Nonlinear Stochastic Modeling of Aphid Population Growth

1. Nonlinear Stochastic Modeling of Aphid Population Growth James H. Matis and Thomas Kiffe Texas A&M University

2. Introduction to Aphid Problem • Deterministic Model • Basic Stochastic Model • Transformed Stochastic Model • Approximate Solutions • Generalized Stochastic Models • 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. 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 • Qualitative characteristics:1) Rapid collapse of aphid count after peak. 2) Considerable variability in aphid count on leaf clusters

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

10. Property: • Fitted Curves: • Parametersb = 2.320 2.540d = 58893 96649tmax = 4.73 4.52λ = 2.320 2.542μ = 0.02357 0.02470N0 = 0.0077 0.0054

11. Critique 1) Prajneshu model fits data well, but it is deterministic and symmetric2) Consider extending model to includea) 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 sizeC(t) = cumulative population size • Assume: Given N(t)=n, C(t)=cProb{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. 01 = 247.502 = 870303 = -151097 10 = 108.120 = 563.430 = -12597

15. N(t) C(t) joint • Consider cumulant functions from exact solution

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. 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 • Claims for cumulant functions of N(t)

18. Marginal distribution of N(t) at critical times: t = 1.8, max variance negative skew. t = 2.3, max expectation moderate skewness 95% pred. int using Normal 108.1 ± 2(23.7) = (61, 156) *consistent with data t = 3.3, max skewness positive skew.

19. Claims for cumulant functions of C(t), (solid line) 01(∞) = 499  02(∞) = 1189  03(∞) = -7860 Distribution of C(∞) is near symmetric95% 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 sizeD(t) = cumulative deathsClearly D(t)=C(t)-N(t) • Compartmental Structure: • Assumptions: Given N(t)=n, D(t)=dProb{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 skewness 95% pred. int using Normal 108.1 ± 2(23.7) = (61, 156) *consistent with data t = 3.3, max skewness positive 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)

25. Claim: Note correspondence between 10 and 01 and deterministic model. Set cumulants of order 4 or more to 0, and solve.

26. solid line – exactdashed line – approx. Results for cumulant approx. for N(t) Mean – adequate Variance – underestimate Skewness – poor (not surprising)

27. solid line – exactdashed line – approx. Results for cumulant approx. for C(t) Mean – excellent Variance – equilibrium is ok Skewness – equilibrium near 0

28. Results for final cumulative count, C(∞) Marginal dist. of C(∞)

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

30. Results: For assumed model, we have relatively simple moment closure approximations with:1) adequate point prediction of peak infestation2) adequate point and interval predictions of final cumulative count

31. 6) Generalized Stochastic Models • Consider the logistic population growth modelN = aN – bNs+1s = 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 2Power-law (dead) : N = λN – μN(N 2+D 2)

32. Results: • Power-law models fit data betterTable of s (Root MSE), using SCoP Cluster 113 - Basic Cluster 113 – P-L Cum

33. 7) Conclusions • Aphids have fascinating population dynamics. Net changes in current count, N(t), depend on cumulative count, C(t). • Relatively simple stochastic birth-death model gives good first approximation for peak infestation. • Moment closure approximations are adequate for interval predictions of final cumulative count. • 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.