Population Growth - Malthus Model

Malthus Model I Let P(t) be the population at time t. I The more people there are present at a given time, the greater the birth rate (more people to breed). I Also, the greater the death rate (more people to die). I Then, the birth rate and death rate are directly proportional to how many people there are presently. In mathematical terms: dP = bP − dP = (b − d)P = kP, P(0) = P0 , dt where bP is the birth rate and dP is the death rate, k = b − d is the constant of proportionality, and P0 is the population at time t = 0. Solving the Malthus Model This is separable (also linear) so we will solve it as a linear equation. dP = kP dt 1 dP =k P dt ln |P| = kt + C P(t) = C1 e kt P(t) = P0 e kt Hence, we get exponential growth for the population. Example Suppose there are 200 bacteria in a culture at the start of the day. After 24 hours the population has doubled. Find the population of the culture after 36 hours from the start of our observation. Solution: P(t) = 200e kt 400 = 200e 24k ln(2) k = 24 P(t) = 200(2)t/24 P(36) = 200(2)36/24 ≈ 565.685 Drawbacks to the Malthus Model The Malthus Model assumes exponential growth forever. It takes into account death due to natural causes, but it does not take into account scarcity of resources leading to deaths due to human interaction. Logistic Growth Model I Still the birth rate is proportional to the population: birth rate = k1 P. I if there are P humans alive, the number of possible ≈ 12 P 2 . interactions between two humans resulting P(P−1) 2 I Then, the death rate due to “unnatural” causes is roughly proportional to P 2 : death rate = k2 P 2 . In Mathematical terms: dP = k1 P − k2 P 2 = −aP(P − b), P(0) = P0 . dt The solution to the IVP is: P(t) = bP0 . P0 − (P0 − b)e abt Example The population of the United States in 1990 was 248,709,873. In 2000, it was 281,421,906. In 2010, it was 308,745,538. Estimate the population in 2020 using the Logistic Growth Model. Solution: P(t) = 385784811.779 1 + 0.551143937817e −0.0396221410t