We consider a certain population P, for which we denote by y(t) the size at time

Question

We consider a certain population P, for which we denote by y(t) the size at time t. A well-known model, the logistic model, assumes that, at any time t, the rate of change per unit of time of the population is proportional (with a positive constant of proportionality) to y(t)(1 - y(t)/r), where r > 0. Considering the sign of the rate of change (depending on the value of y(t)), give a biological inter­pretation for the constant r. Show that y satisfies an ODE of the kind y'(t) = ky(t)(1 - y(t)/r) for some constant number k > 0.

Explanation / Answer

1. y(t) is the current population and and the proportionality constant is positive so the sign of rate of change of population per unit time depends on 1 - y(t)/r. Since y(t)/r has a negative sign so it denotes the decrease in population. Therefore the constant r must denote the constant number of deaths occuring per unit of time.

2. As it is given the rate of change of population per unit time is proportional to y(t){1 - y(t)/r}.

Let the proportionality constant be k and k>0 as it is mentioned in the question.

Population at any time t is denoted by y(t). Therefore change in population per unit time is y '(t).

Hence y '(t) = k y(t) { 1 - y(t)/r }

Related Questions

Navigate

• Browse (All)
• Browse W
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.