diff --git a/00041_Trophic_functions/en_article_proofreading.md b/00041_Trophic_functions/en_article_proofreading.md
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--- a/00041_Trophic_functions/en_article_proofreading.md
+++ b/00041_Trophic_functions/en_article_proofreading.md
@@ -3,147 +3,92 @@ keywords: function, rational fractional function, asymptote
is_finished: False
---
-# Trophic functions in predator-prey models
-
-Mathematical models play an important role in the study of nature. These
-models open the possbilities to predict future development, but they also have
-other important roles.
-
-The use of ecological models is sometimes referred to as physical approach
-in ecology because they study the ecosystem in terms of the evolution of populations and
-using mathematical methods originally
-derived to solve physical problems. The outputs of the models include
-the following information.
-
-* **Predictions** Ability to work with mathematical models of ecosystems
- allows to predict future ecosystem development. This may include the evolution of population in static
- environment as well as an evolution in an environment in which the
- parameters are changing. Knowledge of the model will then reveals how the parameters
- change the ecosystem.
+# Trophic Functions in Predator-Prey Models
+
+Mathematical models play an important role in the study of nature. These models not only make it possible to predict future developments, but they also serve several other important purposes.
+
+
+The use of ecological models is sometimes referred to as a physical approach in ecology, as these models examine ecosystems in terms of population dynamics while employing mathematical methods originally developed for solving physical problems. The outputs of such models can provide the following types of insights:
+
+* **Predictions**
+Working with mathematical models of ecosystems enables scientists to predict future developments. This may include the evolution of populations in a stable environment as well as evolution in environments with varying parameters. Understanding the model will then reveal how the change in parameters
+affects the ecosystem.
+
* **Understanding the principles** Mathematical models allow ecologists and
scientists to examine the interactions between different components of ecosystems and
- understand the dynamics of these systems. They help to identify
- factors that influence the structure and function of these ecosystems.
-* **Optimising decision making** Mathematical modelling of ecosystems can
+ to better understand the dynamics of these systems. Such models help to identify the key
+ factors that influence the ecosystem structure and function.
+* **Optimizsing decision making** Mathematical modelling of ecosystems
can be used to optimize decision-making in areas such as
- biodiversity conservation or forest and fisheries management.
- It helps to identify the best strategies for
- to achieve desired objectives.
-
- One of the fundamental relationships in ecosystems is the relationship between _predator and
- prey_. This relationship can be the only interaction in an ecosystem or
-it can be complemented by other interactions. The importance of predator-prey models will be illustrated with the following historically significant examples.
-
-## Lotka-Volterra model
-
-In 1926, one of the first predator-prey models was published by the Italian
-mathematician Vito Voterra. The motivation to build this model was
-the fact that during the fishing restrictions of World War I in
-the percentage of predatory fish in the catch increased.
-The marine biologist Umberto D'Ancona noticed this phenomenon and was not able to find an explanation. He even expected the opposite: with the restrictions on fishing
-he expected an increase in the percentage of smaller fish species that are
-food for predators. D'Ancona introduced this problem to his father-in-law
-Volterra. Volterra's model explains this behaviour as
-as a result of the simple idea of predator-prey interaction.
-
-The model contains two equations. The first equation
-describing the prey population contains the assumption that the population
-grows naturally, but the growth is slowed down
-by predation. More predators lead to a slower growth of the prey. Too many predators can even lead to the fact that the size of the prey population
-will decline and the prey will die out. The second equation (for predator population)
-contains the assumption that without the presence of prey, the predator population dies out.
-However, the more prey is available to predators, the more likely this extinction
-turns into an increase in the predator population.
-
-In the system described above, cycles occur naturally. Enough prey
-allows the predator population to increase. Many predators then act on the prey population
-negatively and the prey population begins to die off. This
-extinction results in a lack of food for the predators and
-and the predators also begin to die out. Over time, the predator population
-is reduced to the point where the prey feels the presence of the predator
-little. Therefore, the prey population can grow again and multiply to its original state. However, the increase in prey population
-allows the predator population to grow again, closing the cycle. These
-periodic changes in the size of both populations can be seen in the records of fur buying
-of snowshoe hare and lynx furs in the Hudson Bay area.
-
-Volterra's model explains not only the origin of the cycles, but also that
-by decreasing the hunting rate, the equilibrium position around which the predator and
-prey oscillate shifts in favour of the predator and not the prey. This
-phenomenon observed by D'Ancona is called the _Volterra effect_.
+ biodiversity conservation, or forest and fisheries management.
+ It helps identify the effective strategies
+ to achieve specific objectives.
+ One of the fundamental relationships in ecosystems is that between _predator and
+ prey_. This relationship may be the only form of interaction in a given ecosystem or
+may be complemented by other types of interactions. The importance of predator-prey models will be illustrated with the following historically significant examples.
+
+
+
+## Lotka-Volterra Model
+
+
+In 1926, one of the first predator–prey models was published by the Italian mathematician Vito Volterra. The motivation behind this model was the observation that, during fishing restrictions imposed in World War I, the percentage of predatory fish in the catch increased.
+
+The marine biologist Umberto D’Ancona noticed this phenomenon but was unable to explain it. He had actually expected the opposite: with fishing limited, he anticipated an increase in the proportion of smaller fish species that serve as food for predators. D’Ancona introduced this problem to his father-in-law, Volterra, who explained the observation using a simple predator–prey interaction model.
+
+
+The model consists of two equations. The first, describing the prey population, assumes that the population grows naturally but that this growth is slowed down by predation. An increase in predator numbers leads to slower growth of the prey population. If there are too many predators, the prey population may even decline and even die out.
+The second equation, describing the predator population, assumes that in the absence of prey, the predators would eventually die out. However, when prey is abundant, the predator population is more likely to increase.
+
+
+In the system described above, cyclical population dynamics occur naturally. A sufficient prey population allows predators to thrive. As the predator population grows, it negatively affects the prey population, which then begins to decline. This leads to a food shortage for the predators, causing their numbers to drop as well. Over time, the predator population decreases to a level where the prey barely senses their presence. The prey population can then recover and grow toward its original size. The increase in prey population allows the predator population to grow again, closing the cycle.
+
+These periodic changes in the sizes of both populations are reflected in historical records of snowshoe hare and lynx fur purchases in the Hudson Bay area.
+
+
+Volterra’s model not only explains the origin of these cycles but also predicts that reducing the rate of hunting shifts the equilibrium point—around which both predator and prey populations oscillate—in favor of the predators rather than the prey. This phenomenon, observed by D’Ancona, is known as the _Volterra effect_.
The same model as Volterra's has been proposed in 1910 by the American mathematician
-Alfred J. Lotka. For this reason is the model called the Lotka-Volterra model.
+Alfred J. Lotka. For this reason, the model is known today as the Lotka-Volterra model.
+
+
-
+## Spruce Budworm Model
-## Spruce budworm model
-Similar periodic fluctuations as in the Lotka-Volterra model
-can also be observed in Canadian forests. Here, after approximately 30 to 40 years
-mass outbreak of the spruce budworm (_Choristoneura
-fumiferana_) occurs. The population of this butterfly is relatively small most of the time, but some
-years it increases by a factor of thousands and its caterpillars can kill up to 80% of the trees in
-of the forest, virtually destroying it. One of the last mass occurrences was
-since 2006 in Quebec. Here, by 2019, about 9.6 million
-hectares of forest [^1], more than the size of Hungary.
+Similar periodic fluctuations to those described by the Lotka–Volterra model can also be observed in Canadian forests. Approximately every 30 to 40 years, a mass outbreak of the spruce budworm (_Choristoneura Fumiferana_) occurs. For most of the time, the population of this butterfly remains relatively small, but in certain years it increases by a factor of thousands. During such outbreaks, its caterpillars can kill up to 80% of trees in the forest, virtually destroying it. One of the most recent mass outbreaks began in 2006 in Quebec. By 2019, about 9.6 million hectares of forest had been affected[^1], which is more than the size of Hungary.
See [^1]: source at <https://www.nrcan.gc.ca>.
-In 1978, scientists D. Ludwig, D. D. Jones and C. S. Holling proposed a model that
-was not only able to model the evolution of the spruce budworm population, but helped to clarify
-the reasons why the described outbreaks occur. The reason is
-predation. In this case, it is the consumption of caterpillars by birds. Birds
-served as a limiting factor in nature for caterpillar numbers, but only
-up to a certain limit. When the forest grows large enough, it provides enough
-food for the caterpillar population. The caterpillar population then grows to
-to such an extent that the birds reach saturation in their food consumption and
-they are no more able to reduce the caterpillar population. Thus, the role of birds as
-predators becomes less important and the caterpillar population can
-multiply rapidly and then devastate the forest.
-
-In this case, predation is important to limit the population
-caterpillars. Because birds as predators have a much slower
-reproductive cycle than the insect, their population can be considered
-constant. Through saturation, birds can then limit the growth rate
-to a limited extent. However, this limitation from a certain size
-the spruce budworm population is no longer sufficient, and it becomes uncontrollably overpopulated.
-
-## Island fox model
-
-The island fox (_Urocyon littoralis_) is a unique species,
-endemic only to the islands around California. It is as large as a
-cat and trusting due to the absence of natural enemies. As a species, it is
-sensitive and vulnerable due to low genetic variation and low
-resistance to diseases introduced from the mainland. It is one of the smallest
-canids. Unlike other canids, however, it can climb trees.
-
-
-As a result of human activity, the population of the island fox was near to extinction at the end of the millennium. On San Miguel Island, the population has declined from
-450 adults in 1994 to 15 in 1999[^2]. A similar situation
-has been experienced also on the other islands, each of which is inhabited by a separate
-subspecies of the island fox. The cause of the mortality was a chain of events:
-The production of the insecticide DDT in the 1940s resulted in
-the extinction of the bald eagle (_Haliaeetus leucocephalus_) which has been
-replaced by the golden eagle (_Aquila chrysaetos_). A predator feeding
-fish-eating predator was replaced on the island by a mammal-preferring predator. This was
-fatal to the island fox. The foxes, formerly apex predators, suddenly became
-prey, and by the turn of the millennium they were close to extinction. Moreover, unlike the Lotka-Volterra model, there was no hope of a return
-foxes back to their original numbers through oscillations, because the eagles also had
-alternative food in the form of wild pigs and skunks.
+
+In 1978, scientists D. Ludwig, D. D. Jones, and C. S. Holling proposed a model that not only described the dynamics of the spruce budworm population but also helped clarify the underlying cause of these outbreaks: predation. Specifically, the consumption of budworm caterpillars by birds.
+
+Birds naturally serve as a limiting factor on caterpillar numbers, but only up to a certain limit. As forests grow larger, they provide more food for the caterpillar population. The population then grows to such an extent that the birds reach their feeding limit and can no longer reduce the caterpillar population. At this stage, the role of birds as predators diminishes, and the caterpillar population can grow rapidly, eventually devastating the forest.
+
+
+In this case, predation is key to limiting the spruce budworm population. However, since birds have a much slower reproductive cycle than insects, their population can be considered constant. Through saturation, their ability to limit the budworm growth rate is thus restricted. Once the budworm population exceeds a certain threshold, this control mechanism fails, leading to uncontrollable overpopulation.
+
+## Island Fox Model
+
+
+
+The island fox (_Urocyon littoralis_) is a unique species, endemic only to the Channel Islands off the coast of California. It is about the size of a domestic cat and is unusually trusting due to the historical absence of natural predators. As a species, it is particularly sensitive and vulnerable due to low genetic variation and limited resistance to diseases introduced from the mainland. It is also one of the smallest canids, and unlike most others, it can climb trees.
+
+As a result of human activity, the population of the island fox was near to extinction at the end of the millennium. On San Miguel Island, the population declined from
+450 adults in 1994 to just 15 in 1999[^2]. A similar situation occurred also on the other islands, each of which is home to a distinct subspecies of the island fox.
+
+The cause of the decline was a chain of events. In the 1940s, the production of the insecticide DDT led to the local extinction of the bald eagle (_Haliaeetus leucocephalus_), a fish-eating predator. On the island, it was eventually replaced by the golden eagle (_Aquila chrysaetos_), which prefers mammals as prey. This shift proved fatal for the island fox. Once the top predator on the islands, the fox suddenly became prey. By the turn of the millennium, it was on the verge of extinction.
+
+Moreover, unlike in the Lotka–Volterra model, there was no hope of the foxes returning to their original numbers through oscillations, because the eagles also had alternative food sources, such as wild pigs and skunks.
+
[^2]: For source see <https://www.iucnredlist.org/species/22781/13985603>.
-Fortunately, with tremendous effort, the island fox as a species
-has been saved. First, the causes of their
-of the decline was identified. Then it was enough to replenish the fox population and ensure
-the conditions in which the population is stable. This involved excluding
-feral pigs, the relocation of golden eagles to mainland, the return of bald eagles,
-the artificial breeding of foxes, their return to the wild and their vaccination
-against introduced diseases. All of this was achieved in record time, in
-one decade. It was one of the most successful rescues
-mammal rescue programme.
-## Trophic functions
+Fortunately, thanks to tremendous conservation efforts, the island fox was saved from extinction. Once the causes of the population decline were identified, it was possible to restore the population and create conditions for long-term stability. This involved the removal of feral pigs, the relocation of golden eagles to the mainland, the reintroduction of bald eagles, captive breeding of foxes, their reintroduction to the wild, and vaccination against diseases brought from the mainland.
+All of this was achieved in record time—within a single decade. It became one of the most successful mammal rescue programs in history.
+
+## Trophic Functions
An important component of predator-prey models, whether it be
any of the above examples, is the _trophic function_. This function
@@ -152,7 +97,11 @@ the rate at which the growth of prey is slowed by a single predator. If $x$
is the size of the prey population and $y$ is the rate at which one predator
slows the growth of prey down (i.e., the amount of prey taken by a predator in
unit time), we can write this function mathematically in the form
-$$y=f(x).$$
+
+$$
+y=f(x).
+$$
+
We'll try to find natural assumptions that
the trophic function must satisfy. Then we will try to find a suitable
analytical form.
@@ -160,18 +109,18 @@ analytical form.
> **Exercise 1.** Assumptions about the predator's effect on prey are reflected in
> the properties that a trophic function must have.
>
-> 1. A predator in an environment with a poor food supply also has a poor
-> catch. More prey means easier prey reach and thus more
+> 1. A predator in an environment with a poor food supply also has a low
+> catch. More prey means easier access to prey, and thus a higher
> catch.
-> 1. Without food, a predator will not catch anything. If the amount of prey is zero,
-> the amount of prey a predator will catch per unit time is zero.
-> 1. Predators consume food only until they are saturated. If food is
-> is abundant, predators will not catch more food per unit time than
-> their saturation.
+> 2. Without food, a predator will not catch anything. If the prey population is zero,
+> the amount of prey a predator catches per unit time is also zero.
+> 3. Predators consume food only until they are saturated. If food
+> is abundant, predators will not catch more pray per unit time than
+> their saturation level.
>
> Express these characteristics in terms that we use to describe
-> properties of functions. What property of functions does each of the following correspond to
-> points?
+> properties of functions. Which propertyies of a functions correspondss to each of the points (characteristics) above?
+
\iffalse
@@ -186,12 +135,11 @@ is bounded above, its graph has a horizontal asymptote at infinity.
\fi
-
-## Holling's type II trophic function
+## Holling's Type II Trophic Function
The trophic function indicates how much prey one predator kills
per unit time for a given prey population size. It must therefore be defined on
-set of non-negative numbers and the function values will be non-negative (this follows
+a set of non-negative numbers and the function values will be non-negative (this follows
from points 1 and 2 in Exercise 1). In the previous section it was shown that
the trophic function has to pass through the origin
and grow to the horizontal asymptote (growth and boundedness from above). These
@@ -208,102 +156,111 @@ transformed graph of inverse proportionality. (own figure)](2.svg)
>
> 1. Scale the graph $k$ times in the vertical direction. This does not change the monotonicity
> nor the position of the horizontal asymptote, but we can change the growth rate.
-> 1. Flip the graph around the horizontal axis and move $S$ units up. This produces a function
+> 2. Flip the graph about the horizontal axis and translate $S$ units up. This produces a function
> which is positive and increasing for positive $x$ and the function grows to
-> the asymptote of $S$.
-> 1. After the above transformations, the graph has a vertical asymptote at zero and
-> one intersection with the horizontal axis to the right of the origin. Move the graph
-> to the left so that the vertical asymptote is to the left of the vertical axis and
+> the asymptote $y=S$.
+> 3. After the above transformations, the graph has a vertical asymptote at zero and
+> one intersection with the x-axis to the right of the origin. Move the graph
+> to the left so that the vertical asymptote is to the left of the y-axis and
> the intersection with the $x$-axis is shifted to the origin.
\iffalse
_Solution._ A function whose graph is obtained by rescaling the graph of the function
-$y=\frac{1}{x}$ in the vertical direction $k$ times is $$y=\frac{k}{x}.$$
-The inversion is achieved by multiplying the function by a factor of $-1$ and the shift is achieved by
+$y=\frac{1}{x}$ in the vertical direction by a factor of $k$ is
+$$
+y=\frac{k}{x}.
+$$
+The flip about $x$-axis is achieved by multiplying the function by $-1$, and the upward shift is achieved by
by adding the value of $S$. These adjustments give the function $$y=S-\frac{k}{x}.$$ The shift
to the left by $b$ is achieved by substituting the expression $x+b$ for $x$. This gives us
-the function $$y=S-\frac{k}{x+b}.$$ When converted to the common denominator
+the function $$y=S-\frac{k}{x+b}.$$ When converted to a common denominator,
the function takes the form
-$$y=\frac{Sx+Sb}{x+b}-\frac{k}{x+b}=\frac{Sx + (Sb-k)}{x+b}.$$
-To ensure $f(0)=0,$ the followin condition must be satisfied.
-$$Sb-k=0$$
-This condition shows that the three constants are not independent,
-but there is a relationship between them.
+$$
+y=\frac{Sx+Sb}{x+b}-\frac{k}{x+b}=\frac{Sx + (Sb-k)}{x+b}.
+$$
+To ensure $f(0)=0$, the following condition must be satisfied.
+$$
+Sb-k=0
+$$
+This condition shows that the three constants are not independent
+but are related by this equation.
\fi
-*Note.* In the previous exercise we derived an analytical form for
- one of the basic trophic functions. This is an increasing function,
- which initially grows towards the horizontal asymptote and the rate of growth
+*Note.* In the previous exercise, we derived an analytical form for
+ one of the basic trophic functions. This is an increasing function that initially grows toward a horizontal asymptote, and the rate of growth
gradually decreases. Such a function is called the Holling's type II
- function. It is common to write it in the form $$f(x)=\frac {Sx}{x+b},\tag{1}$$
+ function. It is common to write it in the form
+ $$
+ f(x)=\frac {Sx}{x+b},\tag{1}
+ $$
where $S$ is the saturation level and $b$ is a constant. The role of the constant $b$ will be explained
in the following exercise.
-> **Exercise 3.** Show that for a population size equal to $b$
-the value of the trophic function (1) is equal to half the value of
-saturation.
+> **Exercise 3.** Show that for a population size equal to $b$,
+the value of the trophic function (1) is equal to half the
+saturation level.
\iffalse
_Solution._ By setting $x=b$ in (1), we get
-$$f(b)=\frac{Sb}{b+b}=\frac {Sb}{2b}=\frac S2.$$
+$$
+f(b)=\frac{Sb}{b+b}=\frac {Sb}{2b}=\frac {S}{2}.
+$$
This proves the statement.
\fi
-The following exercise shows the reverse process, where from a trophic function of the form
-(1) we derive a form showing the successive transformations of the function $y=\frac 1x$.
+The following exercise shows the reverse process, starting from a trophic function of the form
+(1), we derive a form showing the successive transformations of the function $y=\frac 1x$.
-
-> **Exercise 4.** Modify the formula for the function $$y=\frac {6x}{x+2}$$ into its basic form, i.e. so that we can read the successive transformations of the function $y=\frac 1x$$ on the graph of the given function.
+> **Exercise 4.** Rewrite the equation of the function $$y=\frac {6x}{x+2}$$ into its basic form. Into such a form, so that the successive transformations applied to the graph of the function $y=\frac 1x$ to obtain the given function can be easily read.
\iffalse
-_Solution._ We solve the problem by cleverly modifying the fraction. In the numerator, we create a multiple
-of the denominator, divide the fraction in two and truncate:
-
-$$\frac {6x}{x+2} = \frac {6(x+2)-12}{x+2}=\frac {6(x+2)}{x+2}-\frac {12}{x+2}=6-12\frac 1{x+2}$$
+_Solution._ We can solve this problem by cleverly manipulating the fraction. In the numerator, we create a multiple
+of the denominator, split this expression into two fractions, and cancel $(x+2)$ in the first one:
-This calculation shows that the graph of the above function is obtained by expanding
-the graph of the function $y=\frac 1x$ in the vertical direction twelve times, by inverting it around the horizontal
-axis, shifting six units up and two units to the left.
+$$
+\frac {6x}{x+2} = \frac {6(x+2)-12}{x+2}=\frac {6(x+2)}{x+2}-\frac {12}{x+2}=6-12\frac 1{x+2}
+$$
-The same result can be obtained by dividing the denominator by the numerator.
+This form of the equation of the given function shows that its graph is obtained by vertically stretching
+the graph of the function $y=\frac 1x$ by a factor of $12$, flipping it about the
+$x$-axis, shifting it $6$ units up, and $2$ units to the left.
-\fi
-> **Exercise 5.** Construct a trophic function if you know that the predator saturation is $6$, and that the consumption rate is half of the saturation for a prey population of $210$.
+The same result can also be obtained by polynomial long division of the numerator by the denominator.
+\fi
+> **Exercise 5.** Construct a trophic function given that the predator saturation is $6$, and that the consumption rate is half of the saturation for a prey population of $210$.
\iffalse
-*Solution.* Thanks to the note before the Exercise 3 about the general form of the trophic function, we know that the prescription will be of the form
+*Solution.* From the note before Exercise 3 regarding the general form of a trophic function, we know that
$$
y=\frac{Sx}{x+b},
$$
-where $S$ is the saturation value of the predator, i.e.
+where $S$ is the predator saturation level. In this case,
$$
y=\frac{6x}{x+b}.
$$
-We can tell the value of the parameter $b$ straight away as a resuls of Exercise 3, but we can also quickly compute it from the second condition in the problem. Really, we know that
+We can determine the value of the parameter $b$ straight away using the result of Exercise 3, or compute it directly from the second condition in the problem. Since the consumption rate is half of the saturation for a prey population of $210$, we have:
$$
-3=\frac{6\cdot 210}{210+b}
+3=\frac{6\cdot 210}{210+b}.
$$
-and from there, $b=210$. So the final form of the function is
+Solving this equation gives $b=210$. Therefore, the final form of the sought trophic function is
$$
y=\frac{6x}{x+210}.
$$
-
\fi
## References
### Literature
-
* Wikipedia, <https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations> March 3, 2024
* R Mařík, Dynamické modely populací (in Czech), <https://robert-marik.github.io/dmp/uvod.html>, March 3, 2024
* D. Ludwig, D.D. Jones, C.S. Holling: Qualitative analysis of insect outbreak systems: the spruce budworm and forest, Journal of Animal Ecology 47(1): 315–332, February 1978.
@@ -311,4 +268,5 @@ $$
### Image Sources
* <https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations#/media/File:Lotka_Volterra_dynamics.svg>
-* <https://www.npr.org/sections/thetwo-way/2016/08/11/489695842/once-nearly-extinct-california-island-foxes-no-longer-endangered> Kevork Djansezian/AP
\ No newline at end of file
+* <https://www.npr.org/sections/thetwo-way/2016/08/11/489695842/once-nearly-extinct-california-island-foxes-no-longer-endangered> Kevork Djansezian/AP
+