Many have asked me for a solution for Maths T assignment. Sadly I've handed in mine and now have only scribbles of important points so that I can prepare for my viva.
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.
