THE SINGLE FIELD PROBLEM IN ECOLOGICAL MONITORING PROGRAM

THE SINGLE FIELD PROBLEM IN ECOLOGICAL MONITORING PROGRAM Natalia Petrovskaya School of Mathematics, University of Birmingham, UK Midlands MESS 2011 November 8, 2011 Leicester, UK

I. A brief review of the problems in ecological monitoring program

Introduction: the ecological monitoring program The integrated pest control and management program monitoring of pest insects The information about pest population size is obtained through sampling. Once the samples are collected, the total number of the insects in the field is evaluated. The need in reliable methods to estimate the pest population size in order to avoid unjustified pesticides application and yet to prevent pest outbreaks.

Spatial scales in the monitoring problem Single trap link the trap count to the local population density Single field estimate the pest abundance over a certain area, e.g. a large agricultural field A line of pitfall traps, Kongsfjord. ( c S.J. Coulson) Landscape understand the dynamics of movement between different habitats

The single trap spatial scale

The single trap spatial scale A trap of radius r that has caught n insects after having been exposed for time T. The number n may not always reflect the value of the population density in a vicinity of the trap. How to restore the population density from the information available to us?

The single field spatial scale

The single field spatial scale A system of N traps installed at location r i, i = 1,..., N over the field. The trap counts give us the values u i, i = 1,..., N of the population density at the location of the traps. Evaluate the total number M of insects in the field from the discrete population density.

inoculations per plant probed in comparison to M. persicae made by the aphid vector in exactly the same situation. It is often calculated by dividing the number of successful inoculations per plant probed by the aphid, by that of M. persicae, which has a relative efficiency of 1.00. The landscape spatial scale The Potato Council Levy Payer Network The network was first set up in 2002, and for the first two years consisted of around a dozen traps in 4 geographically distinct regions (Scotland, North Yorkshire, East Anglia and Wiltshire). From 2004 onwards the network has consisted of around 100 traps in 8 regions (Fig 1). The number of sites per region is approximately proportional to the amount of seed grown in each region, hence many more traps in Scotland than in England. The data from all these traps remains accessible to levy payers via the website, http://aphmon.csl.gov.uk. A real-world system: Potato aphid monitoring Fig. 1. Location of regions (boxes) and traps (circles) in 2009. Date is first capture of peachpotato aphid in the region (from Phil Northing, 2009)

The landscape spatial scale A system of K domains (agricultural fields and non-farmed habitats). Each habitat is quantified by a single variable, e.g., the total population size M k, k = 1,..., K. Evaluate the cross-correlations between the pest abundance in different fields. http://www.pestwatch.psu.edu/sweetcorn/tool/tool.html

Control of information flows Single trap SINGLE FIELD Landscape

II. The single field problem: data processing for various spatial patterns

The single field problem The main issue: efficiency vs accuracy Financial and labor resources available for monitoring are always limited. Installment of many traps per a unit agricultural area would by itself bring a considerable damage to the agricultural product. The number N of traps installed in a field cannot be made big.

Introduction: statistical approach to integrate data Statistical analysis of the samples is a conventional approach in ecology. P.M.Davis, Statistics for describing populations. In Handbook of Sampling Methods for Arthropods in Agriculture (L.P. Pedigo & G.D. Buntin, Eds.), 1994, pp. 33-54. Boca Raton: CRC Press. ū 1 N N u i, so that M Aū, A is the area. i=1 N =? A lot of empirical knowledge, yet no rigorous procedure to evaluate N.

Introduction: the computational problem Alternative: numerical integration The problem statement Numerical integration on a coarse (uniform) grid where the number N of grid nodes is small and is fixed. No grid adaptation (a repeated trapping with an increased number of traps is not available in ecological applications because of impossibility to reproduce the initial conditions). We want to understand - what the minimum number N of grid nodes should be to achieve desirable accuracy; - how accurate the results can be on a given coarse grid of N nodes (N < N );

Numerical integration of field data Generate a uniform grid of N N nodes in the unit square. Consider the values u ij at grid nodes. 1 Field data (New Zealand flatworm) Y 0.8 0.6 0.4 0.2 11 10 9 8 7 6 5 4 3 2 1 0 0 0.2 0.4 0.6 0.8 1 X

The methods of numerical integration The Newton-Cotes formulas: Replace u(x, y) with polynomial PK n (x, y) of degree K in the vicinity of node n. Consider where I = 1 1 0 0 u(x, y)dxdy i,j I ij = PK n (x, y)dxdy. I ij = Ĩ, The integration error e = c ij I Ĩ I

Pest population density: field data

Pest population density: field data

Pest population density: field data

Numerical integration of field data Table: The population size I and the integration error e for various integration rules on the fine (N 2 = 121) and the coarse (N 2 = 9) grids. field data (a) (b) (c) (d) (e) (f) I, N 2 = 121 544 459 543 419 651 611 I MR, N 2 = 9 269 319 512 450 531 488 e MR 0.506 0.305 0.056 0.074 0.184 0.202 I TR, N 2 = 9 269 319 512 450 531 488 e TR 0.506 0.305 0.056 0.074 0.184 0.202 I SR, N 2 = 9 247 325 589 544 636 561 e SR 0.545 0.291 0.085 0.299 0.023 0.082 I LS, N 2 = 9 250 350 550 567 600 567 e LS 0.540 0.237 0.013 0.352 0.078 0.073 I stat, N 2 = 9 289 300 456 344 444 411 e stat 0.469 0.346 0.161 0.178 0.317 0.327

HOW DOES THE INTEGRATION ERROR DEPEND ON THE SPATIAL PATTERN?

Spatial patterns in the single field problem Examples of the pest population density distribution Field data (New Zealand flatworm) Ecological model (patchy invasion) Y 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 X (a) 11 10 9 8 7 6 5 4 3 2 1 Y 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 X (b) 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Data generation The Rosenzweig MacArthur model: ( ) ( = D 2 U 1 + 2 U + X 2 Y 2 U(X,Y,T ) T V (X,Y,T ) T ) 4ν (K U 0 ) 2 ( ) = D 2 V 2 + 2 U + κ AUV X 2 Y 2 U+B MV. U(U U 0 )(K U) AUV U+B, U and V are the densities of prey and predator at time T (T > 0) and position (X, Y ). Numerical simulation: u(x, y, t) = U(X, Y, T )/K, a rich variety of spatiotemporal patterns. Malchow, H., Petrovskii, S.V. & Venturino, E. 2008 Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulations. Chapman & Hall / CRC Press.

Pest population density distributions Example 1: A smooth distribution of the spatial density u(x, y) over the domain 0.8 0.6 Y 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 X

Pest population density distributions Example 2: A strongly heterogeneous patchy spatial distribution u(x, y) (computationally challenging!) 0.8 0.6 Y 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 X

Pest population density distributions The integration error Y 0.8 0.6 0.4 0.2 a 1 0.85 0.8 0.75 0 0 0.2 0.4 0.6 0.8 1 X A u(x,y) 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 10 0 10-1 10-2 b e 3 5 9 N 17 33 65 129 257 t N

III. The single field problem: numerical integration of high aggregation data

Model case: single peak distribution u(x, y) = U 0 4πσ 2 ( exp (x ˆx)2 + (y ŷ) 2 ) 4σ 2, where ˆr = (ˆx, ŷ) is a random variable Y 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 X u(x,y) 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

Pest population density distributions The integration error 1 a e b Y 0.8 0.6 0.4 u(x,y) 0.75 0.65 0.55 0.45 0.35 0.25 0.15 0.05 10 0 N=3 0.2 10-1 0 0 0.2 0.4 0.6 0.8 1 X 0 1 2 3 4 5 6 7 8 9 10 n r

Grids classification I Fine grids: the asymptotic error estimates are valid. II Coarse grids: no asymptotic error estimates, the accuracy can be evaluated from some additional information about the integrand function. III Ultra-coarse grids: the accuracy can only be evaluated from a probabilistic viewpoint. standard numerical integration problem N=3 III II I numerical integration in ecological applications N

Numerical integration on ultra-coarse grids - reduce the problem to the 1 d case; - consider a peak of the width δ; - consider ultra-coarse grids: the integration error is a random variable (it depends on the location of the peak with respect to the nearest grid node); U(x) U(x) x x

One-dimensional problem: data generation The Rosenzweig MacArthur model: u(x, t) t v(x, t) t = d 2 u + u(1 u) uv x 2 u + p, = d 2 v x 2 + k uv u + p mv. The function u(x, t) is the density of the pest insect (d, k, m, p are parameters). Interaction between reaction and diffusion results in pattern formation. The properties of the pattern depend on the value of dimensionless diffusivity d.

One-dimensional problem: pest population density For an intermediate value of d, the pattern can consist of just one or a few peaks only. U(x) a 1.2 U(x) b 0.3 1.1 1 0.25 0.9 0.8 0.7 0.2 0.6 0.5 0.15 0.4 0.3 0.1 0.2 0 0.2 0.4 0.6 0.8 1 x 0.1 0 0.2 0.4 0.6 0.8 1 x

One-dimensional problem: pest population density The number of humps increases for smaller values of d resulting in oscillations. U(x) 1.1 c 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.2 0.4 0.6 0.8 1 x U(x) d 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.2 0.4 0.6 0.8 1 x Malchow, H., Petrovskii, S.V. & Venturino, E. 2008 Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulations. Chapman & Hall / CRC Press.

Integration error on ultra-coarse grids u(x) g(x) = B A(x x 1 ) 2, x [x 0, x 2 ], I = x 2 x 0 g(x)dx = 2Bh 2Ah 3 3. u(x) u(x) 1 0.8 g(x) 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 x x 0 x 1 x 2 x

Ultra-coarse grids What is the probability of the event e 0.25? x i = x + γ h, γ [0, 1/2]. g(x) a g(x) b x 0 x I x i-1/2 x * x i x II x i+1/2 1 x 0 x I x i-1 x * x i x II x i+1 1

Ultra-coarse grids: one point configuration I = x 2 x 0 g(x)dx = 2Bh 2Ah 3 3, Ĩ = k=1 k=0 ( x k+1 x k p k 0 dx ) = h ( B Aγ 2 h 2). e 0.25 0.75I Ĩ 1.25I γ I (h, δ g ) = δ g 6h 5δg, 2h 6h γ II (h, δ g ) = δ g 2h δg. 2h 2h

Ultra-coarse grids: one point configuration The function γ(h) is shown for various values of the dimensionless diffusivity d. The part of the (h, γ)-plane between the γ I (h, δ g ) and γ II (h, δ g ) curves gives the parameter range where e 0.25. γ(h) a γ(h) b 0.3 γ II d=0.0001 0.3 γ II d=0.00001 0.2 γ I γ II γ I 0.2 γ I γ II 0.1 γ I 0.1 δ 0.2 g 0.4 0.6 0.8 h * h δg h * 0.2 0.4 0.6 0.8 h

Ultra-coarse grids: one point configuration The probability p(h) of having the error e 0.25 for various values of the dimensionless diffusivity d: p(h) = (γ II(h) γ I (h)). (1/2 0) p(h) 0.15 0.1 d=0.0001 0.05 d=0.00001 0 0.2 0.4 0.6 0.8 1 h

Example: Ultra-coarse grids: one point configuration u(x) = Ae 1 2 ( x a σ )2, the peak width δ 0.04 grids h = 1/k, k = 2, 3,..., 15, h > δ, a is a random variable (1000 realizations for each h) u(x) p(h) 50 0.15 40 30 0.1 numerical theoretical 20 0.05 10 0 0 0.2 0.4 0.6 0.8 1 x 0 0.1 0.2 0.3 0.4 0.5 h

Ultra-coarse grids: two point configuration What is the probability of the event e 0.25? x i = x + γ h, γ [0, 1/2]. g(x) a g(x) b x 0 x I x i-1/2 x * x i x II x i+1/2 1 x 0 x I x i-1 x * x i x II x i+1 1

Ultra-coarse grids: two point configuration h = αδ, α [1/2, 1] γ(α) γ II a γ (α) 1/2 b γ II γ I γ^ I γ ^ II γ 0 γ III 1 α^ α 1/2 γ I α α t 1

Ultra-coarse grids: two point configuration α [1/2, α t ], p(α) = 1, α [α t, 1], p(α) < 1, α t 0.84 γ (α) a p(α) b 1 D 1 D 2 0.8 γ II 0.6 γ 0 γ III γ I 0.4 0.2 1/2 α t 1 2 α α t α

Transition from ultra-coarse grids to coarse grids u(x) = A(x x ) 2 + B, δ = 0.06 1 0.8 p theor p num 0.6 p 0.4 0.2 0.05 0.1 0.15 0.2 0.25 0.3 h

Conclusions The information about species abundance is not independent and coupling between different spatial scales is required. Numerical integration provides a reliable alternative to the standard statistical approach in the single field problem. For high aggregation density distributions integration on ultra-coarse grids cannot provide the prescribed accuracy. Instead, the results of the integration should be treated probabilistically by considering the integration error as a random variable.

Future work Investigate how the information about pattern formation can be used to improve the accuracy of integral evaluation. The probabilistic approach applied on ultra-coarse grids should be extended to two-dimensional problems. Investigate how is the accuracy of the integration affected if the density function is known at nodes of an irregular grid? How is the accuracy of the integration affected if an agricultural field has an arbitrary shape? More advanced methods of numerical integration should be applied. Investigate the effect of measurement errors and/or noise. Noise would result in irregular spatial fluctuations in the population density across the domain, in addition to those produced by deterministic factors.