1. 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

2. Problem

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

4. 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

5. 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

6. 𝑼= 𝑵 𝑲
𝒅𝑵=𝑲 𝒅𝑼
𝟏 𝑲(𝑲𝑼 𝟏−𝑼 ) 𝑲𝒅𝑼= 𝒓𝒅𝒕

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

8. Result r0 K0 r K r0 K0 r K error iter 0.5 1000 0.1181969 1032.8 4 1
874.46 2
500
798.42 r0 K0 r K
Error (r<0)
(r=0.118, K=1032.8)
(r<0, K arbitrary)

9. Datapoints

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

11. 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

12. Reference
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