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## Nonlinear Stochastic Modeling of Aphid Population Growth

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**Nonlinear Stochastic Modeling of Aphid Population Growth**James H. Matis and Thomas Kiffe Texas A&M University**Introduction to Aphid Problem**• Deterministic Model • Basic Stochastic Model • Transformed Stochastic Model • Approximate Solutions • Generalized Stochastic Models • Conclusions**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**Pecan orchards:**In West Texas In Mumford, TX, with 12 study plots**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**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**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**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**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**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.**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**Idealized model:λ=2.5 μ=0.01 N(0)=2**• Simulations:**Numerical Solution**• Find Kolmogorov equations with upper limits Nmax= 270, Cmax= 700. This gives about 200K equations. • Bivariate solution at t = 2.28. 01 = 247.502 = 870303 = -151097 10 = 108.120 = 563.430 = -12597**N(t) C(t) joint**• Consider cumulant functions from exact solution**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**1) t max expectation =**2.2810(2.28)=108.120(2.28)=563.430(2.28)=-12597 2) variance is curiously bimodalt max variance = 1.810(1.8)=84.220(1.8)=106530(1.8)=-8906 3) skewness changes signt max skewness = 3.310(3.3)=41.320(3.3)=80530(3.3)=25919 • Claims for cumulant functions of N(t)**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.**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)**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?**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**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.**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)**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)**Claim:**Note correspondence between 10 and 01 and deterministic model. Set cumulants of order 4 or more to 0, and solve.**solid line – exactdashed line – approx.**Results for cumulant approx. for N(t) Mean – adequate Variance – underestimate Skewness – poor (not surprising)**solid line – exactdashed line – approx.**Results for cumulant approx. for C(t) Mean – excellent Variance – equilibrium is ok Skewness – equilibrium near 0**Results for final cumulative count, C(∞)**Marginal dist. of C(∞)**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.**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**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)**Results:**• Power-law models fit data betterTable of s (Root MSE), using SCoP Cluster 113 - Basic Cluster 113 – P-L Cum**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.**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.