New model examines the effects of toxicants on populations in polluted
rivers

Date:
January 11, 2022
Source:
Society for Industrial and Applied Mathematics
Summary:
A new mathematical model describes the interactions between
a population and a toxicant in a river environment, enabling
researchers to study how the way in which a pollutant moves
through a river affects the wellbeing and distribution of the
river's inhabitants.



FULL STORY ==========================================================================
When designing environmental policies to limit the damage of river
pollution, it is paramount to assess the specific risks that particular pollutants pose to different species. However, rigorously testing the
effects of toxicants -- like insecticides, plastic debris, pathogens,
and chemicals -- on entire groups of organisms without severely damaging
their whole ecosystems is simply not feasible. Mathematical modeling can provide a flexible way to assess toxicants' impact on river populations
without endangering the environment.


==========================================================================
In a paper that published today in the SIAM Journal on Applied
Mathematics, Peng Zhou (Shanghai Normal University) and Qihua Huang
(Southwest University, Chongqing) develop a model that describes
the interactions between a population and a toxicant in an advective environment -- a setting in which a fluid tends to transport material in
one direction, like a river. Such a model can help scientists study how
the way in which a pollutant moves through a river affects the wellbeing
and distribution of the river's inhabitants.

Much of the previous experimental research on the ecological risks of
toxicants has been performed on individual organisms in controlled
laboratory conditions over a fairly short-term basis. The design of environmental management strategies, however, requires an understanding
of toxicants' impact on the health of entireexposed natural populations
in the long term. Fortunately, there is an intermediary. "Mathematical
models play a crucial role in translating individual responses to population-level impacts," Huang said.

The existing models that describe the way in which toxicants affect
population dynamics generally ignore many of the properties of
water bodies. But in doing so, they are missing a big piece of the
puzzle. "In reality, numerous hydrological and physical characteristics
of water bodies can have a substantial impact on the concentration
and distribution of a toxicant," Huang said. "[For example], once a
toxicant is released into a river, several dispersal mechanisms --
such as diffusion and transport -- are present that may aid in the
spread of the toxicant." Similarly, the models that mathematicians
often use to portray the transport of pollutants through a river also
do not include all of the necessary components for this study. These
are reaction-advection-diffusion equation models, whose solutions can
show how pollutants distribute and vary under different influences like
changes in the rate of water flow. While such models enable researchers
to predict the evolution of toxicant concentrations and assess their
impact on the environment, they do not consider toxicant influence on
the dynamics of affected populations. Zhou and Huang thus expanded upon
this type of model, adding new elements that allowed them to explore
the interaction between a toxicant and a population in a polluted river.

The authors' model consists of tworeaction-diffusion-advection equations
-- one that governs the population's dispersal and growth under the
toxicant's influence, and another that describes the processes that the toxicant experiences. "As far as we know, our model represents the first
effort to model the population-toxicant interactions in an advective environment by using reaction-diffusion-advection equations," Zhou
said. "This new model could potentially open a [novel] line of research."
The model allows Zhou and Huang to tweak different factors and investigate
the resulting changes to the ecosystem. They tried altering the river's
flow speed and the advection rate -- i.e., the rate at which the toxicant
or organisms are carried downstream -- and observing these parameters' influence on the population persistence and distribution of both the
population and toxicant.

These theoretical results can provide insights that could help inform ecological policies when taken in concert with other information.

One scenario that the researchers studied involved a toxicant that had a
much slower advection rate than the population and thus was not washed
away as easily. The model showed that, intuitively, the population
density decreases with increasing water flow because more individuals
are carried downstream and out of the river area in question. However,
the concentration of the toxicant increases with the increasing flow
speed because it can resist the downstream current and the organisms
are often swept away before they can uptake it.

In the opposite case, the toxicant has a faster advection rate
and is therefore much more sensitive to water flow speed than
the population. Increasing the water flow then reducesthe toxicant concentration by sweeping the pollutants away. For a medium flow speed,
the highest population density occurs downstream because the water
flow plays a trade-off role; it transports more toxicants away but also
carries more individuals downstream.

This demonstrates that a higher sensitivity of a pollutant to water
flow is generally more advantageous to population persistence. "In the
absence of toxicants, it is generally known that the higher the flow
speed, the more individuals will be washed out of the river," Zhou
said. "However, our findings suggest that, for a given toxicant level, population abundance may increaseas flow rate increases." By providing
this model with the parameters for certain species and pollutants, one
may be able to determine criteria regarding the water quality that is
necessary to maintain aquatic life. This outcome could ultimately aid
in the development of policy guidelines surrounding the target species
and toxicants.

"The findings here offer the basis for effective decision-making tools
for water and environment managers," Huang said. Managers could connect
the results from the model with other factors, such as what may happen
to the pollutant after it washes downstream.

Further extensions to Zhou and Huang's new model could make it even more applicable to real river ecosystems -- for example, by allowing the flow velocity and release of toxicants to vary over time, or accounting for
the different ways in which separate species may respond to the same
pollutant.

This mathematical model's capability to find the population-level effects
of toxicants might play a critical part in the accurate assessment of pollutants' risk to rivers and their inhabitants.

========================================================================== Story Source: Materials provided by Society_for_Industrial_and_Applied_Mathematics. Original written by
Jillian Kunze. Note: Content may be edited for style and length.


========================================================================== Journal Reference:
1. Peng Zhou, Qihua Huang. A Spatiotemporal Model for the Effects of
Toxicants on Populations in a Polluted River. SIAM Journal on
Applied Mathematics, 2022; 82 (1): 95 DOI: 10.1137/21M1405629 ==========================================================================

Link to news story: https://www.sciencedaily.com/releases/2022/01/220111153608.htm

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