The Island of Jozie Government has requested a research project to determine the optimal harvest technique of the rare indigenous species of Large Tailed Lemur (Lemurodiea magnacauda var. jozie) which has been found to have cancer curing properties. This proposal will detail how The Capstone Research Company plans to complete this research to harvest the cure from the Large Tailed Lemur, but employ a technique that sustains the population and eliminates the risk of extinction of the only known cure. The Capstone Research Company believes that in six months time to have the most sustainable harvesting technique that will ensure the survival of the cure while maximizing the profit for the Island of Jozie and its inhabitants.
The Island of Jozie has recently discovered a mammal species that only exists on their island named Large Tailed Lemur. One of the Lemur's was tagged and taken to a CDC research facility in Atlanta to study the mammal's habits, DNA, blood, and other physiological characteristics. CDC scientists discovered that when they apply Large Tailed Lemur's white blood cells to a rat with cancer that the tumors begin to shrink without any radiation or chemotherapy. However, when scientists attempted to breed the Lemur's in a laboratory via artificial insemination and harvest the cancer cure it was unsuccessful. Something unidentified in their habitat on the Island of Jozie when processed in their body causes their white blood cells to cure cancer. The Lemur habitat is located in the center of the island where adequate shelter and vegetation are present along with elevation from the mountainous terrain. The Government of Jozie is wanting to maintain control of their indigenous Lemur and seeking the optimal harvest technique so they can ensure the cure does not disappear until it can one day be replicated in a laboratory setting. To aid The Capstone Research Company in their research, the Government of Jozie (GOJ) has provided the following information:
- Initial population of Lemurs on the island
- The rate at which the Lemur population grows (considering sexual maturity, life span, number in a clutch)
- The number of Lemurs the island center can hold without environmental impact (carrying capacity of the island)
- The cost for conservation of the Large Tailed Lemur's habitat and protection to ensure the safety of the lemurs.
In recent decades, there has been a notable increase in research focusing on the interaction between the conservation of natural environment, sustainable development, and economic growth. However, in addition to the direct consumption of renewable resources (i.e. harvesting), there have been identified other factors that may provoke drastic reductions and, possible, extinction of wild species populations .
The single species Gompertz population model, first proposed by Benamin Gompertz  and mentioned by Roberts and Shepherd , served as a model for human populations.
Where P is defined to be the population, T is the time parameter, R is the population. Later, Gompertz model was adapted for fisheries management. "We start by considering a harvested population following a Gompertz growth model with dynamics described by the SDE:
where X(t) is the population size at time t, measured as biomass or as numbers of individuals, r > 0 is a growth parameter, K > 0 is the carrying capacity of the environment, q > 0 is the catchability coefficient, E(t) ≥ 0 is the fishing effort exerted on the population at time t, σ > 0 measures the strength of environmental fluctuations, W(t) is a standard Wiener process and X(0) = x is the population size at time 0, which we assume known. We denote the harvesting rate by H(t) = qE(t)X(t), which is the most traditional form for H (as in Clark (1990)) [4, p. 197]".
There are examples of harnessing natural resources which in turn provide income to the government and its population. At least one-fourth of all mineral lease rentals, royalties, royalty sale proceeds, federal mineral revenue sharing payments and bonuses received by Alaska from is placed in a permanent fund investment. Part of the income from this investment is distributed to qualifying residents . The model for the Alaska Permanent Fund is modeled using differential equations. "The change in the permanent income for the state (dydp) is approximately equal to:
Where i is the real rate of interest, n is the number of years of oil production and is the constant change in real income received by the state [6, p. 141]." The extra income for residents is generally of financial benefit. While households are excessively sensitive to their income tax refunds, there is no evidence to suggest that the seasonal pattern of consumption reacts to the payments from the fund  . Additional there are recognized health benefits for the Alaskan population. It has been documented that the extra income has a significantly positive, but modest effect on birth weight . "The additional income increases birth weight by 17.7g and substantially decreases the likelihood of a low birth weight (a decrease of around 14% of the sample mean) [8, p. 576]".
Using a stochastic differential equation within Gompertz harvesting model  The Capstone Research Company plans to manipulate the equation from fishing to lemur harvesting.
Gompertz Harvesting Model Gompertz Growth:
X(t) is the population size at time t r > 0 is the growth parameter of the population K > 0 is the carrying capacity of the environment q > 0 is the catchability coefficient
0 is the maximum capture effort at time t
σ > 0 measures the strength of environmental fluctuations
W(t) is a standard Wiener process
X(0) is the population size at time 0, which will be defined as the populations before harvesting efforts begin H(t) = qE(t)X(t) is the harvesting rate of the lemurs
is the natural growth rate of the lemurs
Since ) we desire H(t) < f(X) so that the population growth continues or at minimum H(t) = f(X) stays the same. If H(t) > f(X), then the population level will decline which could lead to extinction .
The catchability coefficient is defined in terms of fisheries management and would therefore need to be adopted to meet the needs of capturing lemurs. "The animals are trapped using live-catch Sherman traps baited with banana since this has been found to induce less stress on the animal [10, p. 69]" and any added chemicals from multiple tranquilizers or stress hormones may have a negative impact on the cure. "q can conceptually be considered as the probability of any single fish being caught. Therefore q ranges between 0 and 1 [11, p. 11]" The catchability coefficient using the trapping method for all lemur's should be close to 1, but not exactly 1 since each trap will be up for no longer than 3 days to catch a lemur (checked via GPS location to ensure minimal time in the trap) and the traps may malfunction at times or a non-harvestable lemur may have been trapped. The Capstone Research Company will be able to give an exact value after research has begun and the efficiency of trapping can be evaluated.
This methodology is taking into consideration that Lemurs will not be harvested until after sexual maturity. To ensure lemurs are not being captured before they reach sexual maturity all lemurs will be tagged with a GPS tag with a scan capable bar code. Any lemur who has already passed sexual maturity will be listed as "Okay to Harvest" when the bar code is scanned. This will also help researchers track the number of lemurs on the Island of Jozie to ensure their population is being sustained and not depleting due to over-harvesting or poaching. There are no predators on the Island of Jozie that feed on the Large Tailed Lemur as a primary food source and does not need to be taken into consideration.
Profit for the Island of Jozie
One way to consider calculating the profit for the Island of Jozie is to use Pontryagin's Maximum Principle  to calculate the profit for the CDC and then have the Government of Jozie take a percentage of their profit if done on a continuous time stream revenue. This method has been used in conjunction with Gompertz Harvesting Model for fisheries management in other studies . The Capstone Research Company believes this formula can be altered and applied to the lemur case like Gompertz Model.
θ is the constant cost per unit c is the constant harvest cost per unit effort q is the catchability coefficient
δ is the instantaneous rate of annual discount
Since the Island of Jozie is looking to profit from the cure it is desired to have c < θqX(t) unless the CDC and Island of Jozie will be losing revenue and not creating profit.
Once the maximized profit can be calculated for the CDC, since they are conducting the research on the cure from the Large Tailed Lemur, The Capstone Research Company will decide what is a fair percentage for the Government of Jozie. The percentage to the Government of Jozie will be divided between the inhabitants of the island and the government to conserve the territory and safety of the Large Tailed Lemur.
To help determine the percentage to the Government of Jozie, the differential equation used by the Alaskan government to distribute oil money to its inhabitants will be used to distribute lemur profit to the people of the Island of Jozie. The animals are a part of their culture and they voted to allow the CDC to harvest to Large Tailed Lemur's white blood cells for their cancer cure. Based off of the literature supporting how Alaska has seen gains from this model, The Capstone Research Company is proposing to use this so that the Island of Jozie may see some of those same benefits. Assuming that the government approves the increase, the expected payout to the government and the residents would begin after year five. This follows the processes adopted by Alaska . Assuming an average interest rate of 0.05, the payout per resident would be calculated by the following: 
With yi representing each of the last five years of income growth, r representing the number of qualified residents and E being the expenses.
The animals are a part of the island culture, and the citizens voted to allow the CDC to harvest to Large Tailed Lemur's white blood cells for their cancer cure. Based off of the literature supporting how Alaska has seen gains from this model The Capstone Research Company is proposing to use this so that the Island of Jozie may see some of those same benefits. The cost provided by the Government of Jozie for the conservation and protection of the Large Tailed Lemur will be added and a percentage of the maximized profit for the CDC will be finalized.
By developing our harvesting model for the Lemurs, The Capstone Research Company hopes to be able to answer the following items:
- What is the optimal effort that the CDC needs to put forth in order to maximize their yield for a given population of Lemurs (while at least maintaining the current population of Lemurs)?
- What is the catchability of the Lemurs, so that the CDC knows how much effort to put forth into catching them?
- How will a percentage of the profits be distributed among the residents of the island? What are some of the benefits, financial and otherwise, the residents can expect?
- Can the model guarantee the success of the Lemurs continue?
As analysis of the previously stated questions begins, The Capstone Research Company expects that with careful planning, they can at least maintain the population of Lemurs, and avoid extinction, while maximizing the harvest for the CDC in order to continue to cure cancer. They also expect that their method of trapping will minimize the effort needed to harvest while still giving them their maximum yield and the catchability of the Lemurs will decrease with the avoidance of chemicals and stress. The inhabitants of the island are expected to see a profit from the harvest, and in turn receive additional income and experience additional benefits.
There are notable limitations to this proposal since The Capstone Research Company is proposing to use a method that has been used primarily in fishing with little outside application. In theory this is the best model to use to prevent over-harvesting and depleting the population, but may not be the most efficient in practice since the adaptation of the formula is pure theory. An additional limitation of the proposed research is that the model for distributing profits to the government and the population is based on a previous employed framework for fossil fuels. Since the focus of this part of the study is financial, this is expected to have minimal impact. The oversight for the Alaska Permanent fund is provided by a state government . Considerations would need to be given to the laws and financial organization of the Island of Jozie. If this proposal is chosen and is no longer deemed effective The Capstone Research Company will reevaluate and propose a new strategy with the new information gained during research.
The Capstone Research Company will need to further research the area of impulsive harvesting versus our proposed continuous harvesting. Different environmental factors such as drought and animal tourism may cause varying levels in the population of the Lemurs (interrupting the Lemurs breeding habits), so that continuous harvesting that is presented in our methodology would not be as optimal as pulse harvesting . As a part of further research once the ideal harvest amount has been obtained there would need to be another level of research to determine the break-down of male to female that can be harvested to continue the growth of the population.
The Island of Jozie's Government has requested a research project from The Capstone Research Company to aid the CDC in the optimal harvesting technique of the islands Large Tailed Lemurs while maintaining the current population growth of the lemurs and maximizing the profit for the inhabitants of the island. After reviewing the literature on Gompertz Growth model as applied to fisheries in conjunction with the Pontryagin's Maximum Principle, as well as information on how the dividends are distributed to Alaskans from oil revenues, The Capstone Research Company proposes to develop a Gompertz Growth model for harvesting the Lemurs. This model can aid them in harvesting the Lemurs without over-harvesting the mammal as to promote sustainability in the population as well as maximizing the profit for the inhabitants and the government.
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