Population of Fish in Lake Ontario
This report is our work. Any help, insight, or aid that we have received is explicitly acknowledged. To do otherwise is plagiarism, and we understand that we will receive a zero on this report if it is found that we did not adequately acknowledge the contribution of anybody else.
MA232, Discussion Section 17 Due: September 19, 2013
Mitchell Phillips Date: .
Zachary Rauen Date: .
Devin Winchester Date: .
TABLE of CONTENTS
Title ..…………………………………………..….…….. Page 1
Signatures ………………….….……………….……… Page 1
Table of Contents …………..……………..….…… Page 2
Introduction …………………………..……………… Page 3
Modeling the Population ………………….……. Page 4
Overcrowding and the Carrying Capacity... Page 6
Harvesting and Fish Industry ………………….. Page 7
Population Model Analysis ……………………… Page 7
Conclusions …………….…………………………..… Page 8
Citations ………………………………………………… Page 9
Introduction:
Fish populations in the Great Lakes can be modeled through differential equations. The changes in the rates of birth, death, and the harvesting rates can all be used to represent the populations. With respect to time in years, we are able to analyze tonnage at a given point and are able to explore certain harvesting rates that allow for maintaining a stable population and the fishing industry at long-term levels. The measure of the population of living fish at time t can be modeled by,
dxdt=Birth rate-Death rate-Harvest rate
(1)
Using this model, we can identify a stable harvesting rate with respect to the birth-death mortality rate. Moreover we can find how much of change causes the stability of the system to fail in order to determine a harvesting range. This will allow for the identification of an optimal harvesting number (in tons) that allows for variations in both directions to account for any unforeseen anomalies which can occur in a physical system.
The following technical paper gives an explanation on how to model the aforementioned system. The paper also goes through why and how to choose a harvesting rate that works with a real physical system.
Modeling the Population
The fish population can be modeled at time t by using equation (1). The model accounts for the birth and death rates of fish as well as the rate that which they are harvested. However, the model and total population, x(t), can be described and expanded to,
dxdt=bxt-m+cxtxt-sx(t)
(2)
where the first, second, and third terms correspond to the birth, death, and harvest rates, respectfully. The values of b (birth coefficient), m (mortality coefficient), and s (harvesting coefficient) are all nonnegative proportionality constants. The mortality coefficient is accompanied by cx(t), where c is also a nonnegative proportionality constant, which accounts for overcrowding or the carrying capacity in the population. The quantity of r=b-m can be assigned as well to help simplify the model and is assumed to be a positive number. This can be taken a step further to include s due to all the rates be proportional to x(t), thus the quantity
K=b-m-s,
can be assigned. Based on observations of actual fish populations, it is also assumed that the values of b, m, and c, are all know and the rate can be set using certain parameter arrangements. Using the given conditions, the model is defined as,
dxdt=Kxt-cx2t.
(3)
Analyzing the model
In order to interpret the behavior of the population model, equation (3) can be simplified. This allows the effects of overcrowding and harvesting to be determined and studied. Equation (3) can be represented through a logistic equation (Yao),
x'=Kx1-cKx.
(4)
By applying a logistic model, the population growth can be easily understood and allows for further observation.
Solving the logistic model for x when x’=0, it is determined that
x=0 and x=1c/K.
Based on these solutions, the behavior of the model can be found. If the value of x is greater than 1c/K , x > 1c/K, the rate, x’, is less than zero. If the value of x is between 0 and 1c/K , 0 < x <1c/K , the rate, x’, is greater than 0. If the value if x is less than 0, x < 0, the rate, x’, is also less than 0.
Figure 1: Phase diagram for x'=Kx1-cKx when x’ = 0
Figure 1 lays out a phase diagram for the model of population. It demonstrates how the rate reacts under certain condition as previously stated. Based on figure 1, it can be determined that there is a stable equilibrium solution at x=1c/K. To further represent the nature of the model, a slope field can be used.
Figure 2: Slope field for x'=Kx1-cKx when x’ = 0
Figure 2 shows how the population model behaviors and where there exists equilibrium solutions. Based these findings, it can be concluded that population can only stably exist under the condition that x=1c/K .
Overcrowding and Carrying Capacity
For every population that exists, there is some limit as to how much of that population can survive. This principle can be applied to a population of fish in Lake Ontario. This is modeled using equations 4 and 5. Once an environment reaches carrying capacity, a portion of the population begins to either leave or die. These fish that die or leave due to overcrowding must be considered in our model because without them the values would be very inaccurate. An inaccurate prediction could cause a harvest rate that could end up wiping out the population.
Harvesting and Fish Industry
Fish populations are impacted through human interaction of harvesting and farming. This factor greatly affects all of the other factors. The more that we harvest fish, the less they can reproduce in the future, lowering the birth rate. If more fish are harvested, the overcrowding factor goes down, meaning less fish would die or leave. A harvest rate exactly at the equilibrium would mean the population stays the same every year. As long as the population of fish is the same, a constant yearly harvest could be achieved.
Population Model Analysis
The population of a species of fish can be modeled using equation (1). Birth, death, and harvest rates all account for the population changes. Using the values provided by the University of Wisconsin-Milwaukee and Montana State University a test case can be run and how the model reacts can be studied.
(Use the given numbers to see how everything reacts. Long term behavior. Based on assumptions provide a recommendation for the best value of the harvest parameter, s. General form, not specific values what was included in (2))
Conclusions:
With the aforementioned factors being used and explained it is easy to see why we include these aspects. These aspects cover the overall basis of a given population for any species. There are other factors that can go into a population model but usually these factors count for so little that it is easy to neglect them. External factors can vary anywhere from the food chain to “acts of god.” Usually the food chain aspect is accounted for in the death rate, m(x), in the model analysis but things like new species in the chain or a new order of the chain cannot be perfectly accounted for. “Acts of god” includes natural disasters and any other random occurrence that is not included in the model. These things are not included for two reasons: the instances are either unpredictable or the change in fish caused by these instances is negligible.
The model analysis concludes that a set harvesting rate would have to be set below the difference between the birth and death rate. The harvesting rate can be controlled by giving out harvesting and fishing licenses. It can also be concluded that the harvesting coefficient cannot be close to the difference between the birth and death rate because it is imperative to allow for extra harvesting. This conclusion is reached because it is well known that there will always be people fishing without a license.
In finality, the model of a population of a species can be done quickly and efficiently using basic logistical analysis and using the skills learned in the differential equations course.
(Finish up the analysis! why we consider these aspects, or why do we not consider other effects on the fish population)
Citations
Assessment of an Invasive Lake Trout Population in Swan Lake, Montana, Benjamin Cox, Masters Thesis, Montana State University, Bozeman Montana, July 2010.
Black, Kelly, Guangming Yao. "Differential Equations." MA232. Clarkson Uiversity. Science Center, Potsdam. 9-16 Sept. 2013. Class lecture.
Farlow, Jerry, James Hall, Jean McDill, and Beverly West.Differential equations & linear algebra. 2nd ed. Harlow, England [etc.: Pearson-Prentice Hall, 2007. Print.
"Slope Field Generator and Applet. Create images of custom slope fields."Calculus, Linear Algebra and other Higher Math Lessons--Math Scoop.com. N.p., n.d. Web. 17 Sept. 2013. <http://www.mathscoop.com/calculus/differential-equations/slope-field-generator.php>.
Statement on Pollution and Eutrophication of the Great Lakes, Special Report No. 11, A.M. Beeton, May 1970, Center for Great Lakes Studies, University of Wisconsin-Milwaukee.
Stewart, James.Essential calculus: early transcendentals. Belmont, CA: Thomson Higher Education, 2007. Print.
Wood, Thomas. "Fish Population Modeling."redwoods.edu. N.p., 11 May 2009. Web. 15 Sept. 2013.<msemac.redwoods.edu/~darnold/math55/deproj/sp09/TomWood/Thomas_Wood_Presentation.pdf