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Seasonal Invasion Dynamics in a Spatially Heterogeneous River with Fluctuating Flows

Abstract

A key problem in environmental flow assessment is the explicit linking of the flow regime with ecological dynamics. We present a hybrid modeling approach to couple hydrodynamic and biological processes, focusing on the combined impact of spatial heterogeneity and temporal variability on population dynamics. Studying periodically alternating pool-riffle rivers that are subjected to seasonally varying flows, we obtain an invasion ratchet mechanism. We analyze the ratchet process for a caricature model and a hybrid physical–biological model. The water depth and current are derived from a hydrodynamic equation for variable stream bed water flows and these quantities feed into a reaction-diffusion-advection model that governs population dynamics of a river species. We establish the existence of spreading speeds and the invasion ratchet phenomenon, using a mixture of mathematical approximations and numerical computations. Finally, we illustrate the invasion ratchet phenomenon in a spatially two-dimensional hydraulic simulation model of a meandering river structure. Our hybrid modeling approach strengthens the ecological component of stream hydraulics and allows us to gain a mechanistic understanding as to how flow patterns affect population survival.

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Acknowledgments

The authors thank the anonymous referees for many insightful comments and references that helped in improving the paper. The authors thank Julia Blackburn for discussions and for programming the code used in Figs. 7 and 8. We thank Ed McCauley for inspiring us to undertake much of this research. This work was supported by the MITACS NCE Network for Biological Invasions and Dispersal, the Natural Sciences and Engineering Research Council of Canada (Discovery and Accelerator grants), Alberta Sustainable Resource Development, the Alberta Water Research Institute, Killam Research Fellowship and a Canada Research Chair (MAL).

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Appendices

Appendix 1: Derivation of Single Models from Benthic-Drift Models

(a) Derivation of (1)

Note that in a river or stream, many species of plankton spend most of the time on the benthos and little time in the flow (Walks 2007). We may divide the whole population into two parts: population on benthos and population in flow, and then describe the population dynamics by the benthic-drift system [(see (Pachepsky et al. 2005)]

$$\begin{aligned} \displaystyle \frac{\partial N_\mathrm{{d}}}{\partial t}&= \mu N_\mathrm{{b}}-\sigma N_\mathrm{{d}} +D \frac{\partial ^2 N_\mathrm{{d}}}{\partial x^2}-v \frac{\partial N_\mathrm{{d}}}{\partial x},\nonumber \\ \displaystyle \frac{\partial N_\mathrm{{b}}}{\partial t}&= -\mu N_\mathrm{{b}}+\sigma N_\mathrm{{d}} +f(N_\mathrm{{b}}) N_b, \end{aligned}$$
(25)

where \(N_\mathrm{{b}}\) and \(N_\mathrm{{d}}\) are population densities on the benthos and in the flow, respectively, \(D\) is the diffusion rate, \(v\) is the flow velocity, \(\mu \) is the transfer rate of individuals from the benthos to the flow, and \(\sigma \) is the transfer rate from the flow to the benthos. In a limiting case, when the transfer rates become strong (i.e., \(\sigma , \mu \rightarrow \infty \) with \(\sigma =\tau \mu \)), the second equation in (25) yields \(N_\mathrm{{b}} =\tau N_\mathrm{{d}}\), and hence, (25) can be combined into a single equation:

$$\begin{aligned} \frac{\partial N_\mathrm{{b}}}{\partial t}=\tilde{f}(N_\mathrm{{b}})N_\mathrm{{b}}+\tilde{D}\frac{\partial ^2 N_\mathrm{{b}}}{\partial x^2}-\tilde{v}\frac{\partial N_\mathrm{{b}}}{\partial x} \end{aligned}$$
(26)

with

$$\begin{aligned} \tilde{f}(N_\mathrm{{b}})=\frac{f(N_\mathrm{{b}})}{1+1/\tau }, \tilde{D}=\frac{D}{\tau +1}, \tilde{v}=\frac{v}{\tau +1}, \end{aligned}$$

where \( f\), \(D\), and \(v\) are the parameters of (25) [see Sect. 6 in (Pachepsky et al. 2005) for details]. Model (1) is of the same form as (26).

(b) Derivation of (3)

In a spatially heterogeneous habitat, the benthic-drift model becomes

$$\begin{aligned} \displaystyle \frac{\partial N_\mathrm{{d}}}{\partial t}&= \mu N_\mathrm{{b}}- \sigma N_\mathrm{{d}} + \frac{1}{A(x)}\frac{\partial }{\partial x}\left( D(x)A(x)\frac{\partial N_\mathrm{{d}}}{\partial x}\right) - \frac{Q}{A(x)}\frac{\partial N_\mathrm{{d}}}{\partial x},\nonumber \\ \displaystyle \frac{\partial N_\mathrm{{b}}}{\partial t}&= -\mu N_\mathrm{{b}}+ \sigma N_\mathrm{{d}} +f(N_\mathrm{{b}}) N_\mathrm{{b}}, \end{aligned}$$
(27)

where \(A(x)\) is the cross-sectional area at \(x\) (see (Lutscher et al. 2006) for the model details). Similarly as above, when the transfer rates \(\mu \rightarrow \infty \) and \(\sigma \rightarrow \infty \) with \(\sigma =\tau \mu \), (27) can be combined into a single model, which has the same form as (3).

(c) Derivation of (24)

In a two-dimensional habitat, the benthic-drift model becomes

$$\begin{aligned} \displaystyle \frac{\partial N_\mathrm{{d}}(x,y,t)}{\mathrm{{d}}t}&= \frac{\mu (x,y)}{h(x,y)} N_\mathrm{{b}}(x,y,t) -\sigma (x,y) N_\mathrm{{d}}(x,y,t)\nonumber \\&-\frac{1}{h(x,y)}\left[ \frac{\partial }{\partial x}\left[ v_1(x,y) h(x,y) N_\mathrm{{d}}(x,y,t)\right] \right. \nonumber \\&\left. +\frac{\partial }{\partial y}\left[ v_2(x,y) h(x,y) N_\mathrm{{d}}(x,y,t)\right] \right] \nonumber \\&+\frac{1}{h(x,y)}\left[ \frac{\partial }{\partial x}\left[ D(x,y) h(x,y)\frac{\partial N_\mathrm{{d}}(x,y,t)}{\partial x}\right] \right. \nonumber \\&\left. +\frac{\partial }{\partial y}\left[ D(x,y) h(x,y)\frac{\partial N_\mathrm{{d}}(x,y,t)}{\partial y}\right] \right] ,\nonumber \\ \displaystyle \frac{ \partial N_\mathrm{{b}}(x,y,t)}{\mathrm{{d}}t}&= f(N_\mathrm{{b}}(x,y,t))+\sigma (x,y) N_\mathrm{{d}}(x,y,t) h(x,y) -\mu (x,y) N_\mathrm{{b}}(x,y,t),\nonumber \\ \end{aligned}$$
(28)

where \(h(x,y)\) is the water depth at \((x,y)\), \(v_1\), and \( v_2 \) are the flow velocity in the \(x\) and \(y\) directions, respectively. Similarly as above, when the transfer rates \(\mu \rightarrow \infty \) and \(\sigma \rightarrow \infty \) with \(\sigma =\tau \mu /h\), where \(\tau \) is a constant, we approximately have \(n_\mathrm{{b}} =\tau n_\mathrm{{d}}\). Then, the summation of the first equation multiplied by \(\tau \) and the second equation multiplied by a constant \(\varsigma \ge 0\) yields

$$\begin{aligned} \displaystyle \frac{\partial N_\mathrm{{b}}(x,y,t)}{\mathrm{{d}}t}&= \tilde{f}(N_\mathrm{{b}}) N_\mathrm{{b}}-\frac{1}{h(x,y)}\left[ \frac{\partial }{\partial x}\left[ \tilde{v}_1(x,y) h(x,y)N_\mathrm{{b}}(x,y,t)\right] \right. \nonumber \\&+\left. \frac{\partial }{\partial y}\left[ \tilde{v}_2(x,y) h(x,y) N_\mathrm{{b}}(x,y,t)\right] \right] \nonumber \\&+\frac{1}{h(x,y)}\left[ \frac{\partial }{\partial x}\left[ \tilde{D}(x,y) h(x,y)\frac{\partial N_\mathrm{{b}}(x,y,t)}{\partial x}\right] \right. \nonumber \\&\left. +\frac{\partial }{\partial y}\left[ \tilde{D}(x,y) h(x,y)\frac{\partial N_\mathrm{{b}}(x,y,t)}{\partial y}\right] \right] , \end{aligned}$$
(29)

with

$$\begin{aligned} \tilde{f}(N_\mathrm{{b}})=\frac{\varsigma f(N_\mathrm{{b}})}{1+\varsigma }, \tilde{D}=\frac{D}{\varsigma +1}, \tilde{v}=\frac{v}{\varsigma +1}. \end{aligned}$$

Model (24) has the same form of (29) if \(f(N_\mathrm{{b}})=rN_\mathrm{{b}}(1-N_\mathrm{{b}}/K)\).

Appendix 2: The Upstream and Downstream Extents

Definition 1

Assume that a species is introduced into the river at \(t=0\). The upstream (downstream) extent (or range) \(x_t^{-}\) (\(x_t^{+}\)) at time \(t\) is defined to be the most upstream (downstream) position where individuals are observed at this time.

Remark

In simulations, \(x_t^{-}\) (\(x_t^{+}\)) is represented by the location where the population reaches a threshold detection density \(N_{\mathrm{{thresh}}}\) at time \(t\) in the upstream (downstream) direction. That is, the first \(x\) in the upstream (downstream) direction such that \(N(t,x)=N_{\mathrm{{thresh}}}\) is defined as \(x_t^{-}\) (\(x_t^{+}\)). See Fig. 9.

Fig. 9
figure 9

Upstream and downstream extents of a population with density distribution \(N(t,x)\) in a river

Definition 2

Assume that a species is introduced into the river at \(t=0\). The average speeds of spread of the population in the time interval \([0,t]\) are defined as \((x_0^--x_t^{-})/t\) and \((x_t^{+}-x_0^+)/t\) in the upstream and downstream directions, respectively.

Remark

Note that \((x_0^--x_t^{-})/t \rightarrow c^-\) and \((x_t^{+}-x_0^+)/t \rightarrow c^+\) as \(t\rightarrow \infty \), where \(c^-\) and \(c^+\) are asymptotic spreading speeds as described in Eq. (2). Therefore, we can use \((x_0^--x_t^{-})/t\) and \((x_t^{+}-x_0^+)/t\) to approximate the speeds of spread of the population in the upstream and downstream directions. In the simulations in this paper, we simply use the upstream extent \(x_t^-\) to describe the upstream invasion.

Appendix 3: The Derivation of the Water Depth Eq. (9)

In a gradually varied flow, the water depth \(y\) (unit: m) is non-uniform due to spatially varying slopes of the river bed. The governing equation for the gradually varied flow [see (5–7) in (Chaudhry 1993)] is given as:

$$\begin{aligned} \frac{\mathrm{{d}}y}{\mathrm{{d}}x} = \frac{S_0(x)-S_f(y)}{1-F_r^2(y)}. \end{aligned}$$
(30)

Here, \(S_0\) (unit: m/m) is the slope of the channel bed. It varies in space and is considered as a function of the spatial variable \(x\) (unit: m), i.e., \(S_0=S_0(x)\). Generally, the average values of \(S_0\) are between 0.0002 and 0.008 for big rivers and are slightly larger for small streams (see Table 5). \(S_f\) is the friction slope, i.e., the slope of the energy grade line, or approximation of the water surface slope. It is also spatially varying and can be determined from the Manning equation

$$\begin{aligned} S_f=\frac{n^2 v^2}{k^2 R_h^{4/3}} ~, \end{aligned}$$
(31)

where \(n\) (unit: s/m\(^{1/3}\)) is Manning’s roughness coefficient, varying with basic channel bed mechanism with values in the order of 0.025–0.050 for rivers and representing the resistance to water flows in channels, \(v\) (unit: m/s) is the water flow velocity, \(k=1\) is the conversion factor, \(R_h\) (unit: m) is the hydraulic radius, which is the ratio of wetted area \(A\) (unit: m\(^2\)) and wetted perimeter \(P\) (unit: m), i.e., \(R_h=A/P\). \(F_r\) is the Froude number that is defined as the ratio between the flow velocity and the water wave propagation velocity and is used to determine the resistance of a partially submerged object moving through water. It is a dimensionless parameter. For an arbitrarily shaped channel,

$$\begin{aligned} F_r=\sqrt{\frac{Q^2 W}{g A^3}}, \end{aligned}$$

where \(W\) (unit: m) is the top width at a particular depth, \(Q\) (unit: m\(^3\)/s) is the flow discharge, and \(g=9.8\) (unit: m/s\(^2\)) is the gravitational acceleration.

Table 5 Parameters in this paper

Consider a river (or stream) channel with rectangular cross-sections and fixed width \(B\) along one spatial dimension \(x\). Note that for natural rectangular rivers, the width \(B\) is much larger than the depth \(y\) (i.e., \(B\gg y\)), and therefore, \(R_h= By/(B+2y)\approx y\), so we simply assume \(R_h=y\). It then follows that \(Q = A v = y B v\), \(W\equiv B\),

$$\begin{aligned} S_f=\frac{n^2 v^2}{k^2 R_h^{4/3}}\approx \frac{n^2 v^2}{k^2 y^{4/3}}= \frac{n^2 Q^2}{k^2 B^2 y^{10/3}} \end{aligned}$$
(32)

and

$$\begin{aligned} F_r=\sqrt{\frac{Q^2 W}{g A^3}}=\sqrt{\frac{Q^2 B}{g A^3}}=\frac{Q}{B y\sqrt{g y}}. \end{aligned}$$
(33)

Substituting (32) and (33) into (30), we obtain a first order ODE of the channel depth \(y\):

$$\begin{aligned} \frac{\mathrm{{d}}y}{\mathrm{{d}}x} =\displaystyle \frac{g k^2 B^2 y^{10/3} S_0(x) -n^2 Q^2 g}{k^2 B^2 g y^{10/3}-Q^2 k^2 y^{1/3}} \displaystyle = \frac{g k^2 y^{10/3} S_0(x) -n^2 (Q/B)^2 g}{k^2 g y^{10/3}-(Q/B)^2 k^2 y^{1/3}}. \end{aligned}$$

Appendix 4: The Uniform Flow

Assume that the river has rectangular cross sections and a constant bed slope. If the river channel is long and channel cross sections and the bottom slope do not change with distance, then the flow accelerates or decelerates for a distance until the accelerating and resistive forces are equal (Chaudhry 1993). From that point on, the flow velocity and flow depth remain constant. Such a flow, in which the flow depth does not change with distance, is called a uniform flow, and the corresponding flow depth is called the normal depth, which is actually the critical point of Eq. (9) with a constant bed slope \(S_0\):

$$\begin{aligned} y_n=\left( \frac{Q^2 n^2}{B^2 S_{0} k^2}\right) ^{\frac{3}{10}}. \end{aligned}$$

As a subcritical flow has downstream control, for any downstream boundary condition, in such a flow, the water depth approximates to the normal depth in the upstream end far away from the downstream (see Fig. 10). Therefore, if we just consider the population spreading to the upstream end, then we may assume that the water depth stays at the normal depth, and the associated flow is a uniform flow.

Fig. 10
figure 10

The water depths in a river with a constant bed slope and with different downstream boundary conditions. The solid line represents the normal depth \(y_n=0.5978 m\). Parameters: \(g=9.8~m/s^2\), \(k=1\), \(B=50~m\), \(n=0.03~s/m^{1/3}\), \(S_{0}=0.005\), \(Q_=50~m^3/s\), channel length \(= 700 m\)

Appendix 5: The Periodic Solution to (9)

Existence, Uniqueness and Stability

Note that a periodic solution to (9) with period \(L\) corresponds to a solution to (19), and that a flow is subcritical if and only if \(F_r<1\), which implies that \(y>y_c=(Q^2/(B^2g))^{1/3}\), where \(y_c\) is called the critical depth of the river. We solve the ODE (19) in the half plane of \(y>y_c\). When the bed slope \(S_0(x)\) is given as in (18), (19) has a solution if and only if there exists some \(y_0>y_c\) such that the solutions to the two problems

$$\begin{aligned}&\left\{ \begin{array}{l} \displaystyle \frac{\mathrm{{d}}y_1}{\mathrm{{d}}x} = \frac{g k^2 y_1^{10/3} S_{0r} -n^2 (Q/B)^2 g}{k^2 g y_1^{10/3}-(Q/B)^2 k^2 y_1^{1/3}}\\ y_1(0)=y_0. \end{array}\right. \mathrm{{and}}\\&\left\{ \begin{array}{l} \displaystyle \frac{\mathrm{{d}}y_2}{\mathrm{{d}}x} = -\left( \frac{g k^2 y_2^{10/3} S_{0p} -n^2 (Q/B)^2 g}{k^2 g y_2^{10/3}-(Q/B)^2 k^2 y_2^{1/3}}\right) ,\\ y_2(0)=y_0. \end{array}\right. \end{aligned}$$

satisfy \(y_1(L_1)=y_2(L_2)\), which is true if and only if there exists some \(y_0>y_c\) such that the solutions to the two problems

$$\begin{aligned}&\left\{ \begin{array}{l} \displaystyle \frac{\mathrm{{d}}y_1}{\mathrm{{d}}x} = \frac{g k^2 y_1^{10/3} S_{0r} -n^2 (Q/B)^2 g}{k^2 g y_1^{10/3}-(Q/B)^2 k^2 y_1^{1/3}}\\ y_1(0)=y_0. \end{array} \right. \mathrm{{and}}\\&\left\{ \begin{array}{l} \displaystyle \frac{\mathrm{{d}}y_3}{\mathrm{{d}}x} = -\left( \frac{g k^2 y_3^{10/3} S_{0p} -n^2 (Q/B)^2 g}{k^2 g y_3^{10/3}-(Q/B)^2 k^2 y_3^{1/3}}\right) \cdot \frac{L_2}{L_1},\\ y_3(0)=y_0. \end{array}\right. \end{aligned}$$

satisfy \(y_1(L_1)=y_3(L_1)\).

We actually look for a solution to (19) in the half plane of \(y>y_n\), where \(y_n=\left( Q^2 n^2/(B^2 S_{0r} k^2)\right) ^{3/10}\) is the normal depth of the river where the bed slope is \(S_{0r}\), because otherwise it is easy to see from the above equivalent relations that we cannot have a solution to (19). Let \(f_1\) and \(f_2\) be the right-hand sides of these two equations, respectively. Then \(f_1\) is an increasing function of \(y\), and \(f_2\) is a decreasing function of \(y\). As \(S_{0r}>0\) and \(S_{0p}<0\), we know that \(f_1<f_2\) for small \(y>y_c\) and \(f_1>f_2\) for big \(y>y_c\). This results in that the solution \(y_1(x,y_0)<y_3(x,y_0)\) at small \(x\) and \(y_1(x,y_0)>y_3(x,y_0)\) at big \(x\) provided that \(y_0\) is not such that \(f_1>f_2\). Therefore, by the continuity of \(y_1\) and \(y_3\) with respect to \(x\), for any \(y_0>y_c\) with \(f_1(y_0)<f_2(y_0)\), there exists an \(x_{y_0}\) such that \(y_1(x_{y_0},y_0)=y_3(x_{y_0},y_0)\). By the continuity and monotonicity of \(f_1\) and \(f_2\), there exists a unique \(y_0>y_n\) such that \(x_{y_0}=L_1\), which corresponds to a unique solution to (19).

For any solution to (9) with some downstream boundary condition, we can show that the solution approaches the periodic solution at the very upstream periods by iterating the solution backward from the downstream to the upstream. As an illustration, an example is shown in Fig. 11, where the solid curve represents the periodic solution. Therefore, the periodic solution to (9) is stable for solutions to (9) with all downstream boundary conditions.

Fig. 11
figure 11

The water depths in a periodic pool-riffle river with different downstream boundary conditions. The solid line is the periodic solution with downstream boundary condition \(y(\mathrm{{end}})=0.7652\) m. Parameters: \(g=9.8~\mathrm{{m/s^2}}\), \(k=1\), \(B=50\) m, \(n=0.03~\mathrm{{s/m^{1/3}}}\), \(L_1=100\) m, \(L_2=200\) m, \(S_{0p}=-0.001\), \(S_{0r}=0.005\), \(Q_=50~\mathrm{{m^3/s}}\), channel length\(= 1,500\) m

Numerical Calculation of the Periodic Solution

Recall that subcritical flow has a downstream control, which means that to change the flow conditions in a section, flow conditions must be changed at a downstream location. Consequently, when we solve Eq. (19), we start the computation at a downstream control section and proceed in the upstream direction.

In more details, the idea for solving (19) is as follows. Guessing a boundary value \(y^{(0)}(L)\) at the downstream end \(x=L\), one calculates backward and obtains all values of \(y\) on \([0,L]\), especially \(y^{(0)}(0)\) at the upstream end \(x=0\). Then one iterates by taking a new boundary value at the downstream \(y^{(1)}(L)=y^{(0)}(0)\) and obtains \(y^{(1)}(0)\). Then let \(y^{(2)}(L)=y^{(1)}(0)\) and repeat the process till \(y^{(n)}(0)=y^{(n)}(L)\) at the \(n\)-th step. The reason for integrating backward is that the depth is asymptotic to a constant (called normal depth) proceeding upstream. The fixed point iteration scheme will converge very quickly integrating upstream, while it will diverge if integrated downstream. In each iteration, the equation is solved by the Runge–Kutta method.

Appendix 6: Spreading Speeds for the Models

Spreading Speeds for the Model in a Temporally Constant Flow

The Existence of Spreading Speeds for (3)

Let \(N(t,x)\) be the solution to (3) with initial value \(N_0\in C(\mathbb {R},\mathbb {R})\). Define the solution map \(\Phi _t\) of (3) as

$$\begin{aligned} \Phi _t[N_0](x)=N(t,x), \, t\ge 0. \end{aligned}$$

It follows from the standard theory of solutions to (3) that (3) generates a monotone semiflow \(\{\Phi _t\}_{t\ge 0}\) in the sense that \(\Phi _0[N_0] = N_0\) for all \(N\in C(\mathbb {R}_+,\mathbb {R})\), \(\Phi _t[\Phi _s[N_0]] = \Phi _{t+s}[N_0]\) for all \(t\ge 0\), \(s\ge 0\) and \(N_0\in C(\mathbb {R},\mathbb {R})\), and \(Q[t, N_0] := \Phi _t[N_0]\) is continuous in \((t, N_0)\) for all \(t\ge 0\) and \(N_0\in C(\mathbb {R},\mathbb {R})\). As (3) is a standard parabolic equation, \(\Phi _t\) satisfies all conditions in the abstract theory for spreading speeds for a semiflow defined in a periodic habitat in (Liang and Zhao 2010). Then by ((Liang and Zhao 2010, Theorem 5.2)), we obtain the existence of the upstream and downstream spreading speeds (\(c^-\) and \(c^+\)) [see also e.g., (Berestycki et al. 2005; Liang and Zhao 2007; Weinberger 2002)]. Note that by similar arguments to those in ((Lou and Zhao 2010, Lemma 2.10)), we have \(c^++c^->0\). The results of spreading speeds are included in the following theorem.

Theorem 1

Let \(N(t,x;\varphi )\) be the solution of (3) with \(N(0,x;\varphi )=\varphi (x)\) for all \(x\in \mathbb {R}\). The system (3) admits upstream spreading speed \(c^-\) and downstream spreading speed \(c^+\) in the following sense.

  1. (i)

    For any \(c>c^+\) and \(c^\prime >c^-\), if \(\varphi \in C_K=\{\psi \in C(\mathbb {R},\mathbb {R}),0\le \psi (x)\le K \mathrm{{for all}}x\in \mathbb {R}\}\) with \(\varphi (x)=0\) for \(x\) outside a bounded interval, then

    $$\begin{aligned} \lim \limits _{t\rightarrow \infty , x\ge c t}N(t,x;\varphi )=0, \qquad \lim \limits _{t\rightarrow \infty , x\le -c^\prime t}N(t,x;\varphi )=0. \end{aligned}$$
  2. (ii)

    For any \(c<c^+\) and \(c^\prime <c^-\), there is a positive number \(r\in \mathbb {R}\), such that if \(\varphi \in C_{K}\) and \(\varphi (x)>0\) for \(x\) on an interval of length \( r\), then

    $$\begin{aligned} \lim \limits _{t\rightarrow \infty , -c^\prime t\le x\le c t}(N(t,x;\varphi )-K)=0. \end{aligned}$$

Moreover, the spreading speeds in the upstream and downstream directions are also the minimal wave speeds for spatially periodic traveling waves, respectively. The following result follows from Theorem 5.3 in (Liang and Zhao 2010).

Theorem 2

\(\{\Phi _t\}_{t\ge 0}\) has an \(L\)-periodic rightward traveling wave \(V(x-ct,x)\) connecting \(K\) to \(0\) with \(V(\xi ,x)\) being continuous and non-increasing in \(\xi \in \mathbb {R}\) if and only if \(c\ge c_+\). \(\{\Phi _t\}_{t\ge 0}\) has an \(L\)-periodic rightward traveling wave \(V(x+ct,x)\) connecting \(0\) to \(K\) with \(V(\xi ,x)\) being continuous and increasing in \(\xi \in \mathbb {R}\) if and only if \(c\ge c_-\).

The Estimation of Spreading Speeds for (20)

We can follow the steps in Example 6.2 in (Weinberger 2002) to derive the spreading speeds for (20). The linearized equation for (20) at \(N=0\) is

$$\begin{aligned} \frac{\partial N(t,x)}{\partial t} = \frac{1}{y(x)} \frac{\partial }{\partial x} \left[ D(x) y(x) \frac{\partial N(t,x)}{\partial x} \right] - \frac{Q}{B y(x)} \frac{\partial N(t,x)}{\partial x} + r N(t,x).\qquad \quad \end{aligned}$$
(34)

Let \(N(t,x)=e^{\lambda t-\zeta \rho x}\psi (x)\) with \(\zeta >0\) and \(\rho =\pm 1\), and substitute it into (34). We obtain

$$\begin{aligned} \lambda e^{\lambda t-\zeta \rho x}\psi (x)&= \frac{1}{y(x)} \frac{\partial }{\partial x} \left[ D(x) y(x) \frac{\partial [e^{\lambda t-\zeta \rho x} \psi (x)]}{\partial x} \right] \\&-\frac{Q}{B y(x)} \frac{\partial [e^{\lambda t-\zeta \rho x}\psi (x)]}{\partial x} +r e^{\lambda t-\zeta \rho x}\psi (x), \end{aligned}$$

which can be simplified as

$$\begin{aligned} \lambda \psi (x)&= \displaystyle D(x)\psi ^{\prime \prime }(x) +\left[ \frac{1}{y(x)} \frac{\partial (D(x) y(x))}{\partial x} -2 \zeta \rho D(x)-\frac{Q}{B y(x)}\right] \psi ^\prime (x)\nonumber \\&\displaystyle +\left[ D(x)\zeta ^2-\frac{\zeta \rho }{y(x)}\frac{\partial (D(x) y(x))}{\partial x}+ \frac{Q \zeta \rho }{B y(x)}+r\right] \psi (x). \end{aligned}$$
(35)

Define \(L_\rho \) as

$$\begin{aligned} L_\rho&:= D(x) \frac{\partial ^2}{\partial x^2}\left[ \frac{1}{y(x)} \frac{\partial (D(x) y(x))}{\partial x}-2 \zeta \rho D(x) -\frac{Q}{B y(x)}\right] \\&+\frac{\partial }{\partial x} \left[ D(x) \zeta ^2 -\frac{\zeta \rho }{y(x)}\frac{\partial (D(x) y(x))}{\partial x}+ \frac{Q \zeta \rho }{B y(x)}+r \right] . \end{aligned}$$

It follows that \(L_\rho \) is compact and strongly positive, and hence, it admits a single principal eigenvalue with a positive periodic eigenfunction. Let \(\lambda (\zeta \rho )\) be the principle eigenvalue and \(\psi \) be the associated positive periodic eigenfunction. Then \(\lambda (\zeta \rho )\) and \(\psi \) satisfy (35) with \(\psi (0)=\psi (L)\) and \(\psi ^\prime (0)=\psi ^\prime (L)\). Define

$$\begin{aligned} c^\rho =\inf \limits _{\zeta >0}\frac{\lambda (\zeta \rho )}{\zeta }. \end{aligned}$$

When \(\rho =1\), \(c^\rho \) is the downstream spreading speed and when \(\rho =-1\), \(c^\rho \) is the upstream spreading speed. We can apply the techniques for the principal eigenvalue of Hill’s equations to (35) to obtain \(\lambda \psi (x)\).

This approximation provides a way to estimate spreading speeds for (20). However, we cannot have explicit expressions and will have to follow numerical calculations. This results in difficulties in finding how different factors affect spreading speeds.

Spreading Speeds for the Model in a Time-Varying Flow Environment (22)

To find the spreading speeds for the model (22) with varying water discharge, let \(N(t,x)=V(t,x)\cdot e^{-\eta \rho x}\) and substitute it into the linearized equation of (22) at \(N=0\):

$$\begin{aligned} \displaystyle \frac{\partial N(t,x)}{\partial t} \displaystyle&= -\frac{\partial y(t,x)}{\partial t} \cdot \frac{N(t,x)}{y(t,x)}\nonumber \\&\displaystyle +\frac{1}{y(t,x)} \frac{\partial }{\partial x} \left[ D(t,x) y(t,x) \frac{\partial N(t,x)}{\partial x} \right] \nonumber \\&- \frac{Q(t)}{B y(t,x)} \frac{\partial N(t,x)}{\partial x} + f(0) N(t,x), \end{aligned}$$
(36)

where \(\rho \) represents the direction with \(\rho =1\) in the downstream direction and \(\rho =-1\) in the upstream direction. We then have

$$\begin{aligned} \displaystyle \frac{\partial V(t,x)}{\partial t} \displaystyle&= -\frac{\partial y(t,x)}{\partial t}\cdot \frac{V(t,x)}{y(t,x)} +D(t,x)\frac{\partial ^2 V(t,x)}{\partial x^2}\nonumber \\&\displaystyle +\left[ -2\eta \rho D(t,x)+ \frac{\partial D(t,x)}{\partial x}+\frac{D(t,x)}{y(t,x)}\frac{\partial y(t,x)}{\partial x}-\frac{Q(t)}{B y(t,x)}\right] \frac{\partial V(t,x)}{\partial x}\nonumber \\&\displaystyle + \left[ \eta ^2 D(t,x)-\eta \rho \left( \frac{\partial D(t,x)}{\partial x}+\frac{D(t,x)}{y(t,x)}\frac{\partial y(t,x)}{\partial x}-\frac{Q(t)}{B y(t,x)}\right) +f(0)\right] \nonumber \\&\times V(t,x), \end{aligned}$$
(37)

Let \(L_{\eta \rho }^T\) be the Poincaré map of (37), where \(T\) is the period of the varying flow. Then the theory in (Weinberger 2002) implies that the spreading speed of \(L_{\eta \rho }^T\) is

$$\begin{aligned} c_\rho ^T= \inf \limits _{\eta >0}\frac{\ln \lambda (\eta \rho )}{\eta }, \end{aligned}$$

where \(\lambda (\eta \rho )\) is the principal eigenvalue of \(L_{\eta \rho }^T\) corresponding to a positive periodic eigenfunction, and the sign of \(\rho \) determines the direction (upstream or downstream). It then follows from the theory in (Liang et al. 2006) that the spreading speed of (22) is

$$\begin{aligned} c_\rho =\frac{1}{T} \inf \limits _{\eta >0} \frac{\ln \lambda (\eta \rho )}{\eta }. \end{aligned}$$

\(c_{1}\) is the downstream spreading speed, and \(c_{-1}\) is the upstream spreading speed.

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Jin, Y., Hilker, F.M., Steffler, P.M. et al. Seasonal Invasion Dynamics in a Spatially Heterogeneous River with Fluctuating Flows. Bull Math Biol 76, 1522–1565 (2014). https://doi.org/10.1007/s11538-014-9957-3

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Keywords

  • Pool-riffle rivers
  • Spatial-temporal heterogeneity
  • Environmental flow assessment
  • Invasion ratchet
  • Flow regime