Question

Consider a population N(t) modeled in continuous-time under the following assumptions: (1) the growth rate is propositional to the difference between the available food supply, fa, and the amount of food necessary for sustaining the existing population, fc; and (2) fc is proportional to the size of the population N. Formulate the model under these assumptions, being sure to specify the sign of any constants you use. Once you have formulated the model, notice that it is a type of model we have studied in lecture.

Answer 1

Explanation:

Given that, N(t) is continuous and it is model as

1. The rate of growth of N(t) is proportional to the difference between food supply (fa) and the food sustaining (fc)

Then,

dN(t)/dt ∝ fa - fc

2. fc is proportional to the size of the population N.

i.e.

fc ∝ N(t)

We have two proportions.

Analysis of each proportion.

1. dN(t)/dt ∝ fa - fc

Let β be the constant of proportionality

Then,

dN(t)/dt = β(fa - fc). Equation 1

Also for the second proportion

fc ∝ N(t)

Let γ be constant of proportionality

Then,

fc = γN(t). Equation 2

Substitute equation 2 into 1

So,

dN(t)/dt = β(fa - γN(t))

Divide both side by β

1/β dN(t)/dt = fa - γN(t)

Therefore, fa = 1/β dN(t)/dt + γN(t)

Note, 1/β is still a constant let Call it β again

Then,

fa = γN(t) + β dN(t)/dt

This is the model formulated

fa = γN(t) + β dN(t)/dt