}\int _{y_0}^yI(s)\,ds\right) }, \end{aligned}$$
(13)
$$\begin{aligned} D= & {} \left( \lim _{y\rightarrow \infty }\frac{1}{y}\int _{y_0}^yI(s)^{-1} \eta (s)^{-1}\,ds\right) ^{-1}\left( \lim _{y\rightarrow \infty }\frac{1}{y} \int _{y_0}^yI(s)\,ds\right) ^{-1}, \end{aligned}$$
(14)
$$\begin{aligned} \varLambda= & {} \frac{\displaystyle \lim _{y\rightarrow \infty }\frac{1}{y} \int _{y_0}^y\lambda (s)I(s)^{-1}\eta (s)^{-1}\,ds}{\displaystyle \lim _{y\rightarrow \infty }\frac{1}{y}\int _{y_0}^yI(s)^{-1}\eta (s)^{-1}\,ds}, \end{aligned}$$
(15)
$$\begin{aligned} M= & {} \frac{\displaystyle \lim _{y\rightarrow \infty }\frac{1}{y}\int _{y_0}^y \mu (s)I(s)^{-2}\eta (s)^{-2}\,ds}{\displaystyle \lim _{y\rightarrow \infty }\frac{1}{y}\int _{y_0}^yI(s)^{-1}\eta (s)^{-1}\,ds}, \end{aligned}$$
(16)
assuming that these limits exist and have the same values as \(y\rightarrow -\infty \).

4 Residence index, invasion criterion, and asymptotic spread rate

It is revealing to recast the coefficients of the homogenized model in terms of the residence index (Turchin 1991). The residence index at y is proportional to the expected time that an organism will spend within a unit area about y (Turchin 1998). The residence index is proportional to the equilibrium density for the model (2) with no reaction term (\(f=0\)) and no large scale drift (\(u=0\)). Here it is given by
$$\begin{aligned} r(y)=I(y)^{-1}\eta (y)^{-1}. \end{aligned}$$
(17)
The residence index, which is positive, increases as the motility decreases or as the value of the integrating factor decreases. Though the former relationship is intuitive, the latter is less so. If \(\eta \) is constant, the integral \(\int _{y_0}^yv(s)/\eta (s)\,ds\) is large and positive at y when the bias v(s) becomes large and positive at the points to the left of y. Furthermore, by assumption, the average value of \(v/\eta \) is zero, so there must be points to the right of y where the bias is large and negative. Thus, the value of this integral at y gives non-local information about the movement bias around y, and it will tend to be largest when movement is biased from both directions toward y, resulting in I being small and the residence index large.
Garlick et al. (2011) applied the method of homogenization to a RD equation and found that the homogenized diffusion coefficient D was equal to \(1/R_{av}\), where \(R_{av}\) is the arithmetic average of the residence index
$$\begin{aligned} R_{av}=\lim _{y\rightarrow \infty }\frac{1}{y}\int _{y_0}^{y}r(s)\,ds. \end{aligned}$$
(18)
Here, we have shown that for the RDA equation (2), D is proportional \(1/R_{av}\)
$$\begin{aligned} D=\left( \frac{1}{R_{av}}\right) \left( \frac{1}{I_{av}}\right) , \end{aligned}$$
(19)
where \(I_{av}\) is the arithmetic average of the integrating factor
$$\begin{aligned} I_{av}=\lim _{y\rightarrow \infty }\frac{1}{y}\int _{y_0}^{y}I(s)\,ds. \end{aligned}$$
(20)
Note that if there is no fast scale bias (\(v=0\)) then \(I(y)=1\), and (19) is simply the harmonic average of the motility \(\eta (y)\), a result that is consistent with Garlick et al. (2011). The residence index, as defined by (17), appears prominently in the formulas for all of the coefficients of the homogenized model (1316). For example, from (15), we can rewrite \(\varLambda \) as
$$\begin{aligned} \varLambda =\left( \lim _{y\rightarrow \infty }\frac{1}{y}\int _{y_0}^{y} \lambda (s)r(s)\,ds\right) \left( \frac{1}{R_{av}}\right) . \end{aligned}$$
(21)
Thus, the homogenized intrinsic growth rate is proportional to the arithmetic average of the heterogeneous reaction rate weighted by the residence index. This result is intuitive, because r(y) reflects the relative amount of time an individual spends near y, where it experiences growth rate \(\lambda (y)\).

4.1 Invasion criterion and asymptotic invasion speed

The homogenized model (11) suggests an approximate invasion criterion. In order for invasion to occur, \(\varLambda \) must be positive, which is equivalent to requiring
$$\begin{aligned} \lim _{y\rightarrow \infty }\frac{1}{y}\int _{y_0}^y\lambda (s)r(s)\,ds>0. \end{aligned}$$
(22)
This establishes a simple framework for exploring how invasibility is affected by environmental heterogeneity and corresponding movement behaviors. This invasion criterion is also intuitive, because it states that the average growth rate, weighted by the residence index, must be positive for invasion to occur.
We may also apply the homogenized model (11) to approximate the asymptotic invasion speed for the population. In the case that there is no large scale drift (\(u=0\)), the homogenized model (11) becomes a constant-coefficient RD model with logistic growth dynamics. As a result, the classical formula for the asymptotic invasion speed (Kolmogoroff et al. 1937; Uchiyama 1978)
$$\begin{aligned} c^{*}=2\sqrt{\varLambda D} \end{aligned}$$
(23)
may be applied. If the large scale drift u is a nonzero constant, the asymptotic invasion speed is simply
$$\begin{aligned} c^{*}=uU+2\sqrt{\varLambda D}. \end{aligned}$$
(24)
As with the invasion criterion (22), these invasion speeds are simple formulas that account for detailed movement characteristics and landscape heterogeneity.

5 Periodic landscapes and fragmented landscapes

5.1 Periodic landscapes

If v, \(\eta \), \(\lambda \), and \(\mu \) are periodic in y with period \(2\ell \), then the coefficients (1316) can be simplified by replacing averages of the form \(\lim _{y\rightarrow \infty }\left( \int _{y_0}^y h(s)\,ds/y\right) \) with averages over a single period, which have the form \(\int _{-\ell }^{\ell } h(s)\,ds/(2\ell )\).

5.2 Habitat fragmentation and interface conditions

The homogenization results may be extended to a landscape with small scale fragmentation, by breaking up the environment into regions within which the coefficients vary smoothly. Suppose that the boundaries between regions are at the points \(y_i\), for \(i=0,\pm \,1,\pm \,2,\ldots \), and that the fast scale model (2) applies within each region [i.e., for \(y\in (y_{i-1},y_i)\)]. Across region boundaries, v, \(\eta \), and f may fail to be differentiable or even continuous, but we assume that u is differentiable over \({\mathbb {R}}\).

At the interfaces of the regions, we require the flux to be continuous so that individuals are conserved as they disperse in the absence of population dynamics. This results in the following interface condition at each boundary:
$$\begin{aligned}&[\varepsilon u(\varepsilon y_i)+v(y_i^{+})]\rho (y_i^{+},\tau ) -\partial _y(\eta \rho )(y_i^{+},\tau )\nonumber \\&\quad =[\varepsilon u(\varepsilon y_i)+v(y_i^{-})] \rho (y_i^{-},\tau )-\partial _y(\eta \rho )(y_i^{-},\tau ). \end{aligned}$$
(25)
Here \(y_i^{+}\) and \(y_i^{-}\) indicate limits taken from the right and left at \(y_i\). Furthermore, we assume that the dynamic level, \(\varGamma (y,\tau )\), is continuous at the interfaces, so that
$$\begin{aligned} \varGamma (y_i^{+},\tau )=\varGamma (y_i^{-},\tau ) \end{aligned}$$
(26)
where \(\varGamma (y,\tau )\) is defined by
$$\begin{aligned} \varGamma (y,\tau )=I(y)\eta (y)\rho (y,\tau )=\rho (y,\tau )/r(y). \end{aligned}$$
(27)
In terms of the dynamic level, the flux interface condition (25) may be written more simply as
$$\begin{aligned} \partial _y(\varGamma )(y_i^{+},\tau )=\partial _y(\varGamma )(y_i^{-},\tau ) \end{aligned}$$
(28)
so that flux is conserved at the interfaces if and only if the dynamic level is continuously differentiable there. Diffusion acts to smooth out variation in \(\varGamma \) (Skellam 1973) within regions, and, thus, interface condition (26) results in \(\varGamma \) being uniform throughout all of \({\mathbb {R}}\) at equilibrium in the absence of population dynamics. The interface conditions (26) and (28) are consistent with those implied by previous analyses (Garlick et al. 2013, 2011; Yurk 2016), though they are not presented explicitly. They are also a special case of more general interface conditions derived from random walk models of detailed movement behaviors at interfaces (Ovaskainen and Cornell 2003; Maciel and Lutscher 2013, 2015), extended to an RDA setting.
Scaling up to the slow scale and regarding x and y as independent, we obtain the two-scale model (9) within the regions [for \(y\in (y_{i-1},y_i)\)], and the interface conditions (25) and (26) become
$$\begin{aligned}&u(x)\rho (x,y_i^{+},t)-(I(y_i^{+}))^{-1}(\varepsilon ^{-1}\partial _y+\partial _x) (I\eta \rho )(x,y_i^{+},t)\nonumber \\&\quad =u(x)\rho (x,y_i^{-},t)-(I(y_i^{-}))^{-1}(\varepsilon ^{-1} \partial _y+\partial _x)(I\eta \rho )(x,y_i^{-},t) \end{aligned}$$
(29)
and
$$\begin{aligned} (I\eta \rho )(x,y_i^{+},t)=(I\eta \rho )(x,y_i^{-},t), \end{aligned}$$
(30)
respectively. With these interface conditions, the homogenization analysis proceeds identically to that presented in “Appendix A”, resulting in the same homogenized model (1112), which applies for \(x\in {\mathbb {R}}\), and with coefficients given by the same Formulas (1316). Interface conditions that are different from (26), representing different movement behaviors at the interfaces, require modification of the homogenization procedure (Maciel and Lutscher 2013).

5.3 Patchy environments

In this section, we consider the case of a patchy environment with uniform conditions in each patch, and biased movement arising within sensing zones around the patch interfaces (see also Kawasaki et al. 2017). For simplicity, we assume that the environment exhibits periodicity at the slow scale (with period 2L), without requiring periodicity at the fast scale. The landscape is divided into patches of length \(\ell =O(\varepsilon )\) with interfaces at \(y_i=y_0+i\ell \) for \(i=0,\pm \,1, \pm \,2,\ldots \pm (L/\ell )\). It is assumed that L is divisible by \(\ell \), and that there is a patch interface at \(y_0=0\). The intrinsic growth rate, the intraspecific competition coefficient, and the motility are assumed to be constant within each patch. Thus, for each i, set
$$\begin{aligned} \lambda (y)=\lambda _i, \quad \mu (y)=\mu _i, \quad \eta (y) =\eta _i, \quad \text {for }\; y\in (y_{i-1},y_i). \end{aligned}$$
(31)
At the patch interfaces we define sensing zones within which an individual is aware of the interface and exhibits biased movement toward one patch and away from the other. For simplicity, we assume that each sensing zone has length \(\sigma \le \ell \) and is symmetric about the interface. We also assume a constant bias (either to the left or right) within each sensing zone. Thus, for each i, we set
$$\begin{aligned} v(y)=c_i, \quad \text {for }\;y\in (y_{i}-\sigma /2,y_i+\sigma /2), \end{aligned}$$
(32)
and we set \(v(y)=0\) outside of the sensing zones. These definitions are extended periodically (with period 2L) for all real values of y.
Even if there is no explicit large scale drift (\(u=0\)), an effective one may arise. Our homogenization analysis relies on the zero-mean Assumption (3). However, if v and \(\eta \) do not satisfy this assumption, then we define an effective large scale drift by
$$\begin{aligned} u_{eff}=\frac{\int _{-L}^{L}v(s)\eta (s)^{-1}\,ds}{\varepsilon \int _{-L}^{L}\eta (s)^{-1}\,ds}, \end{aligned}$$
(33)
and set
$$\begin{aligned} v'(s)=v(s)-\varepsilon u_{eff}, \end{aligned}$$
(34)
so that \(v'/\eta \) has zero mean. Then, we replace u with the constant \(u_{eff}\) and v with \(v'\) in (2) and proceed. It is not surprising that an effective large scale drift would arise if v has nonzero mean (for example, if the fast scale biases tend to be rightward). On the other hand, an effective large scale drift may arise through variation in the motility, even if v has zero (unweighted) mean.
It is important to note that according to our earlier concept of fragmentation (see Sect. 5.2), each patch identified here consists of three regions within which the model coefficients are constant. At interfaces between adjacent regions we assume that flux is conserved (25), and that the dynamic level is continuous (26). Since I(y) is continuous, the dynamic level condition (26) becomes
$$\begin{aligned} \eta (y_i^+)\rho (y_i^+,\tau )=\eta (y_i^-)\rho (y_i^-,\tau ), \end{aligned}$$
(35)
where \(y_i\) is the location of the ith region interface. At sensing zone boundaries, there is no change in the value of \(\eta \). Thus, the solution is continuous at these interfaces. At interfaces between favorable and unfavorable patches, \(\eta (y)\) is discontinuous if \(\eta _1\ne \eta _2\), resulting in a jump discontinuity in the solution across the interface.
In a fragmented environment with slow scale periodicity, the invasion criterion (22) becomes
$$\begin{aligned} \sum _{i=-L/\ell }^{L/\ell }\lambda _iR_{av}^i>0, \end{aligned}$$
(36)
where \(R_{av}^i\) is the average residence index for patch i
$$\begin{aligned} R_{av}^i=\frac{1}{\ell }\int _{y_{i-1}}^{y_1}r(s)\,ds. \end{aligned}$$
(37)
Thus, for invasion to occur the sum of the patch-specific intrinsic growth rates, weighted by the corresponding average residence indices must be positive. It is similarly possible to simplify the formula for the asymptotic invasion speed (23) in this case.

5.4 Patchy environments with fast scale periodicity

We also explore the special case of a patchy environment that is periodic at the fast scale with two alternating patch types (1 and 2). We assume that the patch interface at \(y_0=0\) has a type 1 patch to the right and a type 2 patch to the left. Thus, the interval \((y_{i-1},y_i)\) corresponds with a patch of type 1 (type 2) when i is odd (even). The growth and movement parameters depend on the patch type, and thus vary periodically with period \(2\ell \). The values are
$$\begin{aligned} \lambda _i=\lambda _1, \quad \mu _i=\mu _1, \quad \eta _i=\eta _1, \qquad c_i=-c, \quad \text {for }i\text { odd}, \end{aligned}$$
(38)
and
$$\begin{aligned} \lambda _i=\lambda _2, \quad \mu _i=\mu _2, \quad \eta _i=\eta _2, \quad c_i=c, \quad \text {for }i\text { even}. \end{aligned}$$
(39)
If the advection speed speed c is positive, then movement within the sensing zones is biased toward type 1 patches and away from type 2 patches. With these choices the zero mean Assumption (3) is satisfied, and, thus, the effective large scale drift is zero.
The residence index is periodic with the same period as the environment, \(2\ell \). Over a single period, it is given explicitly by
$$\begin{aligned} r(y)=\left\{ \begin{array}{ll} \exp (-c(y+\ell )/\eta _2)/\eta _2, &{}\quad y\in (-\ell ,-\ell +\sigma /2]\\ \exp (-c(\sigma /2)/\eta _2)/\eta _2, &{}\quad y\in (-\ell +\sigma /2,-\sigma /2]\\ \exp (cy/\eta _2)/\eta _2, &{}\quad y\in (-\sigma /2,0]\\ \exp (cy/\eta _1)/\eta _1, &{}\quad y\in (0,\sigma /2]\\ \exp (c(\sigma /2)/\eta _1)/\eta _1, &{}\quad y\in (\sigma /2,\ell -\sigma /2] \\ \exp (-c(y-\ell )/\eta _1)/\eta _1, &{}\quad y\in (\ell -\sigma /2,\ell ]. \end{array}\right. \end{aligned}$$
(40)
Note that the residence index at location y depends only on the local movement parameters at y, which are determined by the type of patch type at y. Figure 1 shows the residence index for four choices of movement parameters for a single period. In the top-left panel, there is no bias, and the motilities are the same in the both patch types, and the residence index is uniform. In the top-right panel, there is no bias, but the patches of type 1 have lower motility than patches of type 2. As a result, individuals spend more time in those patches, and the residence index is higher there. In the bottom-left panel, there is no difference in the patch motilities, but there is bias within the sensing regions toward type 1 patches. In this case, the directed movement alone is sufficient to create substantial variation in the residence index, with values increasing as the interiors of patches of type 1 are approached. In the patch interiors, the residence index is constant, with higher values in type 1 patches. If bias toward type 1 patches is present, and if the motility is lower in those patches, then the previous pattern is accentuated (bottom-right panel).
Fig. 1

Residence index for a single period for four parameter choices: no bias (top row), bias toward type 1 patches (bottom row), constant diffusion coefficient (left column), different diffusion coefficients in type 1 and type 2 patches (right column). Parameter values are \(\ell =1\), \(\eta _1=\eta _2=1\) in the left column; \(\eta _1=2/3\), \(\eta _2=2\) in the right column; \(c=0\) in the top row; \(c=2\) in the bottom row. In all cases \(\sigma =0.3\)

The homogenized model (11) has constant coefficients and admits smooth traveling wave solutions p at the slow scale. The leading order solution, \(\rho _0\), is given by Eq. (12). Since I and \(\eta \) are periodic at the fast scale, traveling wave solutions p create periodic traveling wave solutions \(\rho _0\), and if \(\eta _1\ne \eta _2\), then \(\rho _0\) will be discontinuous at the patch interfaces.

5.4.1 Invasion criterion

The population will always persist if \(0<\lambda _2\le \lambda _1\), and it will always decline if \(\lambda _2\le \lambda _1<0\). More interestingly, if \(\lambda _1>0>\lambda _2\), then the population grows locally within favorable (type 1) patches and declines within unfavorable (type 2) patches, and large scale growth or decline then depends on the relative amounts of time spent in favorable and unfavorable patches. In the present case of a periodic, fragmented environment, the invasion criterion (22) becomes
$$\begin{aligned} \lambda _1R_{av}^{(1)}+\lambda _2R_{av}^{(2)}>0, \end{aligned}$$
(41)
where \(R_{av}^{(i)}\) is the average residence index in a patch of type i
$$\begin{aligned} R_{av}^{(i)}=\frac{1}{\ell }\int _{\varOmega _i}r(y)\,dy, \end{aligned}$$
(42)
with \(\varOmega _1=(0,\ell )\) and \(\varOmega _2=(-\ell ,0)\). The invasion criterion (41) states that, in order for invasion to occur, the weighted average of the patch-specific intrinsic growth rates must be positive. In this case, the weights are given by the patch-specific average residence indices.
We rewrite criterion (41) in the following useful form:
$$\begin{aligned} -\frac{\lambda _1}{\lambda _2}>\frac{R_{av}^{(2)}}{R_{av}^{(1)}}, \end{aligned}$$
(43)
assuming that neither denominator is zero. The ratio on the left side of (43) is the ratio of the patch-specific growth and decay rates. We will explore how the ratio of average residence indices on the right side of (43), which we will call the residence ratio, depends on the movement parameters. In particular, we may identify contours, \(R_{av}^{(2)}/R_{av}^{(1)}=\alpha \), which represent invasion thresholds, so that whenever \(-\lambda _1/\lambda _2>\alpha \), invasion occurs, and whenever \(-\lambda _1/\lambda _2<\alpha \), extinction occurs. First, we find an explicit formula for the residence ratio. Evaluating the integrals (42), we find that
$$\begin{aligned} R_{av}^{(1)}=\frac{2}{c\ell }\left( -1+\left[ \frac{c\ell -c\sigma }{2\eta _1} +1\right] \exp \left( \frac{c\sigma }{2\eta _1}\right) \right) , \end{aligned}$$
(44)
and
$$\begin{aligned} R_{av}^{(2)}=\frac{2}{c\ell }\left( 1+\left[ \frac{c\ell -c\sigma }{2\eta _2} -1\right] \exp \left( \frac{-c\sigma }{2\eta _2}\right) \right) . \end{aligned}$$
(45)
Thus, the residence ratio is given by the formula
$$\begin{aligned} \frac{R_{av}^{(2)}}{R_{av}^{(1)}}=\frac{1+\left[ \frac{c\ell -c\sigma }{2\eta _2}-1\right] \exp \left( \frac{-c\sigma }{2\eta _2}\right) }{-1+\left[ \frac{c\ell -c\sigma }{2\eta _1}+1\right] \exp \left( \frac{c\sigma }{2\eta _1}\right) }. \end{aligned}$$
(46)
Note that either of the limits, \(c\rightarrow 0\) or \(\sigma \rightarrow 0\), represent pure diffusion, and in these limits the residence ratio becomes, \(R_{av}^{(2)}/R_{av}^{(1)}\rightarrow \eta _1/\eta _2\).
Figures 2 and 3 are examples of contour plots of the residence ratio. Over most of the motility values in Fig. 2 an increase in motility in patch type 1 requires an increase in motility in patch type 2 or an increase in the ratio \(-\,\lambda _1/\lambda _2\) to maintain persistence. Increasing \(\eta _1\) tends to reduce the residence index in patches of type 1, so a corresponding reduction must occur in the residence index for patches of type 2, or the growth rate in type 1 patches must increase relative to the decay rate in type 2 patches in order to compensate. In Fig. 3, \(\eta _1\) is fixed, but a similar relationship is observed. Over much of the range of \(\eta _2\) and c values, if \(\eta _2\) is decreased at the threshold (typically increasing the residence time in type 2 patches), then the bias toward patches of type 1 or the ratio \(-\lambda _1/\lambda _2\) must be decreased to maintain persistence.
Fig. 2

Contour plot of the residence ratio, \(R_{av}^{(2)}/R_{av}^{(1)}\) as a function of \(\eta _1\) and \(\eta _2\). In the region above the line \(\eta _2=\eta _1\) (dashed), motility is lower in favorable patches (type 1) than in unfavorable patches (type 2). Below this line motility is higher in the favorable patches. Movement is biased toward favorable patches (\(c=2\)). Other parameter values are \(\sigma =0.3\), \(\ell =1\)

Fig. 3

Contour plot of the residence ratio, \(R_{av}^{(2)}/R_{av}^{(1)}\) as a function of \(\eta _2\) and c. \(\eta _1=1\) is fixed. The line \(\eta _2=\eta _1=1\) (dashed) is shown for reference. To the left of this line the motility is lower in the unfavorable patch than it is in the favorable one. Other parameter values are \(\sigma =0.3\), \(\ell =1\)

It is interesting that increasing motility in the type 2 patches first causes the residence ratio to increase before it decreases (Figs. 2, 3). This may be counterintuitive, since in the absence of bias, this would always decrease the residence index in these patches and, hence, the residence ratio. However, with bias toward type 1 patches there is a competing effect that accounts for the initial increase. In this case, the relative contribution of bias in type 2 patches decreases as the motility increases (diffusion becomes more dominant). This favors higher residence times in the type 2 patches (and a higher residence ratio), because individuals have a weaker tendency to be directed out of them.

5.4.2 Asymptotic invasion speed

In the case of a periodic, fragmented environment, the approximate asymptotic invasion speed (23) is given by the simple formula
$$\begin{aligned} c^{*}=\frac{2}{R_{av}}\sqrt{\frac{\lambda _1R_{av}^{(1)} +\lambda _2R_{av}^{(2)}}{2I_{av}}} \end{aligned}$$
(47)
for a population that satisfies the invasion criterion (41). Note that \(R_{av}\), defined by (18), can be written as the average of the patch-specific average residence indices, \(R_{av}=(R_{av}^{(1)}+R_{av}^{(2)})/2\), using the Formulas (44) and (45), and \(I_{av}\), defined by (20), can be written explicitly as
$$\begin{aligned} I_{av}= & {} \frac{1}{c\ell }\left( (\eta _1-\eta _2)+\left[ \frac{c\ell -c\sigma }{2} -\eta _1\right] \exp \left( \frac{-c\sigma }{2\eta _1}\right) \right. \nonumber \\&+\left. \left[ \frac{c\ell -c\sigma }{2}+\eta _2\right] \exp \left( \frac{c\sigma }{2\eta _2} \right) \right) . \end{aligned}$$
(48)

6 Numerical comparisons

We compared the analytical homogenization results to numerical solutions for the two patchy environment scenarios described above (with and without periodicity at the fast scale). Specifically, we compared the leading order solutions and corresponding invasion speed approximations obtained through the homogenization analysis to those obtained through numerical solution of the full model (2).

For the full numerical solutions, the RDA equation (2) was solved with interface conditions (26) and (28) and periodic boundary conditions. For this, the model was recast in terms of the dynamic level, and a second-order implicit method was used to solve the resulting partial differential equation. This approach simplifies the numerical scheme by taking advantage of the continuity and continuous differentiability of the dynamic level at the interfaces. When solution comparisons were made, numerical solutions of the homogenized model (11) were obtained using a similar second-order implicit method.

6.1 Aperiodicity at the fast scale

In the first case, we simulated a random patchy environment at the fast scale by drawing patch-specific intrinsic growth rates (\(\lambda _i\)’s in (31)) from a probability distribution. Next, the advection velocities within sensing zones were set to be proportional to the difference in the patch-specific growth rates
$$\begin{aligned} c_i=\beta (\lambda _{i+1}-\lambda _i). \end{aligned}$$
(49)
If the value of \(\beta \) is positive, then movement in the sensing zone is directed toward patches with higher growth rates. The motility in patch i was set based on the intrinsic growth rate within the patch, with lower motility in patches with higher growth rates. There are many functional forms that may describe this relationship. The following was chosen, because it is able to caricature a variety of relationships:
$$\begin{aligned} \frac{1}{\eta _i}=\left( \frac{1/\eta _m+1/\eta _M}{2}\right) +\left( \frac{1/\eta _m-1/\eta _M}{\pi }\right) \;\arctan (a+b\lambda _i), \end{aligned}$$
(50)
where \(0<\eta _m\le \eta _M\), and \(b>0\). With this formulation, the motility approaches its maximum value, \(\eta _M\), as \(\lambda _i\) becomes large and negative, and it approaches its minimum value \(\eta _m\), as \(\lambda _i\) becomes large and positive. As a function of \(\lambda _i\), \(1/\eta _i\) is symmetric through its inflection point at \((a,(1/\eta _m+1/\eta _M)/2)\). The constant b is a shape parameter that determines the rate at which \(1/\eta _i\) changes with respect to \(\lambda _i\). If \(a=0\), and if the expected value of \(c_i\) is zero, then the expected value of \(c_i/\eta _i\) is 0, satisfying (3) in the long run.

We performed a single simulation, drawing the \(\lambda _i\)’s from a normal distribution with mean zero and standard deviation equal to 1 (Fig. 4a). The patches had width \(\ell =0.5\), the sensing zones had width \(\sigma =0.25\), and \(\mu \) was set equal to 1 in all patches. Advection velocities were set according to (49), with \(\beta =1\), and the motilities were set according to (50), with \(\eta _m=2/3\), \(\eta _M=2\), \(a=0\), and \(b=5\). Thus, the expected values of \(c_i\) and \(c_i/\eta _i\) were both zero. For this simulation, the effective large scale drift \(u_{eff}\), given by (33) was negligible, so there was no need for correction.

We computed the homogenized coefficients (1316) for the simulation parameters, obtaining the values \(U=0\), \(D=0.904\), \(\varLambda =0.461\), and \(M=1.295\). The homogenized model (11), was solved numerically with these coefficients to obtain the approximate slow scale solution, and the approximate invasion speed was computed using Formula (23), to obtain \(c^{*}=1.292\).

Figure 4b–d, compares the numerical solutions of the full model (2) with the homogenized model (11) at three different times. The initial condition was Gaussian at the slow scale. The homogenized model captures the large scale behavior of the heterogeneous model, accurately reflecting both the average population density and the spread rate of the traveling wave. Figure 5 shows the velocities (signed speeds) of the rightward and leftward traveling fronts for the full numerical solution with the approximate invasion speed calculated using the homogenized coefficients. Despite the effective drift being approximately zero over the entire domain, regions of nonzero effective drift are apparent at intermediate scales within the random environment. This suggests the possibility of making intermediate-scale corrections to improve the approximation, by allowing \(u_{eff}\) to vary at these scales. However, this increases the complexity of the homogenized model and decreases numerical efficiency. The current approach minimizes this complexity, yielding a single constant-coefficient model that still describes the large-scale behavior well.
Fig. 4

a Patch-specific growth rates drawn from a normal distribution with mean zero and standard deviation equal to 1. bd Numerical solutions of the full, heterogeneous model (solid lines) for growth rates given in a, along with the slow-scale solution of the homogenized model (dotted lines) at times \(t=2.5\) (b), \(t=12.5\) (c), and \(t=50\) (d)

Fig. 5

Numerical invasion velocities for right-moving (solid) and left-moving (dashed) fronts with homogenization predictions (horizontal lines) plotted against time. Numerical invasion velocities are averages over time intervals of length 5, tracking the rightmost and leftmost positions where the numerical solution exceeded a threshold value of 0.001. Random environment is the same as Fig. 4

Fig. 6

Numerical solution (black) and leading order approximation with fast-scale variation (gray). Inset: zoomed in to show more detail

6.2 Periodicity at the fast-scale

We explored several parameter ranges for patchy environments with periodicity at the fast scale. Figure 6 compares detailed numerical solutions to the leading order solution \(\rho _0\) for on set of parameter values, \(c=4\), \(\lambda _1=1\), \(\lambda _2=-1\), \(\eta _1=0.75\), \(\eta _2=1.5\), \(\mu =1\), \(\ell =0.5\), and \(\sigma =0.25\). The leading order approximation closely matches the numerical solution, including the fast-scale details. This level of agreement was also present for each of the other parameter sets that were explored.

The approximate asymptotic invasion speed, given by Eq. (47) was compared to front speeds obtained from full numerical solutions under a variety of parameter ranges (Figs. 7, 8). Figure 7 shows how the asymptotic invasion speed varies as c varies for different motilites (Fig. 7a) and for different sensing zone widths (Fig. 7b). In Fig. 7a, b, the values \(\lambda _1=1\), \(\lambda _2=-1\), \(\mu =1\), \(\ell =0.5\) are fixed. Additionally, \(\sigma =0.25\) is fixed in Fig. 7a, and \(\eta _1=0.75\) and \(\eta _2=1.5\) are fixed in Fig. 7b. Figure 8 shows how the invasion speed varies with \(\lambda _1\) for two choices of motilities. In this figure, the values \(c=4\), \(\lambda _2=-1\), \(\mu =1\), \(\ell =0.5\), and \(\sigma =0.25\) are fixed. In all of the simulations, there was close agreement between the homogenization predictions and the numerical invasion speeds.
Fig. 7

Numerical invasion speeds (markers) and approximate invasion speed obtained from homogenization analysis (lines) versus sensing zone advection speed, c. In a, \(\eta _1=2/3\), \(\eta _2=2\) (triangle markers, dash-dotted line); \(\eta _1=3/4\), \(\eta _2=3/2\) (circle markers, dashed line); and \(\eta _1=1\), \(\eta _2=1\) (square markers, solid line). In b, \(\sigma =0.125\) (triangle markers, dash-dotted line), \(\sigma =0.25\)(circle markers, dashed line); and \(\sigma =0.5\) (square markers, solid line). Other parameter values are listed in the text

Figure 7a shows that asymptotic invasions speeds are maximized at intermediate values of c. As the bias increases, individuals are directed toward the interiors of patches with positive intrinsic growth rates. This results in both a higher large scale growth rate, \(\varLambda \), and a lower large scale motility, D. The former increases the invasion speed, while the latter decreases it. Initially, this results in increasing invasion speeds as the bias is increased, followed by a decline. This figure also shows the effect of increasing the difference between the patch-specific motilities, \(\eta _1\) and \(\eta _2\). Increasing this difference accentuates the difference in the residence index in the two patch types, as seen in Fig. 1. This increases \(\varLambda \), but may also decrease D, with competing effects on the invasion speed. For the range of parameters considered in Fig. 7a, increasing this difference increases the invasion speed.

Figure 7b shows that increasing the width of the sensing zone shifts the maximum invasion speed to lower values of c. This result is intuitive, since increasing \(\sigma \) while keeping c fixed, increases the extent, and therefore the overall effect, of directed movement toward the interiors of patches with positive growth rates. The same qualitative effect is obtained by increasing the value of c with fixed sensing width. Thus, increasing the sensing zone has the effect of horizontally compressing the invasion speed curves.

Figure 8, shows the relationship between invasion speed and \(\lambda _1\) for different motilities, with all of the other parameter values fixed. Note that \(\varLambda \) increases monotonically with \(\lambda _1\), whereas D does not depend on \(\lambda _1\). Thus, the approximate invasion speed always increases as \(\lambda _1\) increases.
Fig. 8

Numerical invasion speeds (markers) and approximate invasion speed obtained from homogenization analysis (lines) versus intrinsic growth rate \(\lambda _1\) in patches of type 1. Two sets of motilities were used: \(\eta _1=2/3\), \(\eta _2=2\) (triangle markers, dash-dotted line); and \(\eta _1=1\), \(\eta _2=1\) (square markers, solid line). Other parameter values are listed in the text

7 Discussion

In previous work (Yurk 2016), we applied the method of homogenization to study a RDA model of directed movement with in a heterogeneous environment with spatially varying bias and growth rates. Here the analysis is extended to include variation in motility and large scale drift.

The coefficients of the resulting homogenized model are interpreted in terms of the residence index, a concept introduced by Turchin (1991). Garlick et al. (2011) connected homogenized coefficients of an RD model to the residence index, but their model did not incorporate directed movement. When directed movement is introduced through an advection term, the residence index incorporates non-local information about habitat conditions and movement behaviors, and bias near interfaces can have a large impact on the the residence index within patch interiors. It is not surprising that the residence index would feature prominently in these homogenization analyses, since it is proportional to the equilibrium density for the pure movement model (with no reaction term). In these analyses, the reaction rate is assumed to be small at the fast scale (\(O(\varepsilon ^2)\)), and the formal homogenization procedure relies on scale separation due to rapid equilibration of dispersal at the fast scale before the population growth dynamics are resolved. In fact, the two lowest order governing equations are equilibrium equations that do not involve the reaction term.

The present results apply to both periodic and aperiodic environments, and fragmented habitats. The specific interface conditions considered here are a special case of more general conditions studied by Ovaskainen and Cornell (2003) and Maciel and Lutscher (2013). The more general interface conditions account for a variety of movement behaviors at the interfaces, and they affect the size of the jump discontinuity in the population density at the patch boundaries. Although they have been incorporated into homogenization analyses for RD models (Maciel and Lutscher 2013), extending the analyses to RDA models with general interface conditions is a topic for future work.

We have developed simple formulas for approximate invasion criteria and approximate invasion speeds. Kawasaki et al. (2012) developed persistence criteria for an RDA model of directed movement within a periodic environment. Their result, which is based on the linearized model, gives an exact, but relatively complicated procedure for finding invasion thresholds. It is limited to the case of a periodic environment with particular assumptions about movement behavior, including constant motility and piecewise constant bias. Though the persistence results obtained here are approximate, they may be applied to much more general environments and movement scenarios. Additionally, they are much simpler and are directly tied to the biologically meaningful concept of residence index.

The invasion speed results obtained here, both using homogenization analysis and full numerical simulations, demonstrated that invasion speeds are highest at intermediate bias levels. This pattern has been observed previously in an RDA setting (Kawasaki et al. 2012; Shigesada et al. 2015; Yurk 2016). Interestingly, this effect has also been noted when bias is limited to the patch interfaces in an RD setting with interface conditions (Maciel and Lutscher 2013, 2015). This highlights two competing effects that fast-scale bias has on slow-scale invasion speeds. Bias reduces the motility, while bias toward favorable conditions enhances growth rates. Both of these effects are apparent from the formulations of the homogenized coefficients.

In both real and simulated habitats, effective drift may emerge at intermediate to large scales in response to fast-scale variation in the movement parameters. The present work begins to explore this effect, but it has not been fully developed. In particular, this suggests the possibility of incorporating variation at intermediate scales through effective drift, but additional the model complexity must be balanced against solution efficiency and the interpretability of the theoretical results.

The use of an RDA model to describe population growth and dispersal assumes that reproduction and movement occur simultaneously and continuously in time. However, many organisms reproduce and move during separate time intervals, and for these an integrodifference model (Kot and Schaffer 1986; Neubert et al. 1995; Van Kirk and Lewis 1997; Powell and Zimmermann 2004; Kawasaki and Shigesada 2007; Dewhirst and Lutscher 2009; Neupane and Powell 2016) may be more appropriate. In this case, the movement phase is described by a dispersal kernel, which may be given by the solution of a RD or RDA model of dispersal with appropriate assumptions about movement characteristics and settling rates (Neubert et al. 1995; Van Kirk and Lewis 1997; Powell and Zimmermann 2004; Neupane and Powell 2016). The homogenization approach developed here is useful for finding approximate dispersal kernels to describe movement within heterogeneous environments in which both the random and directed components of movement vary spatially. This may be achieved by incorporating a reaction term f into Eq. (1) that describes the settling rate and solving the homogenized model (see Powell and Zimmermann (2004) and Neupane and Powell (2016) for examples of this approach using RD models of dispersal and settling). An additional limitation of RDA models is that population density is assumed to be a continuous quantity. As a result, population density may become arbitrarily small over a large hostile patch, with small fractions of individuals contributing to the advancement of the traveling wave beyond the patch. In this case, a model of invasion that takes into account demographic stochasticity may be more appropriate (see, e.g., Jerde and Lewis 2007).

Though the present analysis is conducted for a model with a logistic reaction term, it is easily extended to other reaction terms satisfying \(f(\rho ,y)=f(\rho _0,y)+O(\varepsilon )\). An important future direction is to extend the homogenization analysis to a population with an Allee effect. This has been explored by Maciel and Lutscher (2015) when movement bias is restricted to patch interfaces, but it has not been studied when directed movement occurs over a sensing region with nonzero width. Other possible future work includes incorporating time dependence in the model coefficients, extensions to higher spatial dimensions, and density-dependent movement behaviors.

Notes

Acknowledgements

I would like to thank Darin R. Stephenson and Charles A. Cusack for useful discussions relating to this work. This work also benefited from discussions during a meeting of the SQuaRE, “Homogenization techniques in ecological and epidemiological models,” supported by the American Institute of Mathematics SQuaREs program. Finally, I would like to thank two anonymous reviewers for their useful feedback on this manuscript.

Compliance with ethical standards

Conflict of interest

The author declares that he has no conflict of interest.

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsHope CollegeHollandUSA

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