Reference no: EM132269249

Part 1.

The application of mathematical models to biology has a long history. Ludwig von Berta-lanffy proposed the following simple model for the growth of various organisms in 1934:

dm/dt = k₁E - k₂m

where m is the mass of the organism at time t, E is its surface area, and k₁ and k₂ are constants.

Question 1. This differential equation is often presented in the following form:

dm/dt = am^2/3 - bm

Explain why this is equivalent.

Question 2. Note that this is a Bernoulli differential equation. Solve it analytically.

Question 3. What does lim_t→∞ m(t) represent, and how is it related to the constants a and b?

Question 4. Denoting the length of the organism by L = L(t), and using the assumption of the von Bertalanffy model that M is proportional to L³, show that your solution for M is equivalent to

L(t) = L_∞ (1- e^-Kt)

where K = b/3 is a positive constant.

Question 5. Now write k = e^-k, known as Ford's growth coefficient. Prove that the length of the organism at age t +1, i.e. L(t + 1), may be written as

L(t +1) = (1- k)L_∞ + kL(t)

Question 6. From (5), a plot of L(t + 1) versus L(t) should yield a straight line; this is called a Walford plot. What is the slope of this line? Where should this line intersect the diagonal L(t + 1) = L(t)?