Abstract
Managing invasive species in rivers can be assisted by appropriate adjustment of flow rates. Using a partial differential equation (PDE) model representing an invasive population in a river, we investigate controlling the water discharge rate as a management strategy. Our goal is to see how controlling the water discharge rate will affect the invasive population, and more specifically how water discharges may force the invasive population downstream. We complete the analysis of a flow control problem, which seeks to minimize the invasive population upstream while minimizing the cost of this management. Using an optimality system, consisting of our population PDE, an adjoint PDE, and corresponding optimal control characterization, we illustrate some numerical simulations in which parameters are varied to determine how far upstream the invasive population reaches. We also change the river’s cross-sectional area to investigate its impact on the optimal control.
Highlights
The need to develop methods for managing invasive species in rivers is growing [1,2,3], and modelling can give insight into management strategies
partial differential equation (PDE), and corresponding optimal control characterization, we illustrate some numerical simulations in which parameters are varied to determine how far upstream the invasive population reaches
We have that a unique optimal control exists for our optimality system
Summary
The need to develop methods for managing invasive species in rivers is growing [1,2,3], and modelling can give insight into management strategies. With a more realistic model, Jacobsen et al concluded that longer stream length and lower flow rates increase the chance of a species persisting while higher flow streams must be even longer to maintain persistence [4]. They showed that flow velocity variability can produce persistence similar to the mean velocity [4]. Jin and Lewis again found that the periodic dispersal kernel and the weighted time-averaged dispersal kernel both yield the same spreading speeds. They found that prevalent winds were vital in determining the spreading speeds as well as the fragmentation into non-buoyant and buoyant pieces was the most important feature in determining spread [7]
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