Logistic Growth

Abstract

Logistic growth is a simple model for predicting the size P (t) of a population as a function of the time t. In the most naive model of population growth, each couple produces β offspring (for some constant β) and then dies. Thus over the course of one generation β P (t) 2 children are produced and P (t) parents die so that the size of the population grows from P (t) to P (t + t g) = P (t) + β P (t) 2 − P (t) = β 2 P (t), where t g denotes the lifespan of one generation. The rate of change of the size of the population per unit time is P (t+t g)−P (t) t g = bP (t) where b = β−2 2t g is the net birthrate per member of the population per unit time. If we approximate P (t+t g)−P (t) t g ≈ dP dt (t) we get the differential equation P ′ (t) = bP (t) Logistic growth adds one more wrinkle to this model. It assumes that the population only has access to limited resources. As the size of the population grows the amount of food available to each member decreases. This in turn causes the net birth rate b to decrease. In the logistic growth model b = b 0 1 − P K , where K is called the carrying capacity of the environment, so that P ′ (t) = b 0 1 − P (t) K P (t) We can learn quite a bit about the behaviour of solutions to differential equations like this, without ever finding formulae for the solutions, just by watching the sign of P ′ (t). For concreteness we'll look at solutions of the differential equation dP dt (t) = 6000 − 3P (t) P (t) We'll sketch the graphs of four functions P (t) that obey this equation. • For the first function, P (0) = 0. The sketchs will be based on the observation that dP dt (t)      = 0 if P (t) = 0 > 0 if 0 < P (t) < 2000 = 0 if P (t) = 2000 < 0 if P (t) > 2000