These are just guides. There'll be no detailed explanations. After all, this assignment is meant to hone your analytical and critical skills.
Maths T Assignment: Logistic Growth Model
Question 1
Just find the answer in the internet. It's only a definition, you may rephrase it if you want.
Question 2
You simply need to play with the logistic growth equation given. dP/dt = rP (1 - P/k) is the growth rate, so for the growth rate to be increasing, the sign must be positive and for the growth rate to be decreasing, the sign must be negative.
Since rP will never give you a negative value (r is the positive constant and P, the population, would never be smaller than zero), for dP/dt to have a negative sign, (1 - P/k) must be negative.
Hence if the population exceeds the carrying capacity (P > k), then P over k (P/k in the equation) would be bigger than 1, and therefore (1 - P/k) < 0, suggesting the growth rate is decreasing. If the population does not exceed the carrying capacity (P < k), then P/k < 0 and hence (1 - P/k) > 0
Question 3
3(a)
Basically you change your answer in question 2 into word form. Describe how the population growth will vary. If the population exceeds the carrying capacity, then the population growth rate will decrease and subsequently the population will drop. If the population equals the carrying capacity, the growth rate is zero and if the population is smaller than the carrying capacity, then the growth rate is positive and the population ins increasing.
3(b)
What would the value of P for constant growth? And what would the values be if the growth rate is increasing and decreasing? You should already know the answers if you've done the previous 2 questions.
Question 4
Differentiate the logistic growth equation once, and you'll get r/k (k - 2P). For dP/dt to be maximum, the derivative of the first population must equal to zero. Hence for r/k (k - 2P) = 0, (k - 2P) must equal to zero. You'll establish a relation between the carrying capacity and the population here.
However, you need to verify that the P value you get is indeed the maximum value of dP/dt. To verify it, you differentiate its first derivative once again to get d3P/dt3, the second derivative of the logistic growth equation.
When you've verified it, find the maximum value of dP/dt by substituting the value you found into the logistic growth equation.
Your final answer: dP/dt is max when P = k/2, and maximum value of dP/dt = rk/4.
Question 5
You need to use differential equation to find the general solution of the logistic growth equation so that you can express P explicitly. While doing this, you might need to perform integration by partial fractions. Everyone would possibly get a different equation because their constants might be different, but the final answer should not vary much.
You'll get one equation here. But if you wanna express it more clearly, you can form two equations by considering one where P > k and one where P < k. One should be enough.
The equation is a fraction.
Do read the question carefully: "plot, on the same axes, a few graphs to show the behaviour of P against t", meaning there involves only one graph paper and within a graph paper there are a few lines.
You need to use arbitrary values and construct a table. Use actual values to show the behaviour. For example you may use r = 10, k = 200 and P = 100, 200, 300....to draw the graph.
Positive constant, r, would not vary. If it does all physics equations need to be remodelled. There are at least 4 cases: one where the carrying capacity is fixed and the population varies (P > k, P < k), and one where the population is fixed and the carrying capacity varies (k > P, k < P). There should be at least 5 lines.
Question 6
Find the growth rate within the time interval, and find the average of the population within the time interval. E.g. 1/2 (40 + 77) for the first one. Then plot a graph, you should get a bell curve with a maximum point.
Find the value of dP/dt when your population is half the carrying capacity (carrying capacity is the value where the population growth rate is zero). Once the value of P is determined, you'll find the value of the carrying capacity and the positive constant.
For those who have Oxford Fajar book, there's an example of assignment which is somehow similar to this assignment at page 224 and there's a sample answer. You may refer to it. They are almost similar. Use that as a guide.
Good luck.
Please help... For Q5, my solution stops here. How to continue further from here? :
ReplyDeleteP/(k - P) = Ae^rt , P/(k - P) > 0
- Ae^rt , P/(k - P) < 0
Both the equations for Q5 are the same even though pk.
ReplyDeleteOkay, my bad, I initially formed two equations, but to simplify it I ended up had only one equation.
ReplyDeleteAnyway there's a solution here: http://www.scribd.com/doc/127779847/STPM-Mathematics-T-2013-Assignment-B-Mathematical-Modeling-by-Stephen-P-Y-Bong
Gosh, I was waiting for your reply for more than an hour. Anyway, thanks. The new answer is more like it. The conditions for my five lines are (p < k , p > k, p = k, k < p and k > p ). Is that correct?
ReplyDeletePresumably. I don't know. Follow what your teacher says. If he/she doesn't say anything then I presume 5 lines are enough. I'm not a teacher and am a student like you so I can't give you a definite answer. =)
ReplyDeletehow do u write the website for references?
ReplyDeleteIt's an assignment....I don't think references are needed. Only projects require references.
ReplyDeletehow to interpret the max value in question 4?is that means the max population of mosquitoes?
ReplyDeletewhen dp/dt equals to zero. sorry i don't remember much 'cause it's long time ago to me ==
ReplyDeletefor question 4...why dont use the method of "completing the square"?
ReplyDeletehow?
ReplyDeletethat is very simple method tat we hav ady learned since form 4... because i gt exactly the same max value with ur answer...but now i m confused...because tat method never mention abt any d2P/dt2 or d3p/dt3...is it necessary to use your method? because this project emphasized calculus
ReplyDeleteanyway thanks... today surprisingly found ur blog... ur blog is very nice... i hope ur asthma will recover soon... i got asthma too but now still under controlled... jiayou... you are a very talented person ^_^
ReplyDeletefor question 4,what is the answer for d3P/dt3?
ReplyDeletesock ying shi yun: I supposed, since we learn that the maximum value of a function is the value where its first derivative would be zero and its second derivative would be negative, so differentiation might be needed. By the way, thanks, I don't like my asthma but I'm living with it. =) All the best to you too!
ReplyDeleteunknown: can't remember, sorry. just remember it's a negative value.
i get d3P/dt3=r/k(k-2),dont have P to substitude.maybe i was wrong?dono how to prove its negative @@
ReplyDeleteermmm...sorry for disturb...for question 6, i wan to ask izit the the value of k should be around 1200 something??
ReplyDeleteif i not mistaken, my project is around 300 to 350..thats my teacher said
Deletemine one is 800
Deleteyup! Finally i found it around 800... my teacher said tat 700-900 for the value of k is acceptable...
DeleteThis comment has been removed by the author.
ReplyDeletefor question 5,when p=k the graph is a straight line instead of an S-shape curve right?
ReplyDeletewhat can i write in conclusion???
ReplyDeletehttp://www.scribd.com/doc/133661588/STPM-Mathematics-T-Assignment-B-Mathematical-Modelling-2013
ReplyDeleteunknown: ya.
ReplyDeletePeyton: just give an overall conclusion. briefly describe what you've done.
hey can you tell me briefly how to write methodology?? thanks
ReplyDeletefor question 5,my graph looks weird cause when i substitute the t inside the equation,it touches the horizontal asympptote,but based on the sample answer given in the text book,it shouldn't touches the horizontal asympptote,i just left with this graph only,hope u can reply my msg asap,thanks^^
ReplyDeleteMaybe you can make ur "t" value to be smaller a bit (eg. 0.1, 0.2, 0.3....)
Deletesori 4 disturbing...cn i noe hw to figure out wat value tat shud b substituted in the equation for the graph part in ques 5~ *totally lost* :((
DeleteArbitrary values, meaning any values you like as long as it demonstrates the changes you wanna prove. For example, P = 100, 200, 300 while k is 250 and r is 10. Any values.
DeleteMy teacher told me Methodology is actually the method you use to obtain the final answer (result). So the steps you use, in my teacher's opinion, is Methodology.
ReplyDeleteSorry can't help with the graph. I can't remember anything about it anymore. ==
for question 6, can i find dP/dt by dividing the difference between the time interval and the difference between the population??is it necessary to plot graph of P versus t to find dP/dt??
ReplyDeleteanybody can gimme pitures of their full assignment?
ReplyDeleteWhat is the shape of the graph for Q.6
ReplyDeletegraph on question 6 is actually like a maximum curve.
Deletedoes the Positive constant, r, vary? since someone told me that r can be vary, so does anyone here has any idea about it?
ReplyDeleteNot likely. Physics students would say a positive constant would never vary because it simply makes no sense. Numerical constant is there to only compensate for the discrepancies between a relation found.
ReplyDeletedo anyone know wat to write for the intro, methodology n conclusion? :)
ReplyDeleteJust write anything you like. According to my teacher, methodology is actually your steps, so I didn't have it. Find something about differential equation and its application on internet for introduction, and conclude briefly about every questions in conclusion.
Deletethnks ya~ ;)
Deletesori to disturb again~u hv done ur viva rite~wat kind of ques tat will b ask other thn the calculations~? :)
she asked what do i understand about the maximum value of a derivative.
Deletemy teacher asked what does a differential equation describes and who were the founders of calculus..
ReplyDeletecrazy ar? ==
Delete