# A Simple Stochastic Process Model for River Environmental Assessment Under Uncertainty

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## Abstract

We consider a new simple stochastic single-species population dynamics model for understanding the flow-regulated benthic algae bloom in uncertain river environment: an engineering problem. The population dynamics are subject to regime-switching flow conditions such that the population is effectively removed in a high-flow regime while it is not removed at all in a low-flow regime. A focus in this paper is robust and mathematically rigorous statistical evaluation of the disutility by the algae bloom under model uncertainty. We show that the evaluation is achieved if the optimality equation derived from a dynamic programming principle is solved, which is a coupled system of non-linear and non-local degenerate elliptic equations having a possibly discontinuous coefficient. We show that the system is solvable in continuous viscosity and asymptotic senses. We also show that its solutions can be approximated numerically by a convergent finite difference scheme with a demonstrative example.

## Keywords

Regime-switching stochastic process Model uncertainty Environmental problem Viscosity solution## 1 Introduction

This paper focuses on a population dynamics modeling of nuisance benthic algae on riverbed under uncertain environment: a common environmental problem encountered in many rivers where human regulates the flow regimes [1]. Blooms of nuisance benthic algae and macrophytes, such as *Cladophora glomerata* and *Egeria densa*, in inland water bodies are seriously affecting aquatic ecosystems [2, 3]. Such environmental problems are especially severe in dam-downstream rivers where the flow regimes are often regulated to be low, with which the nuisance algae can dominate the others [4, 5].

It has been found that the benthic algae are effectively removed when they are exposed to a sufficiently high flow discharge containing sediment particles [6]. Supplying sediment into a river environment can be achieved through transporting earth and soils from the other sites, as recently considered in Yoshioka et al. [7] focusing on a case study in Japan with a high-dimensional stochastic control model.

Assume that we could find a way to supply the sediment into a river environment where the nuisance algae are blooming. Then, a central issue is to what extent the algae bloom can be suppressed in the given environment. Hydrological studies imply that river flows are inherently stochastic and can be effectively described using a Markov-chain [8]. In the simplest case, we can classify river flow regimes into the two regimes: a high-flow regime where the nuisance algae can be effectively removed from the riverbed and a low-flow regime where they are not removed from the riverbed at all. In this view, the algae population dynamics can be considered as a stochastic dynamical system subject to a two-state regime-switching noise. To the best of our knowledge, such an attempt has been least explored despite its high engineering importance.

We approach this issue both mathematically and numerically. We formulate the algae population dynamics as a system of piecewise-deterministic system subject to a Markovian regime-switching noise [9] representing a dynamic river flow having high- and low-flow regimes. This is a system of stochastic differential equations (SDEs, Øksendal and Sulem [10]) based on a logistic model subject to the detachment during the high-flow regime [7] but with a simplification for better tractability. The model incorporates our own experimental evidence that a sudden detachment of the algae occurs when the flow regime switches from the high-flow to the low-flow. This finding introduces a non-locality into the model.

Our focus is not only on the population dynamics themselves, but also on statistical evaluation of the dynamics that can also be important in engineering applications. Namely, another focus is the evaluation of statistical indices such as a disutility caused by the population, which are given by conditional expectations of quantities related to the population. Unfortunately, it is usually difficult to accurately identify model parameters in the natural environment due to technical difficulties and poor data availability. In such cases, we must operate a model under the assumption that it is incomplete and thus uncertain (or equivalently, ambiguous). We overcome this issue by employing the concept of multiplier robust control [11], which allows us to analyze SDEs having uncertainty and further to statistically evaluate their dynamics in a worst-case robust manner. This methodology originates from economics and has been employed in finance [12] and insurance [13], but less frequently in environment and ecology [14]. With this formulation, we demonstrate that the stochastic dynamics having model uncertainty can be handled mathematically rigorously as well as efficiently.

We show that the robust evaluation of a statistical index related to the population dynamics ultimately reduces to solving a system of non-linear and non-local degenerate elliptic equations: the optimality equation having a possibly discontinuous source term. This is the governing equation of the statistical index under the worst-case. Our goal is therefore to solve the equation in some way. We show that solutions to the optimality equation are characterized in a viscosity sense [15], and that it admits a continuous viscosity solution by a comparison theorem [16]. We present an analytical asymptotic estimate of the solution as well. We finally provide a demonstrative computational example with a convergent finite difference scheme [5, 14] to show the validity of the asymptotic estimate and to deeper comprehend the behavior of the model.

## 2 Mathematical Model

### 2.1 System Dynamics

The stochastic nature of the detachment has been justified from our experimental results. We found that the amount of sudden detachment is different among the experimental runs even under the same experimental setting (sediment supply and water flow). This is considered due to inherently probabilistic nature of the sediment supply and microscopic difference of biological conditions (length, density, rock shape, etc.,) of the algae population. The same would be true in real river environment.

### 2.2 Performance Index

### 2.3 A Model with Uncertainty

## 3 Mathematical Analysis

### 3.1 Viscosity Solution

Boundedness and continuity of the value function are analyzed. We firstly prove unique solvability of the system (11).

### Proposition 1.

*The system (**11**) admits a unique strong solution such that* \( 0 \le X_{t} \le 1 \) \( (t \ge 0) \).

### Proof:

A similar contradiction argument to that in the proof of Theorem 2.2 of Lungu and Øksendal [17] applies in our case. We can get unique existence of the system having coefficients extended to be Lipschitz continuous over \( {\mathbf{\mathbb{R}}} \), by Theorem 2.1 of Yin and Zhu [9]. With a contradiction argument [17], we obtain that the strong solution to this modified problem is bounded in \( \left[ {0,1} \right] \).□

By Proposition 1, we get a continuity result of the value function, with which an appropriate definition of viscosity solutions to the optimality equation (19)–(20) is found.

### Proposition 2.

Assume that \( f \) is Hölder continuous in \( \left[ {0,1} \right] \). Then, we get \( \Phi _{i} \in C\left[ {0,1} \right] \), \( (i = 0,1) \).

### Proof:

Combine the strong solution property and the boundedness result in Proposition 1 with the Hölder continuity of \( f \).□

Now, we define viscosity solutions. Set the space of upper-semicontinuous (resp., lower-semicontinuous) functions in \( \left[ {0,1} \right] \) as \( USC\left[ {0,1} \right] \) (resp., \( LSC\left[ {0,1} \right] \)).

### Definition 1.

*A pair* \( \Psi _{0} ,\Psi _{1} \in USC\left[ {0,1} \right] \) *is a viscosity sub*-*solution if for all* \( x_{0} \in \left[ {0,1} \right] \) *and* \( i_{0} = 0,1 \)*, and for all* \( \varphi_{0} ,\varphi_{1} \in C^{1} \left[ {0,1} \right] \) *s.t.* \( \varphi_{i} -\Psi _{i} \) *is locally minimized at* \( x = x_{0} \) *and* \( i = i_{0} \) *with* \( \varphi_{{i_{0} }} \left( {x_{0} } \right) =\Psi _{{i_{0} }} \left( {x_{0} } \right) \)*, the following hold ((**23**) for* \( i_{0} = 0 \)*, (**24**) for* \( i_{0} = 1 \)*):*

*A pair*\( \Psi _{0} ,\Psi _{1} \in LSC\left[ {0,1} \right] \)

*is a viscosity super*-

*solution if for all*\( x_{0} \in \left[ {0,1} \right] \)

*and*\( i_{0} = 0,1 \)

*, and for all*\( \varphi_{0} ,\varphi_{1} \in C^{1} \left[ {0,1} \right] \)

*s.t.*\( \varphi_{i} - \varPsi_{i} \)

*is locally maximized at*\( x = x_{0} \)

*and*\( i = i_{0} \)

*with*\( \varphi_{{i_{0} }} \left( {x_{0} } \right) =\Psi _{{i_{0} }} \left( {x_{0} } \right) \)

*, the following hold ((*

*25*

*) for*\( i_{0} = 0 \)

*, (*

*26*

*) for*\( i_{0} = 1 \)

*):*

*A pair* \( \Psi _{0} ,\Psi _{1} \in C\left[ {0,1} \right] \) *is a viscosity solution if it is a viscosity sub*-*solution as well as a viscosity super*-*solution.*

### Proposition 3.

*Assume that* \( f \) *is Hölder continuous in* \( \left[ {0,1} \right] \) *. Then, the value function is a viscosity solution.*

### Proof:

Apply the Dynkin’s formula and the dominated convergence theorem.□

### Proposition 4.

*For any viscosity sub*-*solution* \( u \) *and a viscosity super*-*solution* \( v \), \( v \ge u \) *in* \( \Omega \)*. Moreover, the optimality equation* *(*19*)*–*(*20*) admits at most one viscosity solution.*

### Proof:

Apply a contradiction argument with the help of the monotonicity of the nonlinear terms. [Proof of Proposition 3.3 in 14] and [Proof of Theorem 11.4 in 16]. □

Notice that the uniqueness result in Proposition 4 holds true for both continuous and discontinuous \( f \). A consequence of Propositions 3 and 4 is the next theorem.

### 3.2 Asymptotic Solution

The optimality equation is uniquely solvable, but its exact solution cannot be found analytically. Instead, we construct an asymptotic solution that close to the solution for small \( x \). The asymptotic solution thus applies to the situation where the population is sufficiently small, which would be encountered during a high-flow regime. The next proposition can be checked by a direct calculation. The asymptotic worst-case uncertainties can also be calculated using this proposition.

### Proposition 5.

*Assume* \( \Phi _{i} \in C^{2} \left[ {0,\varepsilon } \right) \) \( (i = 0,1) \) *with* \( 0 < \varepsilon < 1 \)*. Assume that for small* \( 0 < x < \varepsilon \), \( f\left( x \right) = x^{m} + O\left( {x^{m + \upsilon } } \right) \) *and* \( D\left( x \right) = d + O\left( {x^{\upsilon } } \right) \), \( d \ge 0 \), \( m,\upsilon > 0 \), \( \delta > 0 \) *such that*

*Then, we have the following asymptotic expansions for small*\( x > 0 \):

*with the positive constants*

## 4 Numerical Computation

We numerically discretize the optimality equation (19)–(20) because it is not analytically solvable and the asymptotic solution presented in the previous section can be utilized only under limited conditions. The employed numerical method here is the finite difference scheme with the Newton iteration [5], which can handle the decay and first-order differential terms of the degenerate elliptic differential equations. The non-linear and non-local terms are handled with the interpolation technique of [14]. This scheme is a version of the non-standard finite difference scheme based on local exact solutions to linearized problems [18], with which monotone, stable, and consistent discretization is established. It is thus convergent in a viscosity sense [19]. The non-local term is linearized at each iteration step to enhance computational stability.

## 5 Conclusions

A simple stochastic model for the algae bloom under uncertain river environment was analyzed both mathematically and numerically. Our framework would provide a mathematically rigorous and computationally feasible framework for resolving the engineering problem. We believe that the framework can also contribute to achieving Sustainable Development Goals (SDGs) related to water environmental and ecological issues.

A future topic would be computing the probability density functions and analyzing stability of the population dynamics. Such analysis can be carried out using the derived or computed worst-case uncertainties as an input. Another topic is a partial observation modeling based on the presented model. For example, one can often directly measure water flows relatively easily (using machines), but only indirectly the algae population.

## Notes

### Acknowledgements

JSPS Research Grant 18K01714 and 19H03073, Kurita Water and Environment Foundation Grant No. 19B018, a grant for ecological survey of a life history of the landlocked ayu *Plecoglossus altivelis altivelis* from the Ministry of Land, Infrastructure, Transport and Tourism of Japan, and a research grant for young researchers in Shimane University support this research.

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