A New Ecological Model for Insect Population Size
with Local Collapse
J.H. Matis, T.R. Kiffe, T.I. Matis, G. J. Michels, Jr.
Aphid Problem
Solutions
Linear Approximation
• Consider the Green Bug (GB)
1. Prajneshu (JISAS 1998)
• How well does Taylor series model predict Omax and Ocum ?
N (t ) = a exp(−bt )(1 + d exp(−bt ) )
−2
• Estimate F(∞), tmax, δ for each curve.
• Finding: As water increases, F(∞) and tmax, decrease, but δ increases.
• Profile plot of the mean δ vs. water:
Omax for n=27 GB cases
a, b, d are functions of λ, δ, No
2. Our Alternative 1 (JABES 2006)
let Nmax= peak count, tmax= time of peak
N (t ) =
GB is one of 4000 aphid species. Aphids are very destructive
Data
4 N max exp(− b(t − t max ) )
[1 + exp(− b(t − tmax ) )]2
(3)
3. Our Alternative 2
• 12 years of GB counts, approx. weekly, at TAMU Bushland
Station
• Multiple regression equation, with R2=0.98:
b(t − t max ) ⎤
N (t ) = N max sech 2 ⎡
2 ⎥⎦
⎢⎣
log(Omax)=-0.48+0.60λ-0.48log(δ)
• Note δ increased linearly with water in 2003 (a dry year)
• Also, λ and δ decreased with nitrogen in 2004 (not a dry year)
2. Relate λ and δ for GB to ladybug (predator) abundance.
(4)
• Plot of ladybug, GB, corn leaf aphid abundance for 3 years
Ocum for n=27 GB cases
• Fit (3) or (4) to data. Could link to (2) using:
λ = b(d − 1) (d + 1)
2
δ = b 2N
max
•Each curve has local collapse. Max ranges from 20 to 2000.
We want population size model for GB (and other species)
Logistic Model
No =
4dN max
(1 + d ) 2
where d = exp(bt max )
Model is symmetric with greater tails than the Normal
• Let N(t) = current count
• Assume
N′(t)=(r-sN)N
(1)
Goodness-of-Fit for GB
• Multiple regression equation, with R2=0.99:
log(Ocum)=0.68+0.38λ-0.98log(δ),
1st
Claim 2: Linear regression model, based on order Taylor
series approx, gives exceptional predictions of two
endpoints, Omax and Ocum, based on mechanistic parameters
λ and δ for GB. Same is true for other aphids. This
approach using λ and δ is new and transparent.
Claim 3: Growth rate parameters λ and δ are implicit functions of
ambient conditions.
Summary
⎛K−N⎞
⎟N
• Alternative with carrying capacity K=s/r, N ' (t ) = r ⎜
⎝ N ⎠
• Solution:
K
N (t ) =
(K − N 0 ) ⎤ exp(−rt )
1+ ⎡
N 0 ⎥⎦
⎢⎣
Implementation
• postulate λ and δ
• predict Omax and Ocum
• Equilibrium solution is K
• Fact: λ and δ explain almost all variability in Omax and Ocum
(R2 > 0.95 in all such aphid studies)
2. New approach to prediction:
• observe local conditions
• Implication: λ and δ are implicit functions of local weather,
predator, management, etc. conditions
Logistic model doesn’t fit aphid data!
Biological Principles of New Model
1. Aphids are parthenogenetic (virgin birth) and viviparous (live
birth)
Fit is near perfect
Claim 1: Model fits GB data very well. Same is true for
pecan, mustard, cotton and corn leaf aphids
• Assume birth rate = λN
2. Aphid population growth is constrained by the “cumulative
size” of past population
• Let F(t)=∫N(t)=cumulative density
Assume death rate = δFN
New mechanistic model:
N ' (t ) = (λ − δF ) N
(2)
1. New mechanistic model gives near-perfect fit for aphid data
Approach for Prediction
• Model predicts the full curve.
Consider predicting only two endpoints:
y1=Omax (peak count) and y2=Ocum (cum density)
• Using full model, assume yi=fi[λ,δ,N(0)]+εi ,
from whence log(yi)=gi[λ,log(δ),N(0)]+ε′i .
Taylor series approx: log(yi)=c0i+ c1i λ+ c2i log(δ)+ε′′i
• Our current research seeks to:
1. Relate λ and δ to environmental variables.
Example, consider cotton aphid abundance in 2 factor exp
(water, nitrogen) in split-plot design in Texas in 2003.
Addendum
• Cotton aphid: another of the 4000 species