Nonlinear Stochastic Modeling of Aphid Population Growth

James H. Matis and Thomas Kiffe

Texas A&M University

2

1) Introduction to Aphid Problem2) Deterministic Model3) Basic Stochastic Model4) Transformed Stochastic Model5) Approximate Solutions6) Generalized Stochastic Models7) Conclusions

3

1) Introduction Aphids are group of small, sap-sucking

insects which are serious pests of agricultural crops around the world.

The main economic impact of aphids in Texas is on cotton, e.g. $400 M crop loss in 91-92 in Texas.

Our study is on a pecan aphid, the black-margined aphid, Monellia caryella

4

Pecan orchards:

In West Texas In Mumford, TX, with 12 study plots

5

Four (4) adjacent trees were selected from the middle of each plot, and four (4) leaf clusters were sampled from each tree

Number of nymphs and adults were counted weekly

6

Qualitative characteristics: 1) Rapid collapse of aphid count after peak. 2) Considerable variability in aphid count on leaf

clusters

Mean number of nymphs and adults/cluster (n=192) from May to Sept., 2000

Number of nymphs on 4 clusters in Plot 1, Tree 1

7

Two general objectives:1) Predict peak infestation2) Predict cumulative aphid count

Useful facts about aphids:1) plants have chemical defense mechanism against aphids2) aphids secrete honeydew, which covers leaves and attracts other insects

8

Aphids have a fascinating life-cycle

9

2) Deterministic Model

Prajneshu (1998) develops an analytical model. Logic: honeydew ‘forms a weak cover on the leaf… and so causes starvation… The area covered at t is proportional to the cumulative (aphid) density.’

Model:

Solution:

where

0( )

tN N N N s ds

2( ) (1 ) ,bt btN t ae de

20

2

2 1/ 20

(1 )

2 /

( 2 )

N a d

b d a

b N

10

Property: Fitted Curves:

Parametersb = 2.320 2.540d = 58893 96649tmax = 4.73 4.52

λ = 2.320 2.542μ = 0.02357 0.02470N0 = 0.0077 0.0054

1max logt b d

11

Critique

1) Prajneshu model fits data well, but it is deterministic and symmetric

2) Consider extending model to include a) stochastic (demographic) variability b) asymmetric curves, with rapid collapse after peak value.

12

3) Basic Stochastic Model

Recall Prajneshu model: Let N(t) = current population size

C(t) = cumulative population size Assume: Given N(t)=n, C(t)=c

Prob{unit increase in N and C in Δt}=λnΔtProb{unit decrease in N}=μncΔt

For simplicity we assume:1) simple linear birthrate 2) no “intrinsic” death rate, as in (μ0n+μ1nc)Δt

13

Idealized model:λ=2.5 μ=0.01 N(0)=2

Simulations:

14

Numerical Solution

Find Kolmogorov equations with upper limits Nmax= 270, Cmax= 700. This gives about 200K equations.

Bivariate solution at t = 2.28.

10 = 108.120 = 563.430 = -12597

01 = 247.502 = 870303 = -151097

15

Consider cumulant functions from exact solution

N(t) C(t) joint

16

Comparison of deterministic solution with mean value function.

deterministic, N(t) mean, 10(t)

tmax 2.195 2.28peak 127 108.1shape symmetric right skewed

17

Claims for cumulant functions of N(t)

1) t max expectation = 2.28 10(2.28)=108.1 20(2.28)=563.4 30(2.28)=-12597

2) variance is curiously bimodalt max variance = 1.8 10(1.8)=84.2 20(1.8)=1065 30(1.8)=-8906

3) skewness changes signt max skewness = 3.3 10(3.3)=41.3 20(3.3)=805 30(3.3)=25919

18

Marginal distribution of N(t) at critical times:

t = 1.8, max variance negative skew.

t = 2.3, max expectation moderate skewness95% pred. int using Normal108.1 ± 2(23.7) = (61, 156)*consistent with data

t = 3.3, max skewnesspositive skew.

19

Claims for cumulant functions of C(t), (solid line)

01(∞)

= 499

02(∞)

= 1189

03(∞)

= -7860

Distribution of C(∞) is near symmetric 95% pred. int using Normal499 ± 2(34.5) = (430, 568)

20

Results:

For assumed stochastic model with assumed parameter values:

1) peak infestation is approximately normal2) final cumulative count is approximately normal3) peak infestation prediction is roughly consistent with data

Question:How can we implement this in practice?

21

4) Transformed Stochastic Model Let N(t) = current population size

D(t) = cumulative deaths

Clearly D(t)=C(t)-N(t) Compartmental Structure:

Assumptions: Given N(t)=n, D(t)=d

Prob{unit increase in N in Δt}=λnΔtProb{unit shift from in N to D in Δt}=μn(n+d)Δt

Two forces of mortality:crowding from live aphids (logistic type)

= μn 2

cumulative effect of dead aphids = μnd

22

Exact cumulant functions for N(t), same as before

t = 1.8, max variance negative skew.

t = 2.3, max expectation moderate skewness95% pred. int using

Normal108.1 ± 2(23.7) = (61,

156)*consistent with data

t = 3.3, max skewnesspositive skew.

23

Cumulant functions for D(t), dashed curves, lag those of C(t).

01(∞)

= 499

02(∞)

= 1189

03(∞)

= -7860

Distribution of D(∞) is same as that of C(∞)

95% pred. interval is (430, 568)

24

5) Approximate Solutions

Consider moment closure approximations for basic modelLet joint moment

moment gen. funct.

Claim:

Find diff. eq. for moments, mij(t)Transform to diff. eq. for cumulants, ij(t)

( ) [ ( ) ( )] ( , )i j thijm t E N t C t i j

1 2 1 2, , 0

( , , ) ( ) / ! !i jij

i j

M t m t i j

1 2 1

2

1 1 2

( 1) ( 1) .M M M

e et

25

Claim:

Note correspondence between 10 and 01 and deterministic model.

Set cumulants of order 4 or more to 0, and solve.

10 10 10 01 11

01 10

20 10 20 11 10 11 21 01 10 20

02 10 11

11 10 20 11 10 02 01 11 12

( )

( 2 ) ( 2 2 ( 2 ))

( 2 )

( ) ( )

26

Mean – adequate

Variance – underestimate

Skewness – poor (not surprising)

Results for cumulant approx. for N(t)

solid line – exactdashed line – approx.

27

Results for cumulant approx. for C(t)

Mean – excellent

Variance – equilibrium is ok

Skewness – equilibrium near 0

solid line – exactdashed line – approx.

28

Results for final cumulative count, C(∞)

Measure Exact Approx Error

mean, 01 498.6 498.7 0.02%variance, 02 1189.3 1159.5 2.5%skewness, 03 -7860 3493 –

95% predict. int.(Normal approx)

(431, 566) (432, 565) –

Marginal dist. of C(∞)

29

mgf:

cumulant equations:

Transformed model has: 1. more complex cumulant structure, however 2. approximations of cumulant counts are very close (±5%) to basic model.

1 1 2

2 2

21 1 1 2

( 1) ( 1) ,M M M M

e et

210 10 10 01 10 11 20

201 10 01 10 11 20

220 10 20 10 11 20 01 10 20

10 11 20 21 30

202 10 11 20 10 02 11 01 10 11

12 21

11 11 11

( )

( )

( 2 ) ( 2 )

2 ( 2 ) 2 2

2 2 ( 2 ) ( 2 )

2 2

220 10 10 11 02

01 10 11 20 10 20 12 30

( )

( ) 2

Consider approximations for transformed model

30

Results:

For assumed model, we have relatively simple moment closure approximations with:

1) adequate point prediction of peak infestation

2) adequate point and interval predictions of final cumulative count

31

6) Generalized Stochastic Models Consider the logistic population growth model

N = aN – bNs+1

s = 1 called ordinary logistic models > 1 called power law logistic model

Some past studies have suggested s > 1, e.g.1) empirical data on muskrat population growth2) theoretical considerations for Africanized bees, ‘r-strategists’

Consider similar models for aphidsBasic model : N = λN – μNCPower-law (cumulative) : N = λN – μNC 2

Power-law (dead) : N = λN – μN(N 2+D

2)

32

Results:

Power-law models fit data betterTable of s (Root MSE), using SCoP

Cluster 112 Cluster 113Basic 8.75 6.16P-L Cum 7.91 4.50P-L Dead 7.83 4.43

Cluster 113 - Basic

Cluster 113 – P-L Cum

33

7) Conclusions

1. Aphids have fascinating population dynamics. Net changes in current count, N(t), depend on cumulative count, C(t).

2. Relatively simple stochastic birth-death model gives good first approximation for peak infestation.

3. Moment closure approximations are adequate for interval predictions of final cumulative count.

4. Generalized, power-law dynamics give improved model with more rapid population collapse after peak.

34

Future Research

Expand study to other data- pecan aphids in other years, plots- cotton and other aphids

Explore statistical properties of power-law models.

Investigate moment closure approximations of power-law models.

Develop time-lag models, incorporating nymph and adult stages with minimum parameters.

Couple these models with degree-day models for predicting infestation onset and dynamic rates, λ and μ.