How Mathematical Model is used to see species interactions to ensure stability of the ecosystem?
By : Suphia Zhang
The earth, home to millions of people and millions of species and the home to many homes, is one of the most important planets. Dinosaurs and cavemen once roamed the earth but as time went on they vanished and some survived and evolved into animals that we see now, even though some may have gone extinct due to the global issue. Many scientists and other major people have tried to fix the problem but due to the constant disruption of the change in the world, as progress also takes time to happen it's complicated to predict the future of the world as well as keeping up with the issue.
Species interactions is one of the main importance for maintaining the ecosystem as the ecosystem faces disturbances in the world such as global warming, climate change, pollution and so much more. Understanding the species interaction can allow predictions to be made such as the behavior of a species, species evolution, the impact on the future and much more. These disturbances are caused by human action as well as the change in the animal behavior, therefore when trying to solve the global crisis it is not just about what we humans should do such as recycling more often and ect we also have to care about other things, species interaction.
Why is species interaction so important and how does that impact the ecosystem?
That is one of the questions that many ecologists and others wonder as they see an impact on the ecosystem that species interaction has over the time period. How does species interact with the change that is happening constantly, what do they do, how will they act? The multiple question that has been going on, has no right answer as any answer can cause the issue to become worse or better depending on the issue. Issues such as deforestation, climate change, global warming, water and air pollution and so much more can be fixed but more issues will occur due to the change of the species' behavior.
Species interaction is also known as species thoughts and behavior to the change of their surroundings whether it is against their own species due to change, or other species . This is very important as knowing one species' interaction with another species can change this will cause an impact to biodiversity, ecological balance, ecosystem service and much more. Understanding species interaction will allow information to be gained by ecologists that will see a pattern that will help them maintain ecosystem stability, biodiversity, human well-being and others. With this, ecologists will be able to maintain and predict the potential impacts of climate change and environmental changes on biodiversity and the ecosystem, ensuring a much better way for ecosystem stability. With the ability to be able to predict the impact from the global crisis, ecologists are able to help stabilize the ecosystem as well as be able to improve it. Floods and storms are natural causes that happen but due to global warming and other issue these cause become worse and it is hard to help get it back to normal and so many species are impacted by it causing them to do things differently and unbalancing the system.
(Flood in Pakistan in 2023 from Climate Change.)
A very common question that would appear would be what would help predict species interaction that will allow prediction to ensure the ecosystem stability?
The answer to that is “Mathematical model”.
What is a mathematical model?
A mathematical model is a representation of a physical model of a mathematical concept or a representation of reality. Mathematical models can essentially represent anything either about the biological world, natural world, technological, mathematical, anything that can be described as mathematical expression. Mathematical models therefore can represent a model of the oceans pattern, the soil pattern, animal’s population, as well as many other patterns that are an impact to the ecosystem.
( Ocean Wave Math Model)
Mathematical models can be used in many different ways and for different reasons as it depends on the purpose of use, the data given, the system of the study, the formula and much more. There are many different types of mathematical models as math can be represented in different units and different dimensions etc, however the most commonly used mathematical models are equations and graphs. Mathematical models are a really powerful tool for ecologist to predict species interaction as they use that for different reasons such as :
Predicting how populations of two or more species would change over time.
Mathematical models can use population to predict the birth rate and death rate allowing ecologists to know when a species could possibly go extinct and how long they have to save that species, the model will also allow ecologists to test multiple hypotheses that will allow them to identify a solution. (Bison Population)
Predicting how climate change will alter the species interaction such as their behavior towards a new habit or a new surrounding.
For example, “Darwin’s moth.” Darwin’s moth was a discovery that happened due to the air pollution of many factories and because of the mass air pollution trees surrounding it changed colors and because of that, pepper moths that hid within the trees to blend in from the predators (birds) were soon unable to do so. Many moths were eaten due to the fact they were unable to blend in and so the one that did survive and gave birth gave birth to black moths however the population was very low during that time. This event happened in the 1800 while the mathematical model was invented in 1900 therefore because if the mathematical model was created in the 1800 or the 1700 ecologist or like Darwin could have been able to predict the other natural causes as well, not just the extinction of white pepper moth but other things too.
Predicting two species interaction as well as how global issues impact that interaction will also allow ecologists to ensure stability of the ecosystem with the use of the Lotka - Volterra model.
The Lotka - Volterra model is a type of mathematical model that is considered a function and numerical response which allows ecologists to predict the population of the predator and the prey by the interaction they have. Understanding that can also allow ecologists to know how to limit the population as overpopulation of some animals can cause more issues to the ecosystem. For example a experiment was conducted throughout from 1845 to 1935 where they study the interaction between snow hare and lynx population. As the figure is shown below ecologists could see a pattern that throughout time as the lynx population increases the hare population decreases but it doesn’t disappear as when it reaches a limit it inverts.
Mathematics have created functions to represent the growth and decay of the population allowing a graph to be created which is known as a model. In this case the population of snow hare is:
dH/dt = a1H(t)
dH/dt is the slope or the rate of the hare population over time
a1 is know as the growth rate of the hare
H(t) is the function for hare
However due to the the Lync their is a decline in the population creating a final function of:
dH/dt = a1H(t) - a2H(t)L(t)
a2 is a coefficient of the probability of predation per unit time and per individual when a lynx and a hare encounter which is symbolized as H(t)L(t)
The population of lynx is:
dL/dt = b2H(t)L(t)
dL/dt is the slope or the rate of the lynx population over time
b2 is the rate of the conversion of hares into the lynx population because as the lynx devour the hare more the lynx population grows
In this rate its not just L(t) because the Lynx depend on the hare to grow and so H(t) is also added on
However eventually the lynx will soon outgrow the hare which will cause a reverse malthusian growth meaning the population has outgrown their food supply causing a starvation and decline which creates a final function for the rate of the lynx with the impact included:
dL/dt = -b1L(t) + b2H(t)L(t)
-b1 represents the rate of the lynx as it suffers from the absence of the hare.
The snow hare in this case would also suffer from overpopulation but it is not negatively impacted as the lynx, as overpopulated snow hare can cause impact vegetation these hare also eat meat and so there isn't really a limit to their food supply but because of this it causes an unbalance to the ecosystem. Therefore with the two functions mathematics graph them and are able to predict what the behavior is and keep at a certain level as to keep the hare and lynx from overpopulation or underpopulation.
Conclusion:
Mathematical models are one of the main important tools for ecologists, scientists, engineers, tech, and so many other fields to test hypotheses and predict certain events. In this research we are coming to understand how simple math models can be used for such a big issue such as helping the ecosystem stay balanced through the understanding of species interaction. This idea of mathematical model was invented in the 1900’s as a way to allow ecologists and other to be able to maintain the limitation of certain species as well as others species that can cause an unbalanced to the ecosystem, therefore with the model we can predict the population of two or more species as they impact each other allowing us to maintain a balance in the ecosystem as their are many other issue disturbing the ecosystem. Although yes we can help balance the ecosystem with this model there are many implications that aren’t counted for.
Implications such as human action of hunting, over eating, pollution, carbon increase, war and so much more. In the future 10, 20, 30 years later these problems may get worse as more and more technology advances which cause more issues, wars and their nuclear bombs causing destruction to many animals' homes changing their behavior. The model can predict the future population of many species based on the current environment and issue but it can't predict the changes of the future as we humans don't know for ourselves either.
Work Cited
Fishwick, Paul. “Lotka-Volterra Model.” Lotka-Volterra Model - an Overview | ScienceDirect Topics, 2008, www.sciencedirect.com/topics/earth-and-planetary-sciences/lotka-volterra-model#:~:text=The%20Lotka%E2%80%93Volterra%20model%20assumes,of%20prey%20in%20the%20environment
Mahaffy, Joseph M. “Lotka-Volterra Models Predator-Prey.” Logisticde, 2010, jmahaffy.sdsu.edu/courses/f09/math636/lectures/lotka/qualde2.html
Raj, Vishal. “Study of Two Species Interactions Using Lotka Volterra Model.” Study of Two Species Interactions Using Lotka Volterra Model, 31 Mar. 2012, complexnt.blogspot.com/2012/03/study-of-two-species-interactions-using.html
The Editors of Encyclopedia Britannica. “Mathematical Model.” Encyclopædia Britannica, Encyclopædia Britannica, inc., 4 June 2024, www.britannica.com/science/mathematical-model