THE SINGLE FIELD PROBLEM IN ECOLOGICAL MONITORING PROGRAM
|
|
- Bridget Gallagher
- 5 years ago
- Views:
Transcription
1 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
2 I. A brief review of the problems in ecological monitoring program
3 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.
4 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
5 The single trap spatial scale
6 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?
7 The single field spatial scale
8 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.
9 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 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, A real-world system: Potato aphid monitoring Fig. 1. Location of regions (boxes) and traps (circles) in Date is first capture of peachpotato aphid in the region (from Phil Northing, 2009)
10 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.
11 Control of information flows Single trap SINGLE FIELD Landscape
12 II. The single field problem: data processing for various spatial patterns
13 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.
14 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 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.
15 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 );
16 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 X
17 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 = 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
18 Pest population density: field data
19 Pest population density: field data
20 Pest population density: field data
21 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 = I MR, N 2 = e MR I TR, N 2 = e TR I SR, N 2 = e SR I LS, N 2 = e LS I stat, N 2 = e stat
22 HOW DOES THE INTEGRATION ERROR DEPEND ON THE SPATIAL PATTERN?
23 Spatial patterns in the single field problem Examples of the pest population density distribution Field data (New Zealand flatworm) Ecological model (patchy invasion) Y X (a) Y X (b)
24 Data generation The Rosenzweig MacArthur model: ( ) ( = D 2 U U + X 2 Y 2 U(X,Y,T ) T V (X,Y,T ) T ) 4ν (K U 0 ) 2 ( ) = D 2 V 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 Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulations. Chapman & Hall / CRC Press.
25 Pest population density distributions Example 1: A smooth distribution of the spatial density u(x, y) over the domain Y X
26 Pest population density distributions Example 2: A strongly heterogeneous patchy spatial distribution u(x, y) (computationally challenging!) Y X
27 Pest population density distributions The integration error Y a X A u(x,y) b e N t N
28 III. The single field problem: numerical integration of high aggregation data
29 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 X u(x,y)
30 Pest population density distributions The integration error 1 a e b Y u(x,y) N= X n r
31 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
32 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
33 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.
34 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 x x
35 One-dimensional problem: pest population density The number of humps increases for smaller values of d resulting in oscillations. U(x) 1.1 c x U(x) d x Malchow, H., Petrovskii, S.V. & Venturino, E Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulations. Chapman & Hall / CRC Press.
36 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) g(x) x x 0 x 1 x 2 x
37 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
38 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 I 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
39 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 γ(h) a γ(h) b 0.3 γ II d= γ II d= γ I γ II γ I 0.2 γ I γ II 0.1 γ I 0.1 δ 0.2 g h * h δg h * h
40 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) d= d= h
41 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) numerical theoretical x h
42 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
43 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
44 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 γ II 0.6 γ 0 γ III γ I /2 α t 1 2 α α t α
45 Transition from ultra-coarse grids to coarse grids u(x) = A(x x ) 2 + B, δ = p theor p num 0.6 p h
46 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.
47 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.
Biological turbulence as a generic mechanism of plankton patchiness
Biological turbulence as a generic mechanism of plankton patchiness Sergei Petrovskii Department of Mathematics, University of Leicester, UK... Lorentz Center, Leiden, December 8-12, 214 Plan of the talk
More informationBiological Invasion: Observations, Theory, Models, Simulations
Biological Invasion: Observations, Theory, Models, Simulations Sergei Petrovskii Department of Mathematics University of Leicester, UK. Midlands MESS: Leicester, March 13, 212 Plan of the Talk Introduction
More informationTuring and Non-Turing patterns in diffusive plankton model
Computational Ecology and Software 05 5(): 6-7 Article Turing and Non-Turing patterns in diffusive plankton model N. K. Thakur R. Gupta R. K. Upadhyay National Institute of Technology Raipur India Indian
More informationSpatiotemporal pattern formation in a prey-predator model under environmental driving forces
Home Search Collections Journals About Contact us My IOPscience Spatiotemporal pattern formation in a prey-predator model under environmental driving forces This content has been downloaded from IOPscience.
More informationEntomology in Focus 1. Cláudia P. Ferreira Wesley A.C. Godoy Editors. Ecological Modelling Applied to Entomology
Entomology in Focus 1 Cláudia P. Ferreira Wesley A.C. Godoy Editors Ecological Modelling Applied to Entomology Chapter 8 Computational Methods for Accurate Evaluation of Pest Insect Population Size Natalia
More informationLocal Collapses in the Truscott-Brindley Model
Math. Model. Nat. Phenom. Vol. 3, No. 4, 28, pp. 4-3 Local Collapses in the Truscott-Brindley Model I. Siekmann a and H. Malchow a a Institut für Umweltsystemforschung, Universität Osnabrück, 4976 Osnabrück,
More informationMathematical Theory of Biological Invasions
VI Southern-Summer School on Mathematical Biology Mathematical Theory of Biological Invasions Sergei Petrovskii Department of Mathematics, University of Leicester, UK http://www.math.le.ac.uk/people/sp237..
More informationGary G. Mittelbach Michigan State University
Community Ecology Gary G. Mittelbach Michigan State University Sinauer Associates, Inc. Publishers Sunderland, Massachusetts U.S.A. Brief Table of Contents 1 Community Ecology s Roots 1 PART I The Big
More informationApplication of POD-DEIM Approach on Dimension Reduction of a Diffusive Predator-Prey System with Allee effect
Application of POD-DEIM Approach on Dimension Reduction of a Diffusive Predator-Prey System with Allee effect Gabriel Dimitriu 1, Ionel M. Navon 2 and Răzvan Ştefănescu 2 1 The Grigore T. Popa University
More informationLECTURE 14 NUMERICAL INTEGRATION. Find
LECTURE 14 NUMERCAL NTEGRATON Find b a fxdx or b a vx ux fx ydy dx Often integration is required. However te form of fx may be suc tat analytical integration would be very difficult or impossible. Use
More informationPredation. Vine snake eating a young iguana, Panama. Vertebrate predators: lions and jaguars
Predation Vine snake eating a young iguana, Panama Vertebrate predators: lions and jaguars 1 Most predators are insects Parasitoids lay eggs in their hosts, and the larvae consume the host from the inside,
More informationHIGH-ORDER ACCURATE METHODS BASED ON DIFFERENCE POTENTIALS FOR 2D PARABOLIC INTERFACE MODELS
HIGH-ORDER ACCURATE METHODS BASED ON DIFFERENCE POTENTIALS FOR 2D PARABOLIC INTERFACE MODELS JASON ALBRIGHT, YEKATERINA EPSHTEYN, AND QING XIA Abstract. Highly-accurate numerical methods that can efficiently
More informationStability, dispersal and ecological networks. François Massol
Stability, dispersal and ecological networks François Massol June 1 st 2015 General theme Evolutionary ecology of fluxes o Evolution & ecology of dispersal o Spatial structure, networks of populations
More informationAn Introduction to Numerical Methods for Differential Equations. Janet Peterson
An Introduction to Numerical Methods for Differential Equations Janet Peterson Fall 2015 2 Chapter 1 Introduction Differential equations arise in many disciplines such as engineering, mathematics, sciences
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Numerical Methods for Partial Differential Equations Finite Difference Methods
More informationx n+1 = x n f(x n) f (x n ), n 0.
1. Nonlinear Equations Given scalar equation, f(x) = 0, (a) Describe I) Newtons Method, II) Secant Method for approximating the solution. (b) State sufficient conditions for Newton and Secant to converge.
More informationIntegration, differentiation, and root finding. Phys 420/580 Lecture 7
Integration, differentiation, and root finding Phys 420/580 Lecture 7 Numerical integration Compute an approximation to the definite integral I = b Find area under the curve in the interval Trapezoid Rule:
More informationOn Multigrid for Phase Field
On Multigrid for Phase Field Carsten Gräser (FU Berlin), Ralf Kornhuber (FU Berlin), Rolf Krause (Uni Bonn), and Vanessa Styles (University of Sussex) Interphase 04 Rome, September, 13-16, 2004 Synopsis
More informationIntroduction to Spatial Data and Models
Introduction to Spatial Data and Models Sudipto Banerjee 1 and Andrew O. Finley 2 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. 2 Department of Forestry
More informationCS 365 Introduction to Scientific Modeling Fall Semester, 2011 Review
CS 365 Introduction to Scientific Modeling Fall Semester, 2011 Review Topics" What is a model?" Styles of modeling" How do we evaluate models?" Aggregate models vs. individual models." Cellular automata"
More informationScientific Computing: Numerical Integration
Scientific Computing: Numerical Integration Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course MATH-GA.2043 or CSCI-GA.2112, Fall 2015 Nov 5th, 2015 A. Donev (Courant Institute) Lecture
More informationIntroduction to Spatial Data and Models
Introduction to Spatial Data and Models Sudipto Banerjee 1 and Andrew O. Finley 2 1 Department of Forestry & Department of Geography, Michigan State University, Lansing Michigan, U.S.A. 2 Biostatistics,
More informationCOURSE Numerical integration of functions
COURSE 6 3. Numerical integration of functions The need: for evaluating definite integrals of functions that has no explicit antiderivatives or whose antiderivatives are not easy to obtain. Let f : [a,
More informationHamburger Beiträge zur Angewandten Mathematik
Hamburger Beiträge zur Angewandten Mathematik Numerical analysis of a control and state constrained elliptic control problem with piecewise constant control approximations Klaus Deckelnick and Michael
More informationInverse Transport Problems and Applications. II. Optical Tomography and Clear Layers. Guillaume Bal
Inverse Transport Problems and Applications II. Optical Tomography and Clear Layers Guillaume Bal Department of Applied Physics & Applied Mathematics Columbia University http://www.columbia.edu/ gb23 gb23@columbia.edu
More informationComplex Population Dynamics in Heterogeneous Environments: Effects of Random and Directed Animal Movements
Int. J. Nonlinear Sci. Numer. Simul., Vol.13 (2012), pp. 299 309 Copyright 2012 De Gruyter. DOI 10.1515/ijnsns-2012-0115 Complex Population Dynamics in Heterogeneous Environments: Effects of Random and
More information[2] (a) Develop and describe the piecewise linear Galerkin finite element approximation of,
269 C, Vese Practice problems [1] Write the differential equation u + u = f(x, y), (x, y) Ω u = 1 (x, y) Ω 1 n + u = x (x, y) Ω 2, Ω = {(x, y) x 2 + y 2 < 1}, Ω 1 = {(x, y) x 2 + y 2 = 1, x 0}, Ω 2 = {(x,
More informationarxiv: v1 [math.ds] 14 May 2015
A two-patch prey-predator model with dispersal in predators driven by the strength of predation Yun Kang 1, Sourav Kumar Sasmal, and Komi Messan 3 arxiv:1505.0380v1 [math.ds] 14 May 015 Abstract Foraging
More informationApplied Numerical Mathematics. High-order numerical schemes based on difference potentials for 2D elliptic problems with material interfaces
Applied Numerical Mathematics 111 (2017) 64 91 Contents lists available at ScienceDirect Applied Numerical Mathematics www.elsevier.com/locate/apnum High-order numerical schemes based on difference potentials
More informationBifurcations and chaos in a predator prey system with the Allee effect
Received 11 February Accepted 5 February Published online May Bifurcations and chaos in a predator prey system with the Allee effect Andrew Morozov 1,, Sergei Petrovskii 1,* and Bai-Lian Li 1 1 Ecological
More informationHIGH-ORDER ACCURATE METHODS BASED ON DIFFERENCE POTENTIALS FOR 2D PARABOLIC INTERFACE MODELS
HIGH-ORDER ACCURATE METHODS BASED ON DIFFERENCE POTENTIALS FOR 2D PARABOLIC INTERFACE MODELS JASON ALBRIGHT, YEKATERINA EPSHTEYN, AND QING XIA Abstract. Highly-accurate numerical methods that can efficiently
More information1 Assignment 1: Nonlinear dynamics (due September
Assignment : Nonlinear dynamics (due September 4, 28). Consider the ordinary differential equation du/dt = cos(u). Sketch the equilibria and indicate by arrows the increase or decrease of the solutions.
More informationTransition to spatiotemporal chaos can resolve the paradox of enrichment
Ecological Complexity 1 (24) 37 47 Transition to spatiotemporal chaos can resolve the paradox of enrichment Sergei Petrovskii a,, Bai-Lian Li b, Horst Malchow c a Shirshov Institute of Oceanology, Russian
More informationPhysics 115/242 Romberg Integration
Physics 5/242 Romberg Integration Peter Young In this handout we will see how, starting from the trapezium rule, we can obtain much more accurate values for the integral by repeatedly eliminating the leading
More informationStochastic Spectral Approaches to Bayesian Inference
Stochastic Spectral Approaches to Bayesian Inference Prof. Nathan L. Gibson Department of Mathematics Applied Mathematics and Computation Seminar March 4, 2011 Prof. Gibson (OSU) Spectral Approaches to
More informationCommunity Structure. Community An assemblage of all the populations interacting in an area
Community Structure Community An assemblage of all the populations interacting in an area Community Ecology The ecological community is the set of plant and animal species that occupy an area Questions
More informationOptimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 1/36
Optimal multilevel preconditioning of strongly anisotropic problems. Part II: non-conforming FEM. Svetozar Margenov margenov@parallel.bas.bg Institute for Parallel Processing, Bulgarian Academy of Sciences,
More informationNUMERICAL STUDY OF CONVECTION DIFFUSION PROBLEM IN TWO- DIMENSIONAL SPACE
IJRRAS 5 () November NUMERICAL STUDY OF CONVECTION DIFFUSION PROBLEM IN TWO- DIMENSIONAL SPACE. K. Sharath Babu,* & N. Srinivasacharyulu Research Scholar, Department of Mathematics, National Institute
More informationLECTURE 16 GAUSS QUADRATURE In general for Newton-Cotes (equispaced interpolation points/ data points/ integration points/ nodes).
CE 025 - Lecture 6 LECTURE 6 GAUSS QUADRATURE In general for ewton-cotes (equispaced interpolation points/ data points/ integration points/ nodes). x E x S fx dx hw' o f o + w' f + + w' f + E 84 f 0 f
More informationBayesian Hierarchical Models
Bayesian Hierarchical Models Gavin Shaddick, Millie Green, Matthew Thomas University of Bath 6 th - 9 th December 2016 1/ 34 APPLICATIONS OF BAYESIAN HIERARCHICAL MODELS 2/ 34 OUTLINE Spatial epidemiology
More informationIterative Methods for Linear Systems
Iterative Methods for Linear Systems 1. Introduction: Direct solvers versus iterative solvers In many applications we have to solve a linear system Ax = b with A R n n and b R n given. If n is large the
More informationFinite difference method for elliptic problems: I
Finite difference method for elliptic problems: I Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore 560065 http://math.tifrbng.res.in/~praveen
More information7 Planar systems of linear ODE
7 Planar systems of linear ODE Here I restrict my attention to a very special class of autonomous ODE: linear ODE with constant coefficients This is arguably the only class of ODE for which explicit solution
More informationOikos. Appendix 1 and 2. o20751
Oikos o20751 Rosindell, J. and Cornell, S. J. 2013. Universal scaling of species-abundance distributions across multiple scales. Oikos 122: 1101 1111. Appendix 1 and 2 Universal scaling of species-abundance
More informationCOURSE Numerical integration of functions (continuation) 3.3. The Romberg s iterative generation method
COURSE 7 3. Numerical integration of functions (continuation) 3.3. The Romberg s iterative generation method The presence of derivatives in the remainder difficulties in applicability to practical problems
More informationApplication of POD-DEIM Approach for Dimension Reduction of a Diffusive Predator-Prey System with Allee Effect
Application of POD-DEIM Approach for Dimension Reduction of a Diffusive Predator-Prey System with Allee Effect Gabriel Dimitriu 1, Ionel M. Navon 2 and Răzvan Ştefănescu 2 1 The Grigore T. Popa University
More informationLecture 9 Approximations of Laplace s Equation, Finite Element Method. Mathématiques appliquées (MATH0504-1) B. Dewals, C.
Lecture 9 Approximations of Laplace s Equation, Finite Element Method Mathématiques appliquées (MATH54-1) B. Dewals, C. Geuzaine V1.2 23/11/218 1 Learning objectives of this lecture Apply the finite difference
More informationExam 3. Principles of Ecology. April 14, Name
Exam 3. Principles of Ecology. April 14, 2010. Name Directions: Perform beyond your abilities. There are 100 possible points (+ 9 extra credit pts) t N t = N o N t = N o e rt N t+1 = N t + r o N t (1-N
More informationGaussian with mean ( µ ) and standard deviation ( σ)
Slide from Pieter Abbeel Gaussian with mean ( µ ) and standard deviation ( σ) 10/6/16 CSE-571: Robotics X ~ N( µ, σ ) Y ~ N( aµ + b, a σ ) Y = ax + b + + + + 1 1 1 1 1 1 1 1 1 1, ~ ) ( ) ( ), ( ~ ), (
More informationPlant responses to climate change in the Negev
Ben-Gurion University of the Negev Plant responses to climate change in the Negev 300 200 150? Dr. Bertrand Boeken Dry Rangeland Ecology and Management Lab The Wyler Dept. of Dryland Agriculture Jacob
More informationAsymptotically safe Quantum Gravity. Nonperturbative renormalizability and fractal space-times
p. 1/2 Asymptotically safe Quantum Gravity Nonperturbative renormalizability and fractal space-times Frank Saueressig Institute for Theoretical Physics & Spinoza Institute Utrecht University Rapporteur
More informationArthropod Containment in Plant Research. Jian J Duan & Jay Bancroft USDA ARS Beneficial Insects Research Unit Newark, Delaware
Arthropod Containment in Plant Research Jian J Duan & Jay Bancroft USDA ARS Beneficial Insects Research Unit Newark, Delaware What we do at USDA ARS BIIRU - To develop biological control programs against
More informationdiscrete variation (e.g. semelparous populations) continuous variation (iteroparous populations)
Basic demographic models N t+1 = N t dn/dt = r N discrete variation (e.g. semelparous populations) continuous variation (iteroparous populations) Where r is the intrinsic per capita rate of increase of
More informationPACKAGE LMest FOR LATENT MARKOV ANALYSIS
PACKAGE LMest FOR LATENT MARKOV ANALYSIS OF LONGITUDINAL CATEGORICAL DATA Francesco Bartolucci 1, Silvia Pandofi 1, and Fulvia Pennoni 2 1 Department of Economics, University of Perugia (e-mail: francesco.bartolucci@unipg.it,
More informationVariance Reduction and Ensemble Methods
Variance Reduction and Ensemble Methods Nicholas Ruozzi University of Texas at Dallas Based on the slides of Vibhav Gogate and David Sontag Last Time PAC learning Bias/variance tradeoff small hypothesis
More informationAnnouncements. Topics: Homework:
Announcements Topics: - sections 7.5 (additional techniques of integration), 7.6 (applications of integration), * Read these sections and study solved examples in your textbook! Homework: - review lecture
More informationAdvanced computational methods X Selected Topics: SGD
Advanced computational methods X071521-Selected Topics: SGD. In this lecture, we look at the stochastic gradient descent (SGD) method 1 An illustrating example The MNIST is a simple dataset of variety
More informationOn second order sufficient optimality conditions for quasilinear elliptic boundary control problems
On second order sufficient optimality conditions for quasilinear elliptic boundary control problems Vili Dhamo Technische Universität Berlin Joint work with Eduardo Casas Workshop on PDE Constrained Optimization
More informationFokker-Planck model for movement of the cara - bid beetle Pterostichus melanarius in arable land: Model selection and parameterization
Fokker-Planck model for movement of the cara - bid beetle Pterostichus melanarius in arable land: Model selection and parameterization Herman N.C. Berghuijs 1,2, Bas Allema 2, Lia Hemerik 1, Wopke van
More informationA population is a group of individuals of the same species occupying a particular area at the same time
A population is a group of individuals of the same species occupying a particular area at the same time Population Growth As long as the birth rate exceeds the death rate a population will grow Immigration
More informationA note on accurate and efficient higher order Galerkin time stepping schemes for the nonstationary Stokes equations
A note on accurate and efficient higher order Galerkin time stepping schemes for the nonstationary Stokes equations S. Hussain, F. Schieweck, S. Turek Abstract In this note, we extend our recent work for
More information(x 3)(x + 5) = (x 3)(x 1) = x + 5. sin 2 x e ax bx 1 = 1 2. lim
SMT Calculus Test Solutions February, x + x 5 Compute x x x + Answer: Solution: Note that x + x 5 x x + x )x + 5) = x )x ) = x + 5 x x + 5 Then x x = + 5 = Compute all real values of b such that, for fx)
More information5 Applying the Fokker-Planck equation
5 Applying the Fokker-Planck equation We begin with one-dimensional examples, keeping g = constant. Recall: the FPE for the Langevin equation with η(t 1 )η(t ) = κδ(t 1 t ) is = f(x) + g(x)η(t) t = x [f(x)p
More informationMathematical Modelling of Plankton-Oxygen Dynamics under the Climate Change. Yadigar Sekerci & Sergei Petrovskii 1
Mathematical Modelling of Plankton-Oxygen Dynamics under the Climate Change Yadigar Sekerci & Sergei Petrovskii Department of Mathematics, University of Leicester, University Road, Leicester LE 7RH, U.K.
More informationNumerical Algorithm for Optimal Control of Continuity Equations
Numerical Algorithm for Optimal Control of Continuity Equations Nikolay Pogodaev Matrosov Institute for System Dynamics and Control Theory Lermontov str., 134 664033 Irkutsk, Russia Krasovskii Institute
More informationBIOS 230 Landscape Ecology. Lecture #32
BIOS 230 Landscape Ecology Lecture #32 What is a Landscape? One definition: A large area, based on intuitive human scales and traditional geographical studies 10s of hectares to 100s of kilometers 2 (1
More informationGAMM-workshop in UQ, TU Dortmund. Characterization of fluctuations in stochastic homogenization. Mitia Duerinckx, Antoine Gloria, Felix Otto
GAMM-workshop in UQ, TU Dortmund Characterization of fluctuations in stochastic homogenization Mitia Duerinckx, Antoine Gloria, Felix Otto Max Planck Institut für Mathematik in den Naturwissenschaften,
More informationBIOS 6150: Ecology Dr. Stephen Malcolm, Department of Biological Sciences
BIOS 6150: Ecology Dr. Stephen Malcolm, Department of Biological Sciences Week 7: Dynamics of Predation. Lecture summary: Categories of predation. Linked prey-predator cycles. Lotka-Volterra model. Density-dependence.
More informationAnnouncements. Topics: Homework:
Announcements Topics: - sections 7.3 (the definite integral +area), 7.4 (FTC), 7.5 (additional techniques of integration) * Read these sections and study solved examples in your textbook! Homework: - review
More informationStatistics: Learning models from data
DS-GA 1002 Lecture notes 5 October 19, 2015 Statistics: Learning models from data Learning models from data that are assumed to be generated probabilistically from a certain unknown distribution is a crucial
More informationStatistics of stochastic processes
Introduction Statistics of stochastic processes Generally statistics is performed on observations y 1,..., y n assumed to be realizations of independent random variables Y 1,..., Y n. 14 settembre 2014
More informationJean-Michel Billiot, Jean-François Coeurjolly and Rémy Drouilhet
Colloque International de Statistique Appliquée pour le Développement en Afrique International Conference on Applied Statistics for Development in Africa Sada 07 nn, 1?? 007) MAXIMUM PSEUDO-LIKELIHOOD
More informationA brief introduction to finite element methods
CHAPTER A brief introduction to finite element methods 1. Two-point boundary value problem and the variational formulation 1.1. The model problem. Consider the two-point boundary value problem: Given a
More informationA first order divided difference
A first order divided difference For a given function f (x) and two distinct points x 0 and x 1, define f [x 0, x 1 ] = f (x 1) f (x 0 ) x 1 x 0 This is called the first order divided difference of f (x).
More informationPrey-Predator Model with a Nonlocal Bistable Dynamics of Prey. Received: 5 February 2018; Accepted: 5 March 2018; Published: 8 March 2018
mathematics Article Prey-Predator Model with a Nonlocal Bistable Dynamics of Prey Malay Banerjee 1, *,, Nayana Mukherjee 1, and Vitaly Volpert 2, 1 Department of Mathematics and Statistics, IIT Kanpur,
More informationSpectral Methods for Reaction Diffusion Systems
WDS'13 Proceedings of Contributed Papers, Part I, 97 101, 2013. ISBN 978-80-7378-250-4 MATFYZPRESS Spectral Methods for Reaction Diffusion Systems V. Rybář Institute of Mathematics of the Academy of Sciences
More informationAreal data. Infant mortality, Auckland NZ districts. Number of plant species in 20cm x 20 cm patches of alpine tundra. Wheat yield
Areal data Reminder about types of data Geostatistical data: Z(s) exists everyhere, varies continuously Can accommodate sudden changes by a model for the mean E.g., soil ph, two soil types with different
More informationMicroscale patterns of habitat fragmentation and disturbance events as a result of chemical applications:
Microscale patterns of habitat fragmentation and disturbance events as a result of chemical applications: effects on Folsomia candida (Collembola) populations Mattia Meli Annemette Palmqvist, Valery E.
More informationA High Order Conservative Semi-Lagrangian Discontinuous Galerkin Method for Two-Dimensional Transport Simulations
Motivation Numerical methods Numerical tests Conclusions A High Order Conservative Semi-Lagrangian Discontinuous Galerkin Method for Two-Dimensional Transport Simulations Xiaofeng Cai Department of Mathematics
More informationDYNAMICS IN 3-SPECIES PREDATOR-PREY MODELS WITH TIME DELAYS. Wei Feng
DISCRETE AND CONTINUOUS Website: www.aimsciences.org DYNAMICAL SYSTEMS SUPPLEMENT 7 pp. 36 37 DYNAMICS IN 3-SPECIES PREDATOR-PREY MODELS WITH TIME DELAYS Wei Feng Mathematics and Statistics Department
More informationErkut Erdem. Hacettepe University February 24 th, Linear Diffusion 1. 2 Appendix - The Calculus of Variations 5.
LINEAR DIFFUSION Erkut Erdem Hacettepe University February 24 th, 2012 CONTENTS 1 Linear Diffusion 1 2 Appendix - The Calculus of Variations 5 References 6 1 LINEAR DIFFUSION The linear diffusion (heat)
More informationWhat is insect forecasting, and why do it
Insect Forecasting Programs: Objectives, and How to Properly Interpret the Data John Gavloski, Extension Entomologist, Manitoba Agriculture, Food and Rural Initiatives Carman, MB R0G 0J0 Email: jgavloski@gov.mb.ca
More informationSchwarz Preconditioner for the Stochastic Finite Element Method
Schwarz Preconditioner for the Stochastic Finite Element Method Waad Subber 1 and Sébastien Loisel 2 Preprint submitted to DD22 conference 1 Introduction The intrusive polynomial chaos approach for uncertainty
More informationMean Field Games on networks
Mean Field Games on networks Claudio Marchi Università di Padova joint works with: S. Cacace (Rome) and F. Camilli (Rome) C. Marchi (Univ. of Padova) Mean Field Games on networks Roma, June 14 th, 2017
More informationRecovery-Based A Posteriori Error Estimation
Recovery-Based A Posteriori Error Estimation Zhiqiang Cai Purdue University Department of Mathematics, Purdue University Slide 1, March 2, 2011 Outline Introduction Diffusion Problems Higher Order Elements
More informationThe Royal Entomological Society Journals
Read the latest Virtual Special Issues from The Royal Entomological Society Journals Click on the buttons below to view the Virtual Special Issues Agricultural and Forest Pests Introduction This virtual
More informationTable of Contents. II. PDE classification II.1. Motivation and Examples. II.2. Classification. II.3. Well-posedness according to Hadamard
Table of Contents II. PDE classification II.. Motivation and Examples II.2. Classification II.3. Well-posedness according to Hadamard Chapter II (ContentChapterII) Crashtest: Reality Simulation http:www.ara.comprojectssvocrownvic.htm
More informationHopf bifurcations, and Some variations of diffusive logistic equation JUNPING SHIddd
Hopf bifurcations, and Some variations of diffusive logistic equation JUNPING SHIddd College of William and Mary Williamsburg, Virginia 23187 Mathematical Applications in Ecology and Evolution Workshop
More informationAsymptotic Analysis of First Passage Time Problems Inspired by Ecology
Bulletin of Mathematical Biology manuscript No. (will be inserted by the editor) Venu Kurella Justin C. Tzou Daniel Coombs Michael J. Ward Asymptotic Analysis of First Passage Time Problems Inspired by
More information7. Statistical estimation
7. Statistical estimation Convex Optimization Boyd & Vandenberghe maximum likelihood estimation optimal detector design experiment design 7 1 Parametric distribution estimation distribution estimation
More information7. E C. 5 B. 1 D E V E L O P A N D U S E M O D E L S T O E X P L A I N H O W O R G A N I S M S I N T E R A C T I N A C O M P E T I T I V E O R M U T
7. E C. 5 B. 1 D E V E L O P A N D U S E M O D E L S T O E X P L A I N H O W O R G A N I S M S I N T E R A C T I N A C O M P E T I T I V E O R M U T U A L L Y B E N E F I C I A L R E L A T I O N S H I
More information3.1 Review: x m. Fall
EE650R: Reliability Physics of Nanoelectronic Devices Lecture 3: Physical Reliability Models: Acceleration and Projection Date: Sept. 5, 006 ClassNotes: Ehtesham Islam Review: Robert Wortman 3. Review:
More information8 Ecosystem stability
8 Ecosystem stability References: May [47], Strogatz [48]. In these lectures we consider models of populations, with an emphasis on the conditions for stability and instability. 8.1 Dynamics of a single
More informationHomogenization Theory
Homogenization Theory Sabine Attinger Lecture: Homogenization Tuesday Wednesday Thursday August 15 August 16 August 17 Lecture Block 1 Motivation Basic Ideas Elliptic Equations Calculation of Effective
More informationGeneral Least Squares Fitting
Chapter 1 General Least Squares Fitting 1.1 Introduction Previously you have done curve fitting in two dimensions. Now you will learn how to extend that to multiple dimensions. 1.1.1 Non-linear Linearizable
More informationA population is a group of individuals of the same species, living in a shared space at a specific point in time.
A population is a group of individuals of the same species, living in a shared space at a specific point in time. A population size refers to the number of individuals in a population. Increase Decrease
More informationChapter 18. Remarks on partial differential equations
Chapter 8. Remarks on partial differential equations If we try to analyze heat flow or vibration in a continuous system such as a building or an airplane, we arrive at a kind of infinite system of ordinary
More informationPattern Formation, Long-Term Transients, and the Turing Hopf Bifurcation in a Space- and Time-Discrete Predator Prey System
Bull Math Biol DOI 10.1007/s11538-010-9593-5 ORIGINAL ARTICLE Pattern Formation, Long-Term Transients, and the Turing Hopf Bifurcation in a Space- and Time-Discrete Predator Prey System Luiz Alberto Díaz
More informationAlex Zerbini. National Marine Mammal Laboratory Alaska Fisheries Science Center, NOAA Fisheries
Alex Zerbini National Marine Mammal Laboratory Alaska Fisheries Science Center, NOAA Fisheries Introduction Abundance Estimation Methods (Line Transect Sampling) Survey Design Data collection Why do we
More informationAM205: Assignment 3 (due 5 PM, October 20)
AM25: Assignment 3 (due 5 PM, October 2) For this assignment, first complete problems 1, 2, 3, and 4, and then complete either problem 5 (on theory) or problem 6 (on an application). If you submit answers
More information