# Integrodifference models for persistence in temporally varying river environments

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## Abstract

To fully understand population persistence in river ecosystems, it is necessary to consider the effect of the water flow, which varies tremendously with seasonal fluctuations of water runoff and snow melt. In this paper, we study integrodifference models for growth and dispersal in the presence of advective flow with both periodic (alternating) and random kernel parameters. For the alternating kernel model, we obtain the principal eigenvalue of the linearization operator to determine population persistence and derive a boundary value problem to calculate it. For the random model, we establish two persistence metrics: a generalized spectral radius and the asymptotic growth rate, which are mathematically equivalent but can be understood differently, to determine population persistence or extinction. The theoretical framework and methods for calculations are provided, and the framework is applied to calculating persistence in highly variable river environments.

## Keywords

Integrodifference Time-varying environment Random advection kernels Persistence## Mathematics Subject Classification (2000)

45C05 45R05 92B05## 1 Introduction

### 1.1 Background

Stream and river ecosystems are shaped by their physical environment of unidirectional water flow. Questions of population persistence in river ecosystems must necessarily consider the effect of the water flow on populations in space and time. To complicate matters, this water flow can vary tremendously with seasonal fluctuations of water runoff and snow melt.

How can populations persist in streams when they are being constantly washed downstream? This so-called *drift paradox* (Muller 1982) has engaged biologists and mathematicians in a series of modeling efforts using reaction–advection–diffusion equations to describe the population densities in space and time. The early paper of Speirs and Gurney (2001) uses a modification of Fisher’s equation that includes advection to show the existence of a critical flow rate in the stream, below which the populations will persist, and above which the population will wash out, much as a chemostat population will persist or wash out in low flow and high flow conditions.

The approach of Speirs and Gurney (2001) employs classical mathematical methods of population spreading speeds and critical domain size. The spreading speed for Fisher’s equation, \(2\sqrt{r\!D}\) where \(r\) is the intrinsic growth rate and \(D\) is the diffusion coefficient (Aronson and Weinberger 1975), yields the critical advection velocity \(v_c\), below which stream populations will persist, and above which they will wash out. This can be understood intuitively: when the advection velocity \(v\) exactly matches the spreading speed \(2\sqrt{r\!D}\), the population is washed downstream by water flow at the same speed it is moving upstream by the combined effects of growth (\(r\)) and diffusion (\(D\)). Speirs and Gurney (2001) show the critical domain size \(L_c\) exists for all advection velocities that lie below the critical value \((0<v<v_c)\) and that the critical domain size approaches infinity as \(v\) approaches \(v_c\). Biologically, this is interpreted as implying that stream populations will persist if the advection speed falls below a threshold value, and there is a sufficiently large stretch of stream available. This theory has been tested empirically by Walks (2007) who related the persistence of plankton in flowing water to stream advection velocities. A mathematical review of the ideas in Speirs and Gurney (2001) can be found in Lewis et al. (2009).

Extensions to the theory have focused on increasingly realistic models for the stream populations. These include stationary and mobile compartments to describe subpopulations on the benthos and in the stream (Pachepsky et al. 2005), non-diffusive dispersal of stream populations that can include long-distance jumps (Lutscher et al. 2005), spatially varying stream environments (Lutscher et al. 2006), spatial interactions between competitors in the stream environment (Lutscher et al. 2007) and periodic fluctuations in environmental conditions (Jin and Lewis 2011, 2012).

Despite these extensions to the theory, the models have been limited to the case where the stream environment is predictable. Although convenient from a modelling perspective, this is inaccurate. For example stream flows not only vary by an order of magnitude between spring and fall seasons (Abrahamsson and Hakanson 1998), they also vary unpredictably from year to year (Anderson et al. 2006). While some models exist for spreading populations in randomly fluctuating (Neubert et al. 2000) environments, none have investigated persistence and spread in environments such as streams, where unidirectional flow predominates.

In this paper we investigate persistence of populations in periodic and randomly fluctuating environments with predominantly unidirectional flow. Our mathematical model is based on a discrete-time and continuous-space dynamical system that takes the form of an integrodifference equation. In the next section we develop a modelling background for integrodifference models.

### 1.2 Integrodifference models

*dispersal event*) while growth is independent of space, but depends on the local population density. Denoting \(n_t(x)\) as the population density at stage \(t\), the growth dynamics are modeled by

We consider a habitat \(\varOmega = [x_0,y_0]\) for some \(x_0 < y_0\). For such a bounded domain \(\varOmega \) the model (3) assumes that the organism can disperse across the boundary, but there is no source term from outside the boundary. This is the case if conditions outside \(\varOmega \) are unfavorable to growth and survival or if the organism cannot disperse back into the habitat \(\varOmega \) once it has left. This would be the case for a stretch of suitable habitat in a stream, surrounded by unsuitable habitat, where the organism cannot survive. A non-aquatic example is of a plant whose seeds are blown across the edge of a field into a parking lot or other unsuitable region.

#### 1.2.1 The dispersal kernel

*dispersal success function*and the

*redistribution function*. The dispersal success function \(s(y)\) indicates the probability that an individual starting at \(y\) successfully settles in the habitat \(\varOmega \) after the dispersal event:

*dispersal success approximation*. To illustrate, Fig. 3 shows the dispersal success approximation (13) compared to the principal eigenvalue \(\lambda _1(K)\) as a function of stream length for a sample kernel.

#### 1.2.2 Including temporal variation

### 1.3 Mathematical setting

We briefly review the mathematical setting and known results for population persistence in the context of integrodifference equations. We restrict our attention to the case where \(K(x,y)=K(x-y)\) is a *difference kernel*, expressed in terms of the difference between the settling location \(x\) and the starting point \(y\). Although this includes the case of symmetric distance kernels where \(K(x,y)= K(|x-y|)\) (such as (4)), we do not require symmetry since we are particularly interested in the case where \(K\) is an asymmetric advective kernel such as (5). We assume the population densities are given by elements of \(C(\varOmega )\), the Banach space of continuous real-valued functions defined on \(\varOmega \). We first discuss the constant kernel case, then review some known results for temporally varying kernels.

#### 1.3.1 Constant environments

*positive*operator, mapping the cone of nonnegative functions \(C_+(\varOmega )\) into itself.

A further assumption regarding the positivity of \(\fancyscript{L}\) allows one to connect the population dynamics of (15) with the spectral properties of \(\fancyscript{L}\). Namely, we say the operator \(\fancyscript{L}\) is *strongly positive* if for any continuous function \(n \ge 0\) there exists a power \(t = t(n)\) such that \(\fancyscript{L}^t(n)(x) > 0\) for all \(x \in \varOmega \). Biologically, this condition implies that on a connected habitat repeated application of the kernel will allow an individual that starts at any point \(y \in \varOmega \) to eventually access all other points \(x\) in \(\varOmega \) (Lutscher and Lewis 2004, Condition A4). In this case, the Krein–Rutman theorem applies, and it follows that \(\fancyscript{L}\) has a principal eigenvalue \(\lambda _1 > 0\) such that \(|\lambda | < \lambda _1\) for all other eigenvalues and \(\lambda _1\) is the only eigenvalue associated with a positive eigenfunction \(\phi \). Our assumptions on \(K\) imply that the zero solution of (15) is linearly stable when \(\lambda _1 < 1\) and unstable when \(\lambda _1 > 1\). Moreover, for \(\lambda _1 > 1\) there exists a nontrivial equilibrium solution of (15) (Hardin et al. 1990, see also Van Kirk and Lewis 1997 for the \(L^2(\varOmega )\) case).

#### 1.3.2 Temporally Varying Environments

### 1.4 Outline of the paper

In this paper, we study the integrodifference equation (15) for population persistence in temporally varying advective environments. In Sect. 2, we study the integro-difference equation with alternating kernels and growth rates in a periodically varying environment and obtain an explicit method to calculate the principal eigenvalue for the two-stage process. We also give the approximation of the principal eigenvalue by virtue of the dispersal success function and the redistribution function. In Sect. 3, we study the model in a randomly varying advective environment, where both the growth rate and the dispersal kernel are random. This contrasts with the earlier work by Hardin et al. (1988a), where only the growth rate fluctuated randomly. We derive the persistence metric \(r\), similar to (23) and obtain its equivalence to the asymptotic growth rate (24). We also provide exact formula for the asymptotic growth rate when kernels take an asymmetric advective form (5) with randomly chosen parameters. This allows us to explicitly calculate population growth rates in randomly fluctuating river environments. Our various methods for calculating persistence and growth metrics are illustrated using numerical examples for models describing randomly fluctuating rivers. A short discussion completes the paper in Sect. 4.

## 2 Alternating kernel model

### *Example 1*

To illustrate how the principal eigenvalue \(\lambda _{1,\mathrm{twostep}}\) of (29) depends on flow velocities we consider the case of variable flow rates \(v_1\) and \(v_2\), but with fixed mean \(\bar{v} = (v_1 + v_2)/2\) for a habitat \(\varOmega =[0,L]\) with \(L=20\), \(R_1=1.2\), \(R_2=1.5\), \(D_1=D_2=1\), \(\beta _1=\beta _2=1\).

For a fixed mean \(\bar{v}\), we can also consider \(\lambda _{1,\mathrm{twostep}}\) as a function of the variation in flow \(|v_1 - v_2|\). Figure 5a illustrates this for the case \(\bar{v} = 1.3\). Note that \(\lambda _{1,\mathrm{twostep}}\) is an increasing function of \(|v_1-v_2|\). For this average flow velocity, the smaller the variation between \(v_1\) and \(v_2\), the smaller the possibility that the population can persist in the river.

If we fix \(v_1\) and vary only \(v_2\), then Fig. 5b shows that \(\lambda _{1,\mathrm{twostep}}\) is a decreasing function of \(v_2\). This coincides with the fact that, when the flow velocity in one step is constant, then the larger the flow velocity in the second step, the harder it is for the population to persist in the river.

### *Example 2*

To illustrate how one can study critical domain size questions in this setting we consider the case of fixing \(v_1 = 0.1\) and determining the critical domain length as a function of \(v_2\) (leaving the other parameters as in Example 1). We can study this by setting \(\lambda = 1\) in (34) and determining conditions on \(L\) for which the fourth-order system admits a nontrivial solution. An example is shown in Fig. 6. As one might expect, as \(v_2\) increases the critical domain length increases, with \(L \) approaching infinity as \(v_2\) tends to some value. Since the critical domain size represents the minimal length of the river such that population can persists, this observation implies that the higher the flow the more difficult it is for the population to persist in the river, consistent with earlier results in Lutscher et al. (2005) and Jin and Lewis (2011).

The differential equation approach in this section for alternating kernels can be generalized to the case of a sequence of \(n\) kernels (Jacobsen and McAdam 2014). However, we will instead turn to the case of random kernels, and in particular, asymmetric Laplace kernels whose parameters are chosen from a given distribution.

## 3 Random kernel model

### 3.1 Persistence metrics

- (C1)
- 1.
For each \(\alpha \in {\mathcal {A}}\), \(K_\alpha (x-y)\) is continuous for \(x,y \in \varOmega \).

- 2.There exists constants \(\underline{K}> 0\) and \(\overline{K}\) such that$$\begin{aligned} \underline{K}\le K_\alpha (x-y)\le \overline{K} \quad \text { for all } \alpha \in \mathcal {A} \text { and } x, y\in \varOmega . \end{aligned}$$

- 1.
- (C2)
- 1.
For any \(\alpha \in {\mathcal {A}}\), \(f_\alpha :\mathbb {R}\rightarrow [0,\infty )\) is continuous with \(f_\alpha (u)=0\) for all \(u\le 0\).

- 2.There exists \(m> 0\), \(\underline{f}>0\), and \(\overline{f}>0\), such that for any \(\alpha \in \mathcal {A}\),
- (a)
\(f_\alpha (u)\) is an increasing function in \(u\).

- (b)
\(0\le f_\alpha (u)\le m\) for all \(u\in C_+(\varOmega )\).

- (c)
If \(0 < v < u\) then \(\frac{f_\alpha (u)}{u}<\frac{f_\alpha (v)}{v}\).

- (d)
\(f_\alpha \) is right differentiable at \(0\). For simplicity, we denote the right derivative as \(f_\alpha ^\prime (0)\).

- (e)
\(\frac{f_\alpha (u)}{u}\rightarrow f_\alpha ^\prime (0)\) as \(u\rightarrow 0^+\), uniformly for \(\alpha \in \mathcal {A}\)

- (f)
\( \underline{f}=\inf \limits _{\alpha \in \mathcal {A}}f_\alpha ^\prime (0)\le f_\alpha ^\prime (0)\le \overline{f}\)

- (g)
For \(b=m \overline{K} |\varOmega |\), there exists \(\underline{f_1}=\inf \limits _{\alpha \in \mathcal {A}}f_\alpha (b)>0\).

- (a)

- 1.
- (C3)
There exists \(\alpha ^*\in \mathcal {A}\) such that \(F_{\alpha }(u)\le F_{\alpha ^*}(u)\) for all \(\alpha \in \mathcal {A}\) and \(u\in C_+(\varOmega )\).

### **Theorem 1**

Assume that \(F_{\alpha _t}\) (\(\alpha _t\in \mathcal {A}\) for \(t\in \mathbb {N}\)) defined by (44) satisfies (C1)–(C3). For nonzero initial data \(n_0\in C_+(\varOmega )\), the population \(n_t\) of (44) converges in distribution to a stationary distribution \(\mu ^*\), independent of \(n_0\), that is either concentrated at \(0 \in C_+(\varOmega )\) (extinction) or supported in \(C_+(\varOmega )\backslash \{0\}\) (persistence).

- (a)
If \(r<1\), then the population will go extinct.

- (b)
If \(r>1\), then the population will persist.

The quantity (45) provides a means to study population persistence for our integrodifference stream model in the context of random dispersal and growth, within the framework of the hypotheses (C1)–(C3). We now show there is an alternate metric for (44), which can also be used to analyze persistence of the population, is numerically easier to work with, and has a clear biological interpretation.

### **Theorem 2**

Let \(r\) and \(\varLambda \) be defined in (45) and (48), respectively. Then \(\varLambda = r\).

### *Remark 1*

It follows from Theorems 1 and 2 that if \(\varLambda >1\), the population will be persistent and if \(\varLambda <1\), the population will go extinct. Thus population persistence or extinction can be studied by computing \(\varLambda \) for the iterates \(n_t\) of the linear model (46) (recalling \(R_{\alpha _t}\) and \(K_{\alpha _t}\) change at each step).

We illustrate applications of Theorem 2 for several examples of (46), using \(\varLambda \) to determine population persistence or extinction. For simplicity, we use (48) to approximate \(\varLambda \).

### *Example 3*

(Random two kernel model) Consider (46) where the kernel \(K_t\) is chosen at random from one of two asymmetric advective kernels \(K_1\) and \(K_2\), with equal probability. For \(K_i\) as in (5), we assume \(v_1=0.1, v_2=1,\) \(D_1 = D_2 = 1\), and \(\beta _1 = \beta _2 = 1\). Since we are effectively flipping a coin to determine the kernel \(K_t\) we call this the “coin-flip kernel model” or CFK model. We assume \(R_t = 1.2\) if \(K_t = K_1\) and \(R_t = 1.5\), if \(K_t = K_2\).

Next, we compare the principal eigenvalue for the alternating kernel model from Sect. 2 with the value of \(\varLambda \) for the random CFK model. Figure 8 shows a plot of the principal eigenvalue \(\lambda _{1,\mathrm{twostep}}\) of the alternating kernel model (27) (using the same parameters from the CFK model) with \(\varLambda \) for the the CFK model (46). The principal eigenvalue of the alternating kernel model appears to match well with \(\varLambda \) for the random model.

### *Example 4*

(Log-normal flow velocities) Our next example considers (46) with the flow rate for kernel \(K_t\) chosen from a log-normal distribution (keeping the other parameters fixed). We consider the relation between the asymptotic growth rate \(\varLambda \) as a function of the variance in flow rate, while maintaining a fixed mean.

First, to illustrate the log-normal distribution, Fig. 9 shows the probability density function for a log-normal distribution with a fixed mean for two different variances.

### 3.2 Explicit calculation for \(\varLambda _t\)

## 4 Discussion

Even though classical ecological models assume environmental uniformity, the true natural environment shows a high degree of temporal variability. While the yearly specifics of the environmental variations rarely can be predicted, the general nature of the variability, as measured over many years, can be described statistically. One emerging challenge in mathematical biology has been to incorporate such measures of environmental variability into mathematical models for population persistence (see, for example, Benaïm and Schreiber 2009; Tuljapurkar 1990; Tuljapurkar et al. 2003; Schreiber 2010; Roth and Schreiber 2014). Although much recent mathematical attention has focused on this challenge, the pioneering work by Hardin et al. (1988a) actually provided mathematical tools to understand variability for integrodifference equations, as long as a quarter of a century ago.

The connection between \(r\) and \(\varLambda \) has allowed us to infer persistence properties of the nonlinear stochastic dynamical system, describing population growth and dispersal in rivers, based on \(\varLambda \). In particular, it means that our explicit calculations of the asymptotic growth rate for river systems with asymmetric exponential dispersal (56) can be rigorously connected to persistence in the associated nonlinear system.

In our analysis we also developed a connection between periodically fluctuating river system, with asymmetric exponential dispersal, and a differential operator describing growth of an associated eigenfunction. Numerical results show a close concordance of persistence thresholds for the *alternating kernel model*, where good and bad years alternate, and those for a related *coin flip kernel model*, where good and bad years are chosen randomly with equal probability (Fig. 8).

The class of models in this paper can be generally applied to river or stream populations, where unidirectional flow dominates. However, particular stream populations are likely to require more detailed and specific models. One advantage of a general model is the ability to draw general conclusions. What can be concluded, in general, from the models in this paper regarding the role of variability in persistence in streams and rivers? First, longer streams (Fig. 3) and lower flow rates (Figs. 4, 5b) increase the likelihood of persistence, and higher flow streams must be longer, providing more habitat, if populations are to persist (Fig. 6). These, by themselves are not new theoretical results, and have been understood theoretically since the work of Speirs and Gurney (2001). However, a closer look at Fig. 4 shows that the variability in the flow velocity, as given in the alternating kernel model, can determine persistence outcomes as much as the mean velocity. Specifically, increased variability gives an increased probability of persistence (Fig. 5a). Here the effects of flow rate variation do not simply average out, and the beneficial effect of a low-flow period more than compensates for the detrimental effect of a corresponding high-flow period. This relationship between flow variability and persistence holds over to the more complex case where the dispersal kernel is chosen from a family where the flow velocity is drawn from a continuous probability density function, such as a log-normal distribution (Figs. 9, 10, 11). We considered variations of the parameters in the positive space for the two-step alternating kernel model and the random model and made numerous simulations for the dependence of \(\lambda _{1, \mathrm{twostep}}\) and \(\varLambda \) on the variance of the flow velocity \(v \). In all our simulations, \(\lambda _{1, \mathrm{twostep}}\) and \(\varLambda \) are increasing functions of the variance of \(v\). While we are not able to theoretically prove this result for these two models, Figs. 4, 5, and 9, 10, 11 were typical numerical examples chosen to illustrate the calculations.

Although our model, with uncorrelated random environments, showed that increasing temporal variations can promote population persistence, this phenomenon may not hold in other settings. For example, it has been shown that for a given average population growth rate, temporal variations in the growth rate may increase the risk of extinction; see e.g., Lewontin and Cohen (1969), Turelli (1978), Lande (1993), Halley and Iwasa (1999). Positive temporal autocorrelations in environmental conditions can decrease or increase extinction risk depending on other features; see, e.g., Schwager et al. (2006), Heino et al. (2000), Ripa and Lundberg (1996). In particular, positive autocorrelations in temporal fluctuations can disrupt predator-prey coexistence (Roth and Schreiber 2014). In more general and realistic situations where there are environmental variations in space and time, the effect of interactions between temporal correlations, spatial heterogeneity and dispersal on population persistence becomes even more complex. For instance, metapopulations whose expected fitness in every patch is less than \(1\) can persist if there are positive temporal autocorrelations in relative fitness, sufficiently weak spatial correlations, and intermediate rates of dispersal between patches (Schreiber 2010). More recently, Roth and Schreiber (2014) develop a coexistence criterion for interacting structured populations in stochastic environments and show, among other applications, that autocorrelations in temporal fluctuations can interfere with coexistence in predator-prey models.

There is much further work that could be done. In this paper, we did not specifically address the critical domain size problem, other than illustrate how our method can be applied for an example with alternating kernels (Example 2). It is not our purpose here to study how the critical domain size is influenced by the variation of different factors, but this could be an interesting avenue for future work, especially for the random model, which would build upon the work for integrodifference equations in Kot and Schaffer (1986) for symmetric dispersal kernels and Hardin et al. (1988a, b, 1990), Van Kirk and Lewis (1997, 1999), Latore et al. (1999) for more general dispersal kernels, including environmental heterogeneity both in space and in time.

## Notes

### Acknowledgments

The authors thank the anonymous referees for many insightful comments and references that helped strengthen the paper.

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