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

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2 1) Introduction to Aphid Problem 2) Deterministic Model 3) Basic Stochastic Model 4) Transformed Stochastic Model 5) Approximate Solutions 6) Generalized Stochastic Models 7) Conclusions

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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

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4 Pecan orchards: In West TexasIn Mumford, TX, with 12 study plots

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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

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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

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7 Two general objectives: 1) Predict peak infestation 2) Predict cumulative aphid count Useful facts about aphids: 1) plants have chemical defense mechanism against aphids 2) aphids secrete honeydew, which covers leaves and attracts other insects

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8 Aphids have a fascinating life-cycle

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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

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10 Property: Fitted Curves: Parameters b = 2.3202.540 d = 5889396649 t max = 4.734.52 λ = 2.3202.542 μ = 0.023570.02470 N 0 = 0.00770.0054

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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.

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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}=λnΔt Prob{unit decrease in N}=μncΔt For simplicity we assume: 1) simple linear birthrate 2) no intrinsic death rate, as in (μ 0 n+μ 1 nc)Δt

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13 Idealized model: λ=2.5 μ=0.01 N(0)=2 Simulations:

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14 Numerical Solution Find Kolmogorov equations with upper limits N max = 270, C max = 700. This gives about 200K equations. Bivariate solution at t = 2.28. 10 = 108.1 20 = 563.4 30 = -12597 01 = 247.5 02 = 8703 03 = -151097

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

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16 Comparison of deterministic solution with mean value function. deterministic, N(t) mean, 10 (t) t max 2.195 2.28 peak127 108.1 shapesymmetric right skewed

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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 bimodal t max variance = 1.8 10 (1.8)=84.2 20 (1.8)=1065 30 (1.8)=-8906 3)skewness changes sign t max skewness = 3.3 10 (3.3)=41.3 20 (3.3)=805 30 (3.3)=25919

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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.

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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 Normal 499 ± 2(34.5) = (430, 568)

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20 Results: For assumed stochastic model with assumed parameter values: 1) peak infestation is approximately normal 2) final cumulative count is approximately normal 3) peak infestation prediction is roughly consistent with data Question: How can we implement this in practice?

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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}=λnΔt Prob{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

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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.

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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)

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24 5) Approximate Solutions Consider moment closure approximations for basic model Let joint moment moment gen. funct. Claim: Find diff. eq. for moments, m ij (t) Transform to diff. eq. for cumulants, ij (t)

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

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

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

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28 Results for final cumulative count, C() MeasureExactApproxError mean, 01 498.6498.70.02% variance, 02 1189.31159.52.5% skewness, 03 -78603493– 95% predict. int. (Normal approx) (431, 566)(432, 565)– Marginal dist. of C()

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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. Consider approximations for transformed model

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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

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31 6) Generalized Stochastic Models Consider the logistic population growth model N = aN – bN s+1 s = 1 called ordinary logistic model s > 1 called power law logistic model Some past studies have suggested s > 1, e.g. 1) empirical data on muskrat population growth 2) theoretical considerations for Africanized bees, r-strategists Consider similar models for aphids Basic model : N = λN – μNC Power-law (cumulative) : N = λN – μNC 2 Power-law (dead) : N = λN – μN(N 2 +D 2 )

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32 Results: Power-law models fit data better Table of s (Root MSE), using SCoP Cluster 112Cluster 113 Basic8.756.16 P-L Cum7.914.50 P-L Dead7.834.43 Cluster 113 - BasicCluster 113 – P-L Cum

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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.

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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.

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