ELEMENTARY SPANISH SPAN 3
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This 8 page Class Notes was uploaded by Daron Kemmer IV on Thursday October 22, 2015. The Class Notes belongs to SPAN 3 at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 20 views. For similar materials see /class/226851/span-3-university-of-california-santa-barbara in Spanish at University of California Santa Barbara.
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Date Created: 10/22/15
LINEARIZATION OF NONLINEAR ODES We have introduced the idea of a mathematical model for some real world situation and used models to find ODEs It is possible to have several models of varying levels of complexity Each gives a different ODE We also have techniques for solving different classes of ODEs in cluding one class called linear39 This lecture ties these two ideas together We started with New ton39s Law of Cooling for temperature y as a function of time t The temperature y 0 represents an equilibrium state We do not know the relation between the temperature y and its rate of change y but we expect it is a function y f for some function f A straight line approximation to f at y 0 give the simplest model That is fy ky so the ODE y ky is an approximation to the true39relationship We can try this with any autonomous ODE y f 2 Example Suppose a one celled organism has the shape of a sphere of radius y which is a function of time t The VON BERTALANFFY growth model assumes that the growth ie the rate of change of the radius with respect to time is in uenced by two things 1 The intake of nutrients which occurs uniformly over the sur face area of the cell which causes the cell to grow The surface area is 47ry2 so the ODE has a term ayz where a gt 0 is the con stant of proportionality which can absorb the 47139 2 The respiration of waste generated which is proportional to the volume gwyg acts to slow the growth So there is also a term ibyg where b gt 0 is some other constant of proportion ality This gives Cl 2 3 dt 7 cm by This equation is AUTONOMOUS we are making the assumption that the growth of the cell depends on the radius but not the time of day For concreteness let39s assume a 2 and b l The ideas of 25 show there is a STABLE EQUILIBRIUM at y 2 1 Z LINEARIZATION OF NONLINEAR ODES The ODE y 212 7 y3 is separable and a lot of work shows the solution is given implicitly by 277 2621 06quot 9 No amount of algebra ever will let you solve for y as a function of t The equilbrium solution y 2 corresponds to O 0 But the formula for the solution does not show that y 2 is stable or show the behavior near the equilibrium What we will do is change the model to a simpler one We approx imate the function f 212 7 y3 by a tangent line approximation Ly near y 2 By Math 3A the derivative of f at 2 is 74 and so Ly 74y 7 2 Figure 1 shows fy and FIGURE 1 State Space or Phase Plane Since fy m Ly near y 2 we expect solutions to y fy should be close to solution to y Ly near y 2 This last ODE is now a LINEAR ODE y 74y7274y8 or y4y8 The solutions to the new ODE are yt 06quotquot 2 The constant solution y 2 corresponding to O 0 is the STEADY STATE solution and the other term 05quot is the TRANSIENT SOLU TION We see that solutions approach the equilibrium with expo nential decay in the linear model and so this is approxiamtely the behavior in the nonlinear model Figure 2 shows the graph of the solution to the nonlinear ODE above and the linear ODE below for the lVP y0 19 LINEARIZATION OF NONLINEAR ODES 3 FIGURE 2 Exact and approximate solutions We said a point yo is an EQUILIBRIUM if fy0 0 We need an extra definition the equilibrium yo is HYPERBOLIC if also f yo 7 0 Summary The proces of changing from a complicated nonlinear ODE to a simpler linear one is called LINEARIZATION It just means find ing the tangent line approximation Ly to f at the equilibrium and replacing the ODE y f with the linear ODE y We can do this at any hyperbolic equilibrium Notice that this is the first real application of the idea of tangent line approximations that you learned for apparently no reason in Math 3A We like linear ODEs for three reasons 1 They are easy to solve 2 They are based on the simplest possible model which we choose in the absence of a better one example Newton39s Law of Cooling 3 Even if we have a more detailed model the linear approxi mation gives better qualitative information than the solution to the original ODE example the cell growth model above Review exercises 1 Linearize each of the following ODEs at each hyperbolic equi librium point and then solve the linear ODE a y 7 y 71y 7 2 b y 7 expy 7 1 C y 7 My Zooplankton Lectures 3 How are zooplankton adapted to succeed in the pelagic environment Characteristics of the Pelagic Environment A Food dilute rare and small B No cover vulnerable to predation C 3dimensional environment with higher concentrations of food at the surface D Unpredictable patchy in space and time Lecture 1 Outline I Taxonomic groups you are responsible for text II Characteristics of the pelagic environment A Food dilute rare and small Adaptations fOr feeding l Ingestion and clearance rates 0 calculation of ingestion rate 0 calculation of clearance rate 2 Herbivores suspension feeders scan and trap copepods cladocerans euphausiids sieve feeders larvaceans salps flux feeders pteropods Zooplankton Groups you are responsible forsee text gtProtozoa Foraminfera Radiolaria gtCnidaria medusa and siphonophores gtCtenophora comb jellies gtChaetognatha arrow worms gtMollusca Class Gastropoda Heteropods Pteropods sea butterflies gtArthropoda Phylum Crustacea Cladocera Copepods Ostracods Amphipods Euphausiids Decapods gtChordata Subphylum Urochordata Tunicates Larvaceans Appendicularia Salps Dominant zooplankton groups Cladoceran Fre sh water Copepods and Chaetognaths 39 4Q quot V CnManan Siphonophore CnMa mi Inedusae odomaD pteropod planktonic snail II Characteristics of the Pelagic Environment 1 Food dilute and scarce Ingestion rate cells removed eatenanimaltime units cellsanimal h or carbon or calories Clearance rate equivalent amount of water from which all food particles removed by animals per unit time units ml animal h Example 1 experiment with 5 copepods feeding in volume of 100 ml for 1 hour measure food concentration at beginning and end 9 Iiiii Ft Time T0 T1 Concentration 50 cellsml 40 cellsml Ingestion rate change in cell conc volume of vessel number animalstime 50 cellsml 40 cellsml 100 ml 5 animals 1 hr 200 cellsanimal h Clearance rate ingestion rate initial food concentration 200 cellsanimal h 50 cellsml 4 ml animal h Example 2 experiment with 5 ctenophores feeding in volume of 2000 ml for 4 hours measure food concentration copepod preyml at beginning and end Time T0 T1 10 preyml 9 preyml Ingestion rate change in cell conc volume of vessel number animalstime 10 copml 9 copml 2000 ml 5 cteno 4 h 100 copepodscteno h Clearance rate ingestion rate initial food concentration 100 copcteno h 10 copml 10 ml cteno h Herbivores suspension feeders scan and trap feeders copepods cladocerans euphausiids sieve feeders larvaceans salps flux feeders pteropods 5 Suspension feeders Ame scan and trap R 2nd maxilla of adult female C alanus pacifqu Suspension feeders scan and trap Euphausiids Euphausia superba feeding basket from thoracic appendages compression filtration Suspension feeders Sieve Feeders Suspension feeders sievergmwn Tunicates Salp Appendicularian Larvacean Several liters h collect food gt 1 pm up to 1 mm in size lOOs mlh Range 25 583 mlan h Size range 05 um 35 um Flux feeders Pteropods Web collects very small particles Stegosonwazmagnum Appendicularian with house made visible with carmine dye house size of walnut Home Work 3 Due on October 20 2008 Two pages Reading assignments don7t have to be turned in H Reading assignment Read chapters 7 8 9 10 and 11 of Dym7s book The lectures in the next few weeks will concern these chapters Reading assignment Read chapters 2 3 and 4 of the class notes posted on the class web site Some typos have been xed and new material has been added so be sure to download the updated notes Show in complete detail that X is a full column rank matrix if and only if XTX is non singular invertible Assume X is a real matrix Show how to construct at least one left inverse for a full column rank matrix and one right inverse for a full row rank matrix If X and Y are full column rank matrices of rank p show that the rank of XYT is equal to p Show that if A is a rankp matrix then you can nd two full column rank matrices X and Y of rank p such that A XYT If X and Y are two full column rank matrices of rank p nd all matrices A for which RX RA and RAT RY Let X and Y be two full column rank real matrices What are the conditions if any on X and Y such that there exsists a real matrix A such that AX Y and YTA XT Find all such A when the conditions are satis ed Show that if f R 7 R is a function that satis es the following conditions 7 fv20for allvER 7 f1 0 iff 1 0 7 ow lai v for allaERand allvER 7 The set 1 f1 S 1 is convex then f de nes a norm on R
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