By Yu Jin and Suzanne Lenhart
Flow regimes can change significantly over time and space and strongly impact all levels of river biodiversity, from the individual to the ecosystem. Invasive species in rivers—such as bighead and silver carp, as well as quagga and zebra mussels—continue to cause damage. Management of these species may include targeted adjustment of flow rates in rivers, based on recent research that examines the effects of river morphology and water flow on rivers’ ecological statuses. While many previous methodologies rely on habitat suitability models or oversimplification of the hydrodynamics, few studies have focused on the integration of ecological dynamics into water flow assessments.
Earlier work yielded a hybrid modeling approach that directly links river hydrology with stream population models [3]. The hybrid model’s hydrodynamic component is based on the water depth in a gradually varying river structure. The model derives the steady advective flow from this structure and relates it to flow features like water discharge, depth, velocity, cross-sectional area, bottom roughness, bottom slope, and gravitational acceleration. This approach facilitates both theoretical understanding and the generation of quantitative predictions, thus providing a way for scientists to analyze the effects of river fluctuations on population processes.
When a population spreads longitudinally in a one-dimensional (1D) river with spatial heterogeneities in habitat and temporal fluctuations in discharge, the resulting hydrodynamic population model is
\[N_t=-A_t(x,t)\frac{N}{A(x,t)}+\frac{1}{A(x,t)}\left(D(x,t)A(x,t)N_x\right)_x-\frac{Q(t)}{A(x,t)}N_x+rN\left(1- \frac{N}{K}\right)\tag1\]
\[N(0,t)=0 \text{ on } (0,T), x=0,\]
\[N_x(L,t)=0 \text{ on } (0,T), x=L,\]
\[N(x,0)=N_0(x) \text{ on } (0,L), t=0\]
in \(\Lambda =(0,L)\times (0,T)\). Here, \(N=N(x,t)\) represents population density in the river, \(N_t\) and \(N_x\) are derivatives of \(N\) with respect to \(t\) and \(x\) respectively, \(A(x,t)\) is the cross-sectional area of the river, \(D(x,t)\) is the diffusion coefficient that may include biodiffusion and flow-driven diffusion, \(r\) is the intrinsic growth rate, \(K\) is the carrying capacity, and \(Q(t)\) represents the water discharge rate. We assume that there is no population at the upstream boundary \((x=0)\), as the modeled invasive species is moving from downstream to upstream. However, we do assume a free-flow condition downstream at \(x=L\), meaning that an outside source is not increasing the population.
Figure 1. An invasion ratchet in a meandering river. The colors represent the population density at different locations in the river at different times. The parameters are as follows: flow period is \(T=365\) days, low flow season length is \(T_1=270\) days, low flow is \(1 \textrm{ m}^3/\textrm{s}\), high flow season length is \(T_2=95\) days, and high flow is \(20 \textrm{ m}^3/\textrm{s}\). Image courtesy of [3].
In a 1D river with a constant bottom slope or channels that alternate between riffles (shallow areas) and pools, the water discharge per unit width decreases the upstream spreading speed when it is low but increases this speed when it is high [3]. If the flow fluctuates between high and low, a longer high flow season makes it harder for the invasive population to spread upstream.
When investigating the interplay of spatial heterogeneity and temporal fluctuation, an interesting phenomenon can arise from hybrid hydrodynamic-biological models in 1D and two-dimensional (2D) rivers [3]. We call this an invasion ratchet, wherein a species can persist in a favorable patch during adverse times and traverse unfavorable patches in the upstream direction during opportune times. In the long term, this type of phenomenon can ensure a population’s upstream spread and consequential persistence in the entire river. Figure 1 depicts an invasion ratchet phenomenon in a 2D (longitudinal-lateral) meandering river. The population spreads upstream when the flow is low and retreats when the flow is high, though it spreads both upstream and downstream in the long run. Scientists have calculated the steady-state water flow with a numerical model called “River2D” [7]; one can then implement the population model—the 2D version of (1)—into River2D’s water flow.
An existing conjecture states that “a population can persist at any location in a homogeneous habitat if and only if it can invade upstream” [4]. This conjecture has been verified for some models with temporally-varying flows or spatially-homogeneous habitats — but not necessarily with both [2, 3]. If the time of low discharge in a pool-riffle river with fluctuating flows is not long enough for the population to advance from the riffle to the next upstream pool, the population is then washed back to its foothold in the downstream pool, where it remains until the next low discharge time [3]. The population therefore stalls in the river but cannot spread further upstream, which indicates that the aforementioned conjecture’s assumption of homogeneity in space or time is essential [4].
We have now established that the water discharge rate, which is higher during certain seasons and lower during others, affects the population level at various locations. In addition, the water discharge rates and lengths of different seasons also play important roles in the long-term population persistence or invasion in rivers. To investigate the control of water discharge rate as a potential management strategy, one can use optimal control of the water discharge rate \(Q(t)\) on model (1) to force the invasive population downstream [5]. The objective functional is to minimize the population’s integral (over time and space) with an upstream weight multiplier function to keep the invaders downstream, and with a small cost term to implement control. Thus, the term with the upstream weight dominates the objective. Positive upper and lower bounds on the controls (flow rates) are included in the optimization.
Figure 2. Location of the population in the river for the no control case (red), the constant control case with \(Q\) at its upper bound (blue), and the optimal control case \(Q^*(t)\) (magenta) over time. On the y-axis, 0 represents upstream and 10 represents downstream. Image courtesy of [5].
Figure 2 illustrates the distance of the population’s upstream motion with no control, constant control, and optimal control. Using a detection threshold of 0.5 for the population each time, we find the lowest location \(x\) (more upstream) with \(N(x,t)>0.5\) and plot that river location. Constant control takes the value of the controls’ upper bound. As expected, the population with no control moves further upstream than the population with constant control. Maintaining constant control at the upper bound resulted in the least movement upstream; the population moves further upstream with optimal control than with constant control. Note that optimal control is not at its upper bound for the whole time period due to costs in the objective functional, which balances costs with the desire to keep the population downstream. Our objective functional values for optimal control and constant control cases respectively yield 83 and 74 percent improvements over the value with no control.
A hybrid modeling approach allows us to investigate the impact of river morphology and flow patterns on a population’s spatiotemporal dynamics. Optimal control theory can generate strategies for regulating interested parameters to successfully manage an invasive population in a river. Applying the hybrid model and optimal control strategy to a particular invasive species could help scientists design reasonable management plans to control an invasion. For example, we could apply our model and methods to the invasive zebra mussel (Dreissena polymorpha), for which researchers have used a homogeneous version of model (1) [6] and a coupled continuous-discrete model [1] to investigate the species’ persistence and invasion in a temporally homogeneous habitat. It would be interesting to see how temporal and spatial heterogeneities affect mussel resilience and influence decisions about the invasion’s management.
References
[1] Huang, Q., Wang, H., & Lewis, M.A. (2017). A hybrid continuous/discrete-time model for invasion dynamics of zebra mussels in rivers. SIAM J. Appl. Math., 77, 854-880.
[2] Jin, Y., & Lewis, M.A. (2011). Seasonal influences on population spread and persistence in streams: Critical domain size. SIAM J. Appl. Math., 71, 1241-1262.
[3] Jin, Y., Hilker, F.M., Steffler, P.M., & Lewis, M.A. (2014). Seasonal invasion dynamics in a spatially heterogeneous river with fluctuating flows. Bull. Math. Biol., 76, 1522-1565.
[4] Lutscher, F., Nisbet, R.M., & Pachepsky, E. (2010). Population persistence in the face of advection. Theoret. Ecol., 3, 271-284.
[5] Pettit, R., & Lenhart, S. (2019). Optimal control of a PDE model of an invasive species in a river. Math., 7(10), 975.
[6] Speirs, D.C., & Gurney, W.S.C. (2001). Population persistence in rivers and estuaries. Ecol., 82, 1219-1237.
[7] Steffler, P.M., & Blackburn, J. (2002). River2D, two-dimensional depth averaged model of river hydrodynamics and fish habitat, introduction to depth averaged modeling and user’s manual. Alberta, Canada: University of Alberta.
Yu Jin is an associate professor in the Department of Mathematics at the University of Nebraska-Lincoln. Her research is devoted to mathematical biology, as well as dynamical systems and differential equations. Suzanne Lenhart is a Chancellor’s Professor in the Department of Mathematics at the University of Tennessee. Her research involves partial differential equations, ordinary differential equations, optimal control, and biological models.