Logistic Growth Model for Flower Beetles “They are one of the most common and most destructive insect pests for grain and other food products stored in silos, warehouses, grocery stores, and homes” Ling Lin, Fan Du

Problem

Background Pierre François Verhulst (1844–1845) Why we use logistic growth model Pierre François Verhulst (1844–1845) Population Growth, Biology, Chemistry, Biostatistics, Economics etc..

Variables t: time in Days N(t): population size at time t K: Carrying Capacity r: rate of growth t and N – given data K and r

Deriving PDF Given: 𝑵 𝒕 = 𝑲 𝑵 𝟎 𝑵 𝟎 + 𝑲− 𝑵 𝟎 𝒆 −𝒓𝒕 Design matrix: 𝒅𝑵 𝒅𝒓 = −𝑲 𝑵 𝟎 𝑵−𝑲 𝒕 𝒆 𝒕𝒓 ( 𝑵 𝟎 + 𝑲− 𝑵 𝟎 𝒆 −𝒓𝒕 ) 𝟐   𝒅𝑵 𝒅𝒌 = 𝑲 𝑵 𝟎 𝒆 −𝒓𝒕 𝑵 𝟎 + 𝑲− 𝑵 𝟎 𝒆 −𝒓𝒕 𝟐

𝑼= 𝑵 𝑲 𝒅𝑵=𝑲 𝒅𝑼 𝟏 𝑲(𝑲𝑼 𝟏−𝑼 ) 𝑲𝒅𝑼= 𝒓𝒅𝒕

Initial Fit <- plot(y_obs~t, data=data,main="Fitted Graph", col="black") Initial estimate K~1000

Result r0 K0 r K r0 K0 r K error iter 0.5 1000 0.1181969 1032.8 83247.74 4 1 0.3313653 -0.4683583 792.000001 874.46 951504.16 6004893 2 500 0.4065344 -0.0168056 792.104894 798.42 980314.26 r0 K0 r K Error (r<0) (r=0.118, K=1032.8) (r<0, K arbitrary)

Datapoints

Other Methods Newton-Raphson R-function optim Bounds for parameters

What We Learned Very sensitive to initial estimates Newton – Does not ensure ascent Gauss-Newton –Ascent for Invertible Matrix Bounds are important Table pro con

Reference https://fastly.kastatic.org/ka-perseus- images/69602c1370155fd480bb092161bb963905c5c212.png https://en.wikipedia.org/wiki/Logistic_function