Endangerment of Humpback Whales

Apart from poaching which ha as led to a number of animals becoming extinct, there are also some sea animals which are at risk of being extinct due to the human activities, among them is Humpback Whales. . The essay will discuss issues to do with endangerment of humpback whales, together with how to avoid the endangerment by using mathematical models including the exponential growth graphs and the Lotka-Volterra model.

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In the 18th century, the market value of humpback whale rose up making them most preferred species by the hunters. During the 19th century, various nations began hunting intensively for humpback whales leading to a significant fall in the humpback whale population at the beginning of the 20th century. The number of humpback whales went low in 1966 leading to its ban prompted by Whaling Commission that will prevent the species from extinction. The protection yield some result and the population is slowly recovering, and the number is increasing. There are among the first animal species to be protected by Endangered Species Conservation Act, which was the forerunner of Endangered Species Act (ESA) in 1970. Soon after the passing of ESA in 1973, the humpback whale was recorded among the endangered. Marine Mammal Protection Act also protects them (Barlow, Jay, and Phillip 134).

These astounding whales move all through the seas, sing complex vocalizations and utilize nets of bubbles to catch schools of fish. Currently, there is the decimation of the humpback whale populations through whaling, and they are recorded as the endangered species. The humpback whale, commonly referred to as Megaptera novaeangliae is among the most widespread mammals in the world. They are famous for their stupendous jumps out water, unique tail fins (flukes), as well as their dulcet singing while in the ocean's profundities (Baker, et al., 291).

The global population if humpback whales were initially at 80,000, with about 12,000 of them in the North Atlantic, between 18,00 to 20,000 in the North Pacific as well as 50,000 of them in the Southern Hemisphere, this is a reduction from its population of 125,000 during the pre-whaling before 1966 (Barlow, Jay, and Phillip 178).

The number of humpback whale has recovered due to the effort made by the international conservation efforts such as the Endangered Species Act. Records show that currently, there is about 80,000 humpback whale down from 10,000 to 15,000 (Baker, et al., 331)

The endangerment of this species of whale has prompted the scientist to come up the statistical system that will assist them monitor the population of the whales as well as mortality and reproduction rate. In order to make this effective, they have applied two different methods exponential growth graphs and the Lotka-Volterra model, which represent the data of the humpback whale population, its statistical formula and the representation on the graph. These methods are internationally recognized by various ecology department and it has been practically used in determining the population of the whales in Australia as well as the arctic region (Barlow, Jay, and Phillip 321).

Lotka-Volterra model.

During the interaction of the animal species in a dynamic environment, the species get affected. Normally, there is a network of interaction, referred to as a trophic web, that creates the mechanically complex communities. It involves two species or more, mainly concentrating on the two-species systems. There are three broad categories of interaction (1) when the rate of growth of particular population get reduced while the other increases the populations, then they exist in the predatorprey situation. (ii) if the rate of growth of particular population is decreasing, that is competition. (iii) when the population growth rate of each species is heightened, then it is referred to as symbolisms or mutualism (Matsuda, Hirotsugu, et al. 217).

The LotkaVolterra equations, also referred to as predatorprey equations that depict predator-prey change, consists of the pair of nonlinear, first-order, equations that are often used in describing the evolution of the biological framework whereby two species relate, one as a pray and the other as a predator ((Glockner, Deborah and Spearous 231).. There is the change in population in the course of time depending on the air of the equation.

There is a wide application for Lotka-Volterra in ecology, with equations for capturing predatory-prey interaction are among two species. The equations are as follows.

According to the equation (1), dw/dt means a variation in the predator population W which take place over an infinitesimal little time interval dt. Likewise, in the equation (2) ds/dt is the variation in the stock of prey, S, throughout the small-time interval, dt. Then W is considered as the whales population where S as the sand lance (Ammodytes americanos) that they feed on. The two are differential equations.

In the equation (1) F is the rate of reproduction for prey eaten, Q, is the Predators' mortality rate, and A is the rate of predation, Coefficient/constant or measure of search efficiency. On the other aspects, the rate of reproduction of the whales is measurable in whales per whale for every sand lance eaten concerning a given year. The rate of predation coefficient A may be appraised as 1/whales/year, the stock of sand lance may be in tons of sand lance biomass, and Q in whales per whale per year.

Also, R, the sand lance rate of reproduction may be determinate in tons of sand lance biomass per ton of sand lance for each year. Subsequently, the units of the equation (1) will be as follows.

One can stipulate the units in the form of the dialog box of the stock, converters and flows and STELLA will perform unit checking. Maintaining trails of the units is one of the easiest technique for examining if a model is properly defined. The variations in units denote that a vital component is omitted from the pattern, or that the dimensions of the modules which are in the model dont tally with each other. If the variation in the population of whales are measured with regards to tons of whales biomass in each year while the population of the whale was measured according to the individuals, therefore the correspondence that is there between the number of whales together with the biomass requires some sorts of establishments. For example, a new parameter may be introduced hence reflecting the average biomass foe each whale.

Equation (1) and (2) are made up of the simplifying of the assumptions. Another instance, is when the predation controls the growth of the prey, as predators feed on one species of prey, there is limited consumption of prey, while predator and prey often meet one another in a homogeneous environment. These assumptions are explicitly a simplification of the situations whereby the organism live in a natural world. Though, this does not prevent the use of the Lokta-Volterra model in exploring one means in whereby two species are thought to interrelate with one another.

The Lokta-Volterra equation (1) and (2) offers a suitable commencing point for the model of the addition of predator-prey dynamics in the population model of a humpback whale. Though, before proceeding into using these equations, modification will be needed in one of them to relieve one of the dangerous oversimplification- the absence of a limit on predator population.

The LotkaVolterra equations (predatorprey equations), are a couple of nonlinear, first-order, discrepancy equations often utilized in describing the changing aspects of biological organizations whereby there is the interaction of two species, one as a prey and the other as a predator.

Whereby:

x = number of prey

y = the number of predators

and dydt = the rate of growth for the both population over some time.

t=time

while a, b, g, d are the positive actual parameter denoting the interaction of the both species.

A close example ofTheT LotkaVolterra system is a Kolmogorov model, whereby it is a more general system that model the dynamics of environmental systems with the interaction of the predator and prey, mutualism, disease, and competition.

exponential growth graphs

Exponential growth is when the population has a stable birth rate throughout the year and is not restricted by other factors such as diseases, treats or food. The estimated population growth of humpback whales stands at an annual rate of 6.5% for the adequately studied Humpback Whale in the Gulf of Maine (Wagenaar et.al., p.117).

Census

Year 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

Number of Humpback Whales 1149 1218 1292 1367 1441 1515 1590 1644 1738 1821

1844

The primary exponential function is defined as f (x)=bx whereas the base is steady while the exponent x is an independent variable.

f (x)=bx, where b>0, b1

xy

-2 2-2=0.25

-1 2-1=0.5

0 1

1 2

2 4

3 8

A function in the form of f (x)=abx with b>1 and a>0 is the function of exponential growth, whereby the function increases while the x increases.

Matsuda, Hirotsugu, et al. initially proposed an easy pattern for the predation of a single species by another one to describe the oscillatory stages of some fish catches in the (Wagenaar et.al., p.291).

When the predator species have an alternative food source, then it will not die out because of starvation even in the absence of the preys (Glockner, Deborah and Spearous 219). Therefore, it will require the replacement of -cy term as well as the second equation by the logistic-growth:

: x = (a x r x 2) a xy

y = (b y c y 2) + g xy

z = d z + l yz

The more complicated food-chains can be constructed in a similar manner as the system of additional 2 equations. For instance, given that there is eco-system with 3 species. When species x is an herbivorous, with the isolation population it would respect the logistic equation, while the one that is preyed on by the species y which consequently is the main food source of the species z. Therefore, the corresponding population may be modeled by a three- equation system

Then their respective population might be modeled by the 3-equation system:

x = (a x r x 2) a xy

y = c y + g xy d yz

z = d z + l yz

Conclusion

After 1966, there was a drop in the number of the humpback whale, which attracted the international attention, since it remained only 30% of the initial population. This was contributed by the high market value of its meat that was in high demand. Various parties were involved in trying to formulate the system that will seek to control the declining population of the species. Since, the banning of whaling, the population has been increasing drastically increased. This essay has effectively outlined the methods that were used in saving the declining population of the species.

The presented results in the essay are the long-term estimates used to determine the humpback whales abundance that was found across all the ocean basins. Scientists have made various attempts to have a long-time system of monitoring these whales that are in danger of extinction within their geographical scale.

The exponential growth graphs and the Lotka-Volterra model, are common methods that are used in ecology when determining the population of the given species, also they are used in knowing the factors that could be leading to the declining population of the specified species. The methods involve the formation of the formulas based on the prey and the predator, its plotting to the graph and interpretation. It is demonstrated that there is exponential growth of the humpback whales between 2002 to 2012. The growth has been contributed by various factors, including the banning of whaling, which is protected by various Acts.

Works CIted

Baker, C. S., et al. "Hierarchica...