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[email protected]##FISH###200#################### ########## ######! ######"[email protected][email protected](g<[email protected][email protected]#a(@[email protected],@t{ @uIwi.@u# [email protected][email protected][;[email protected][email protected]@[email protected] ;[email protected][email protected] 2 @xY#B@xuas-@xlQ\[email protected]######## $%##&'##f## ##################################@@luU@)#q+P?##############? ##############################@@luU@)#q+P######@*#Lm@)#q+P######## ##?######?######? ###########################################################################@ ?C=###################################@)#q+P# REPRODUCTION###FISH*REPRODUCTION_RATE################ ##### #### #### ####!#### $%##&'##c#######################?######?###? 9t^?##############?##############################?###? 9t^######?9t^?9t^##########?######?######? ###########################################################################? ###################?################? 9t^##REPRODUCTION_RATE###FISH############X@[email protected]@Yfff [email protected]@i&[email protected][email protected]@u @y#ffffg@|#33334@@@######X? #+# J?#bM??-#?ss ##o? +?`A7L?#$/#?##1&?Q#? # nO<############# ####!#### $%##&'##f## ############################# ####@'>#i@#######?##############? ##############################@'>#i@#############@'#2@'>#i######## ##?######?######? ###########################################################################@ #########################################@'>#i##EXTRACTION## #FISH*EXTRACTION_RATE################### ##### #### #### ####!#### $%##&'##c#####################################?Q#?Q#? ##############?##############################?Q#?Q#######? 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[email protected]###REVENUE###EXTRACTION*PRICE######## ##### ##### #### #### ####!####"###@>######@@ [email protected]@[email protected]@[email protected]}[h@H-@I\[email protected]@[email protected] @@[email protected],@[email protected]:[email protected]@[email protected][email protected] [email protected]@[email protected]###### $%##&' ###################################!###### ###################### #######drmd################ #################################!###### ###################### #####H#####(a s#c##########################@########################### ########?##v#-#############)*#+#,- k#(d#\#m############################@####################### ############S#d#e#v#######)*#+#,- k#(f#]#########################@###################### ##############,#h#v# ##5#############)####]#u#]#*#+)#####T,- k#(c############################@###################### ###############x##############)*#+#,- k#(a######################e#$####@######## # # # # # ######## ###############O##########)*#+#,- k#(a#6#####################@#:####@#################### ##########Z##u#########)*#+#,- k#(D###P########################@###################### ################>#:#b#######)*#+# #Graph 1#,- k#(a#r####################\#| #<####@##############################o########### )*#+#,- k#( t## ##################@######### ###### #########a################)*+##b#aBulicek, Kurt J ECON415 (ESES467) 21 June 2009Homework #1GIVEN: population of variable fish [FISH] & the reproduction rate of variable fish [REPRODUCTION_RATE]. We're asked to deternmine the population of variable FISH over a given time period, initalized @ 20 years.QUESTIONS:1) What is variable FISH @ 20 years with exponential growth?2) What is variable FISH @ 20 years with density-controlled growth?3) What is variable FISH @ 20 years with density-controlled growth & extraction?4) What is variable REVENUES @ 20 years for case 3?SOLUTION: The model is initialized @ 200 fish & a reproduction rate of 5% [FISH = 200, REPRODUCTION_RATE = 0.05]. We're first asked to find the exponential growth of this intial population assuming a healthy environment after 20 years time without any members of the existing population dying. The obvious assumption is that under these conditions, the fish will continue to multiply at a slow, but exponential rate as a function of their small initial population. When the model is run under these parameters, one can see that after year one, we have increased the fish population by 10 [+10 FISH]. After year two, we have increased the population of fish by an additional 11 [+11 FISH since we round up to nearest whole number of fish]. After year three, we increase the previous year's population by an additional 12, etc. This process continues on into year 20, at which point 26 fish [+26 FISH] have been added during a single annual season for a combined total fish population of 531 fish [see graph 1, pg. 1]. For question 2 pertaining to reproduction with density-controlled growth, the number of fish reproduced within the first few years of the model should be considerably higher given the variable REPRODUCTION_RATE = 0.2. Under this model, the number of added fish each year decreases as the total fish population grows. After year one, 31 fish [+31 FISH] have joined the initial population set forth by the model parameters. As the years go on, graph 1 pg. 2 depicts the fish population increasing at a decreasing rate, adding only 3 additional fish [+3 FISH] by year 20 for a combined total of 486 fish within the population.In question 3, the rate at which the fish are harvested for profit is set to 0.03 [EXTRACTION_RATE = 0.03]. Intuitively, the population will continue to increase over the years despite the fact some of them are then taken out (for the puposes of profitability, it would disadvantageous to harvest fish faster than they are able to reproduce since you would ultimately deplete your fish population & go broke). Refer to graph 1, pg.3 for a visual representation of this effect. With the extraction rate factored into the model, the fish population again increases at a decreasing rate, ending in year 20 with only 1 additional fish [+1 FISH] in comparison with the previous calendar year. The accumulated total in that year for the fish population was 394 [see graph 1, pg. 3, line 1] vs. 486 [see graph 1, pg. 2] in the model asked for in question 2. Question 4 operated under the assumption that the revenue generated by this fish harvesting operation [variable REVENUE] would grow slowly since the rate of fish extraction [variable EXTRACTION] from the total population was relatively small. After running the model, we can see that the revenue curve begins @ $30 (assuming that revenue on our graph 1, pg. 3 is expressed in dollars), calculated by multiplying the fish population by the extraction rate, & multiplying that figure by the price [200 FISH * 0.03 EXTRACTION_RATE * 5 PRICE]. At the beginning of the sequence, revenues increase fairly quickly since more fish are added back to the population. However, by the end of the 20 year cycle, revenues slow dramatically, displaying a $2 increase in year 20 for a annual accumulated total of $59 since year 1 [see graph 1, pg. 3, line 2]. ADDITIONAL QUESTIONS/HYPOTHETICAL SITUATIONS:How would fish mortality affect populations, reproduction, & extraction rates? Would more or less fish have to be extracted to maintain the same profitability? Would the cost at which the fish were sold in this model have to be adjusted to account for natural or environmental fluctuations resulting in fewer or more fish within the population (such as diseases, cellular mutation, introduction of predators into the environment, increased abundance of food sources, weather conditions, etc.). What would be the best method of maximizing revenues without reaching the point of overpopulation? How would prices be affected by a greater number of harvested fish? How can we achieve the greatest profit margin under more realistic conditions?CONCLUSION(S):This model would be useful in the determination of revenue or profitability maximization, most pertenantly situations relating to revenue generated from a reproducing population of some kind (wouldn't necessarily have to be fish). It could also be used to model population growth (with slight modifications) of, say, humans in a particular region, as human population growth has profound effects on resource allocation, relative prices, pollution, & many other factors. #,- k#(f#^#Z################### ######@########################## ######)#^#+#^#*#+)#h##h#9,- #########@#####h# ######### k#(d#^############## ##############@######################### #########U##g########)*#+#,- k#(c##a##########################@####################### ##############5############)*#+#,- k#(c##############################@######################## ## #############.#!##########)*#+#,- k#(c###i######################@##################### ############## #_###############)*#+#,- k#(a##X################# ###'####@##############################J#,##]##### ##)*#+#,- k#(a#f#:################# #k######@##############################=#"#_#W###### #)*#+#,- k#(a#4#M############# ### ##+####@################################\#^#}##### ##)*#+#,- k#(a#u#<############# ### #&#l####@##################################e#### ###)*#+#,- k##drmd#################5#4### ##########,663##2n?#^####################^###D##### ##.### ##########.bb 6#n############P############| [email protected][email protected]##################,##FISH REPRODUCTION, EXPONENTIAL#/J######0#J#0####################? 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This note was uploaded on 08/30/2009 for the course ECON Dynamic Si taught by Professor Hannon during the Summer '09 term at University of Illinois at Urbana–Champaign.

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