Intermediate Differential Equations. John A. Burns
|
|
- Elmer Leonard
- 5 years ago
- Views:
Transcription
1 Intermediate Differential Equations Delay Differential Equations John A. Burns Interdisciplinary Center for Applied Mathematics Virginia Polytechnic Institute and State University Blacksburg, Virginia MATH Fall 212
2 Population Dynamics Use growth of protozoa as example A population could be Bacteria, Viruses Cells (Cancer, T-cells ) People, Fish, Cows, Things that live and die t time p( t) b( t) d( t) sec, hrs, days, years. Number of cells at time t Probability that a cell divides in unit time at time t Probability that a cell dies in unit time at time t ASSUME a closed population
3 Population Dynamics Number of new cells on t, t t Number of cell deaths on Change in cell population t, t t b( t) p( t) t d( t) p( t) t p( t t) p( t) b( t) p( t) t d( t) p( t) t p( t t) t p( t) b( t) d( t) p( t) r( t) p( t) TAKE LIMIT AS t Fundamental LAW d of population growth p( t) r( t) p( t) dt Thomas R. Malthus ( )
4 Population Dynamics b( t) d( t) r( t) BIRTH RATE DEATH RATE GROWTH RATE d dt DO AN EXPERIMENT p ( ) p 1 r 1 p(1) 1e 25 Malthus ASSUMED CONSTANT RATES p( t) r p( t) r p( t) e p t b( t) d( t) r( t) r e r b r d 25 1 r ln e b 2.5 r.9163 d ln(2.5)
5 Population After 5 Days p( t) 1 e.9163t
6 Population After 7.5 Days p( t) 1 e.9163t
7 Population After 1 days 1 x p( t) 1 e.9163t 7 6 NOT WHAT REALLY HAPPENS
8 Incubation Period Fundamental LAW d p( t) r( t) p( t) of population growth dt More accurate? -- d p ( t ) r ( t ) p ( t r ) dt r > is an incubation period r() t r b d
9 Example d p ( t ) p ( t ) dt d p t dt ( ) p ( t /2) NOT a good model of a population growth
10 Improved Model COMPETITION FOR FOOD AND SPACE Malthus ASSUMED PLENTY OF SPACE AND FOOD Pierre-Fancois Verhulst ( ) ) ( ) (, ) ( 1 1 t p d d t d p(t) b b t b ) ( 1 1 ) ( 1 ) ( 1 1 t p K r t p r d b r t r ) ( ) ( ) ( ) ( 1 1 t p d b d b t d t b t r
11 Logistics Growth Rate K b 1 r d 1 p( t) r( t) p( t) 1 r( t) r 1 p ( t ) K LE 1 p( t) r 1 p( t) p( t) K K b 1 r d CARRYING CAPACITY 1 NATURAL REPRODUCTIVE RATE IS THIS A BETTER (MORE ACCURATE) MODEL??
12 A Comparison: First 5 Days p( t) ( ) r p t 1 p( t) r 1 p( t) p( t) K p( t) 1 e.9163t p(t) Malthusian LAW of population growth Logistic LAW of population growth
13 A Comparison: First 7.5 Days p( t) ( ) r p t 1 p( t) r 1 p( t) p( t) K , 5, Malthusian LAW of population growth Logistic LAW of population growth
14 A Comparison: First 1 Days p( t) ( ) r p t 1 p( t) r 1 p( t) p( t) K 1 x 14 1, 9 9, Malthusian LAW of population growth Logistic LAW of population growth
15 Logistic Equation: 15 Days HOWEVER WE OFTEN OBSERVE 1 p( t) r 1 p( t) p( t) K p(t) K
16 Periodic Populations: Blowflys DATA Hutchinson s Equation
17 Delayed Logistics Equation r b( t) b b p(t r), d( t) d 1 r b d r( t) b( t) d( t) b d b p( t r) K 1 r( t) r 1 p( t r) K 1 r b 1 HE 1 ( ) 1 ( ) ( ) p t r p t r p t K
18 1 p( t) 1 p( t) p( t) 1 Delayed Logistic 1 p t p t p t 1 ( ) 1 ( /2) ( ) p() 1 p( s) 1, s s
19 Different Initial Functions 1 p t p t p t 1 ( ) 1 ( /2) ( ) p( s) 1, s s p s e s s 5s ( ),
20 Different Initial Functions 1 p t p t p t 1 ( ) 1 ( /2) ( ) p( s) (1 s) / r, s p( s) (1 s) / r, s s s
21 Predator - Prey Models PREDATOR macrophage PREY bacteria
22 Predator - Prey Models PREDATOR Macrophage PREY Ecoli
23 Interacting Species Predator - Prey Models Vito Volterra Model (1925) Alfred Lotka Model (1926) x( t) y( t) a, b, c, d THINK OF SHARKS AND SHARK FOOD Numbers of prey Numbers of predators Parameters d x ( t ) x ( t ) a by ( t ) dt d y ( t ) y ( t ) c dx ( t ) dt
24 Delayed Predator - Prey Models IT TAKES SOME FINITE TIME FOR THE PREDATOR TO NOTE THAT THE FOOD (PREY) HAS INCREASED HENCE THE RATE OF INCREASE IN THE PREDATOR POPULATION IS DELAYED x( t) Numbers of prey y( t) Numbers of predators r delayed response a, b, c, d Parameters d x ( t ) x ( t ) a by ( t ) dt d y ( t ) y ( t ) c dx ( t r ) dt Delayed Logistic Equation
25 Immune System & HIV HIV VIRUS CD4+T
26 Immune System & HIV T(t) concentration of uninfected targeted helper T cells, T * (t) concentration of infected T cells producing virus, V(t) concentration of virus. MORE ACCURATE Perelson, Banks.
27 Cancer Models
28 Background In the U.S. 4% chance for the average person to develop cancer Breast cancer is the 2nd most common cancer among American women Risk factors: incidence in family, oral contraceptives, obesity Normal cells have many checkpoints During checkpoints reproduction is stopped if abnormality is detected Cancer cells don t have these checkpoint. Unmanageable proliferation leads to loss of genetic information Recent cancer cells more mutated than older cancer cells.
29 The Cancer Cell Cycle 4 stages to cell cycle: G1 (presynthetic) S (synthetic) G2 (postsynthetic) mitosis G (quiescent) Immune cells cytotoxic T-cells flow increases to area of tumor cells INTERFACE STAGE Paclitaxel is a common drug used for Breast, Ovarian, Head and Neck Cancer - attack tumor cells during a cell cycle
30 Cell Population Dynamics T () t I -- Population of cells in Interface stage T () M t -- Population of cells in Mitosis stage T () Q t -- Population of cells in Quiescent stage It () ct () -- Population of Immune cells (cytotoxic T-cells) -- Concentration of the drug Paclitaxel G. Newbury, A Numerical Study of a Delay Differential Equation Model for Breast Cancer, MS Thesis, Department of Mathematics, Virginia Tech, Blacksburg, VA, August, 27. M. Villasana and G. Ochoa, Heuristic Design of Cancer Chemotherapies, IEEE Transactions of Evolutionary Computation, 8 (24), R. Yafia, Dynamics Analysis and Limit Cycle in a Delayed Model for Tumor Growth with Quiescence, Nonlinear Analysis, Modeling and Control, 11 (26),
31 Delay Equation Model (1) (2) (3) (4) dtq () t a5ti ( t ) a6tq ( t) d4tq ( t) c5tq ( t) I( t) u1( t) TQ ( t) dt dti () t 2 a4tm ( t) a5ti ( t ) a6tq ( t) c1t I ( t) I( t) d2ti ( t) a1t I ( t ) dt dtm () t dt a T ( t ) d T ( t) a T ( t) c T ( t) I( t) u ( t) T ( t) 1 I 3 M 4 M 3 M 2 M n di() t I( t)[ TQ ( t) TI ( t) TM ( t)] k c n dt [ T ( t) T ( t) T ( t)] Q I M 4 M 6 Q I( t) T ( t) c I( t) T ( t) c I( t) T ( t) d I( t) u () t I( t) I u () t g ( w( t), c( t) ) i i
32 Delay Equation Model (5) (6) dw1 () t dt dw2 () t dt w ( t) c( t), w () w ( t) c( t), w () 2 2 w( t) r w ( t) r w ( t) To compare with existing models. u t 1 k6wt () () k (1 e ) k2wt () k4wt () u () t k (1e ) u () t k (1e ) We also investigated the ODE model: = 3 3
33 Typical Parameters and Inputs concentration c(t) T T [ T (), T (), T (), I()] [.8, 1.3, 1.2, ].9 Q I M T T [ T (), T (), T (), I()] [.7,.8,.6, ].12 Q I M lim It ( ).12 t NORMAL LEVEL PARAMETERS time t c(t) -- Pulsed concentration of drug
34 Typical Short Time Simulation T T [ T (), T (), T (), I()] [.7,.8,.6, ].9 Q I M? IS THE DRUG WORKING?
35 Longer Time Simulation T T [ T (), T (), T (), I()] [.7,.8,.6, ].9 Q I M!! NO!!! SOLUTION GOES TO + AS t +? HOW DO WE KNOW THIS WILL HAPPEN? 3 years
36 { (IVP) Delay Differential Equations (Σ) (IC ) (IC 1 ) x( t) f x( t), x( t r), q x( t ) x R x( s) ( s), r s n n n m n f ( x, z, q) : D R R R R EXAMPLES
37 Example 1 x( t) cos( t) x( t) x( t / 2), x() 1 x( t) sin( t) cos( t / 2) x( t / 2) x( t) cos( t) sin( t) x( t) sin( t) cos( t) cos( t / 2) sin( t / 2) x( t / 2)
38 Example
39 Example 2 x( t) x( t 1), x() x t 2 ( ) ( t 1) / 2 3/ x() t t 1.5 xt ( )
40 Example 2 x( t) x( t 1), x() x t t t t 2 3 ( ) 1 ( 1) / 2 ( 2) /3! x t t t 2 ( ) 1 ( 1) / 2 1 x( t) t ? WHAT IS A REASONABLE INITIAL VALUE PROBLEM?
41 Example 2 x( t) x( t 1), x()
42 Example 2 x( t) x( t 1), x()
43 IVP for Delay Equations x( t) f ( x( t), x( t r)), x() x( s) ( s), r s () s r t r t
44 x( t) f ( x( t), x( t r)) x() Solution x( s) ( s), r s PAST HISTORY t x( t) f ( x( s), x( s r) ds x( t) f ( x( t), x( t r)) n n n f ( x, z) : D R R R
45 Existence n n n f ( x, z) : R R R Theorem D1. Assume f: R n R n ---> R n is a continuous function on R n R n and r >. If R n and () is continuous on the interval [-r,), then there exists at least one solution to the initial value problem (IVP) x( t) f ( x( t), x( t r)) x() x( s) ( s), r s UNIQUENESS IS MORE COMPLEX
46 Some Notation x ( ) :[ r,) R t n xt ( s) x( t s), r s x () t x () s t xt ( s) xt () r s t r t s t
47 xt Retarded Equations ( s) x( t s), r s x ( r) x( t r) t x ( s) x( s) ( s), r s x( t) f ( x( t), x( t r)) x() x( s) ( s), r s x( t) f ( x( t), x ( r)) x() x ( s) ( s), r s t n F( x, ( )) : R C[ r,] R F( x, ( )) f ( x, ( r)) n
48 Retarded Equations x( t) f ( x( t), x( t r)) x() x( t) f ( x( t), x ( r)) x() t x( s) ( s), r s x ( s) ( s), r s F( x, ( )) f ( x, ( r)) x( t) f ( x( t), x( t r)) F( x( t), x t ( )) ( x(), x ( )) (, ( )) n R C[ r,] STATE SPACE OF INITIAL DATA
49 More Retarded Equations x( t) x( t) P( s) x( t s) ds x() r x( s) ( s), r s F( x, ( )) x P( s) ( s) ds F( x( t), x ( )) x( t) P( s) x ( s) ds x( t) P( s) x( t s) ds t r t r r x( t) x( t) P( s) x( t s) ds F( x( t), xt ( )) r
50 Definition n F( x, ( )) : R C[ r,] R n The function F: R n C[-r,] ---> R n is Lipschitzian on the open set R n C[-r,] if there constant > such that F( x, ( )) F( y, ( ) ( x y sup ( s) ( s) ) rs for all ( x, ( ), ( y, ( )) RC[ r,] FUNDAMENTAL UNIQUENESS THEOREM
51 Uniqueness Theorem Theorem D2. Assume F: R n C[-r,] ---> R n is continuous and Lipschitzian on every compact subset R n C[-r,]. If (, ()), then there exists a unique solution to the initial value problem (IVP) ( x(), x ( )) (, ( )) n F( x, ( )) : R C[ r,] R x( t) F( x( t), x t ( )) R C[ r,] n n Jack Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.
52 Applications in Life Sciences POPULATIONS WITH DELAYED RESPONSE TO A RESOURCE d N ( t ) r ( t ) N ( t ) dt 1 r( t) r 1 N( t) K LE d 1 N ( t ) r 1 N ( t ) N ( t ) dt K 1 r( t) r 1 N( t r) K HE d 1 N ( t ) r 1 N ( t r ) N ( t ) dt K
53 Hutchinson s Equation DATA Hutchinson s Equation
54 Delay Predator-Prey Models
55 Delay Predator-Prey Models
56 HIV Models T(t) concentration of uninfected targeted helper T cells, T * (t) concentration of infected T cells producing virus, V(t) concentration of virus. MORE ACCURATE WITH DELAY Nelson, Murray and Perelson
57 HIV Models MORE COMPLEX WITH CONTINUOUS DELAY
58 Delay Glucose-Insulation System
59 Delay Glucose-Insulation System
60 dt () t Delay Cancer Models a T ( t ) a T ( t) d T ( t) c T ( t) I( t) Q (1) 5 I 6 Q 4 Q 5 Q dt (2) dti () t dt 2 a T ( t) a T ( t ) a T ( t) c T ( t) I( t) d T ( t) a T ( t ) 4 M 5 I 6 Q 1 I 2 I 1 I (3) dtm () t dt a T ( t ) d T ( t) a T ( t) c T ( t) I( t) 1 I 3 M 4 M 3 M (4) n di() t I( t)[ TQ ( t) TI ( t) TM ( t)] k n dt [ T ( t) T ( t) T ( t)] Q I M c I( t) T ( t) c I( t) T ( t) d I( t) 4 M 6 Q 1 c 2 I( t) T ( t) I G. Newbury, A Numerical Study of a Delay Differential Equation Model for Breast Cancer, MS Thesis, Department of Mathematics, Virginia Tech, Blacksburg, VA, August, 27.
61 Delay Cancer Models
62 Delay Cancer Models
63 Delay Epidemic & Biochemical Networks
64 Delay Immune Response
Sensitivity Analysis of a Cancer Model with Drug Treatment
Sensitivity Analysis of a Cancer Model with Drug reatment John A. Burns & Golnar Newbury Interdisciplinary Center for Applied Mathematics Virginia Polytechnic Institute and State University Blacksburg,
More informationTheory of Ordinary Differential Equations. Stability and Bifurcation I. John A. Burns
Theory of Ordinary Differential Equations Stability and Bifurcation I John A. Burns Center for Optimal Design And Control Interdisciplinary Center for Applied Mathematics Virginia Polytechnic Institute
More informationMODELS ONE ORDINARY DIFFERENTIAL EQUATION:
MODELS ONE ORDINARY DIFFERENTIAL EQUATION: opulation Dynamics (e.g. Malthusian, Verhulstian, Gompertz, Logistic with Harvesting) Harmonic Oscillator (e.g. pendulum) A modified normal distribution curve
More informationMATH3203 Lecture 1 Mathematical Modelling and ODEs
MATH3203 Lecture 1 Mathematical Modelling and ODEs Dion Weatherley Earth Systems Science Computational Centre, University of Queensland February 27, 2006 Abstract Contents 1 Mathematical Modelling 2 1.1
More informationM469, Fall 2010, Practice Problems for the Final
M469 Fall 00 Practice Problems for the Final The final exam for M469 will be Friday December 0 3:00-5:00 pm in the usual classroom Blocker 60 The final will cover the following topics from nonlinear systems
More informationDynamical Systems and Chaos Part II: Biology Applications. Lecture 6: Population dynamics. Ilya Potapov Mathematics Department, TUT Room TD325
Dynamical Systems and Chaos Part II: Biology Applications Lecture 6: Population dynamics Ilya Potapov Mathematics Department, TUT Room TD325 Living things are dynamical systems Dynamical systems theory
More informationIntermediate Differential Equations. Stability and Bifurcation II. John A. Burns
Intermediate Differential Equations Stability and Bifurcation II John A. Burns Center for Optimal Design And Control Interdisciplinary Center for Applied Mathematics Virginia Polytechnic Institute and
More informationMATH 1014 N 3.0 W2015 APPLIED CALCULUS II - SECTION P. Perhaps the most important of all the applications of calculus is to differential equations.
MATH 1014 N 3.0 W2015 APPLIED CALCULUS II - SECTION P Stewart Chapter 9 Differential Equations Perhaps the most important of all the applications of calculus is to differential equations. 9.1 Modeling
More informationNonlinear Dynamics. Moreno Marzolla Dip. di Informatica Scienza e Ingegneria (DISI) Università di Bologna.
Nonlinear Dynamics Moreno Marzolla Dip. di Informatica Scienza e Ingegneria (DISI) Università di Bologna http://www.moreno.marzolla.name/ 2 Introduction: Dynamics of Simple Maps 3 Dynamical systems A dynamical
More informationA Delayed HIV Infection Model with Specific Nonlinear Incidence Rate and Cure of Infected Cells in Eclipse Stage
Applied Mathematical Sciences, Vol. 1, 216, no. 43, 2121-213 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.216.63128 A Delayed HIV Infection Model with Specific Nonlinear Incidence Rate and
More informationMotivation and Goals. Modelling with ODEs. Continuous Processes. Ordinary Differential Equations. dy = dt
Motivation and Goals Modelling with ODEs 24.10.01 Motivation: Ordinary Differential Equations (ODEs) are very important in all branches of Science and Engineering ODEs form the basis for the simulation
More informationAge (x) nx lx. Population dynamics Population size through time should be predictable N t+1 = N t + B + I - D - E
Population dynamics Population size through time should be predictable N t+1 = N t + B + I - D - E Time 1 N = 100 20 births 25 deaths 10 immigrants 15 emmigrants Time 2 100 + 20 +10 25 15 = 90 Life History
More informationSolutions to Math 53 Math 53 Practice Final
Solutions to Math 5 Math 5 Practice Final 20 points Consider the initial value problem y t 4yt = te t with y 0 = and y0 = 0 a 8 points Find the Laplace transform of the solution of this IVP b 8 points
More informationViral evolution model with several time scales
Viral evolution model with several time scales AA Archibasov Samara National Research University, 4 Moskovskoe Shosse, 4486, Samara, Russia Abstract In this paper a viral evolution model with specific
More informationC2 Differential Equations : Computational Modeling and Simulation Instructor: Linwei Wang
C2 Differential Equations 4040-849-03: Computational Modeling and Simulation Instructor: Linwei Wang Part IV Dynamic Systems Equilibrium: Stable or Unstable? Equilibrium is a state of a system which does
More informationMA 137 Calculus 1 with Life Science Application A First Look at Differential Equations (Section 4.1.2)
MA 137 Calculus 1 with Life Science Application A First Look at Differential Equations (Section 4.1.2) Alberto Corso alberto.corso@uky.edu Department of Mathematics University of Kentucky October 12, 2015
More informationScience One Math. October 23, 2018
Science One Math October 23, 2018 Today A general discussion about mathematical modelling A simple growth model Mathematical Modelling A mathematical model is an attempt to describe a natural phenomenon
More informationMA 138 Calculus 2 with Life Science Applications Handout
.. MA 138 Calculus 2 with Life Science Applications Handout Alberto Corso alberto.corso@uky.edu Department of Mathematics University of Kentucky February 17, 2017 . Example 4 (Lotka-Volterra Predator-Prey
More informationWe have two possible solutions (intersections of null-clines. dt = bv + muv = g(u, v). du = au nuv = f (u, v),
Let us apply the approach presented above to the analysis of population dynamics models. 9. Lotka-Volterra predator-prey model: phase plane analysis. Earlier we introduced the system of equations for prey
More informationMath 128A Spring 2003 Week 12 Solutions
Math 128A Spring 2003 Week 12 Solutions Burden & Faires 5.9: 1b, 2b, 3, 5, 6, 7 Burden & Faires 5.10: 4, 5, 8 Burden & Faires 5.11: 1c, 2, 5, 6, 8 Burden & Faires 5.9. Higher-Order Equations and Systems
More informationOrdinary Differential Equations
Ordinary Differential Equations Michael H. F. Wilkinson Institute for Mathematics and Computing Science University of Groningen The Netherlands December 2005 Overview What are Ordinary Differential Equations
More informationModeling the Immune System W9. Ordinary Differential Equations as Macroscopic Modeling Tool
Modeling the Immune System W9 Ordinary Differential Equations as Macroscopic Modeling Tool 1 Lecture Notes for ODE Models We use the lecture notes Theoretical Fysiology 2006 by Rob de Boer, U. Utrecht
More informationBIOS 5970: Plant-Herbivore Interactions Dr. Stephen Malcolm, Department of Biological Sciences
BIOS 5970: Plant-Herbivore Interactions Dr. Stephen Malcolm, Department of Biological Sciences D. POPULATION & COMMUNITY DYNAMICS Week 10. Population models 1: Lecture summary: Distribution and abundance
More informationMath 232, Final Test, 20 March 2007
Math 232, Final Test, 20 March 2007 Name: Instructions. Do any five of the first six questions, and any five of the last six questions. Please do your best, and show all appropriate details in your solutions.
More informationQuazi accurate photo history
Quazi accurate photo history Evolution- change over time Fossils preserved remains Geologic Time earth s history The evidence shows changes in environment changes in species The Theory of Evolution supported
More informationSTUDY OF THE DYNAMICAL MODEL OF HIV
STUDY OF THE DYNAMICAL MODEL OF HIV M.A. Lapshova, E.A. Shchepakina Samara National Research University, Samara, Russia Abstract. The paper is devoted to the study of the dynamical model of HIV. An application
More informationPopulations Study Guide (KEY) All the members of a species living in the same place at the same time.
Populations Study Guide (KEY) 1. Define Population. All the members of a species living in the same place at the same time. 2. List and explain the three terms that describe population. a. Size. How large
More informationInteractions. Yuan Gao. Spring Applied Mathematics University of Washington
Interactions Yuan Gao Applied Mathematics University of Washington yuangao@uw.edu Spring 2015 1 / 27 Nonlinear System Consider the following coupled ODEs: dx = f (x, y). dt dy = g(x, y). dt In general,
More informationMath 345 Intro to Math Biology Lecture 19: Models of Molecular Events and Biochemistry
Math 345 Intro to Math Biology Lecture 19: Models of Molecular Events and Biochemistry Junping Shi College of William and Mary, USA Molecular biology and Biochemical kinetics Molecular biology is one of
More informationB2 Revision Questions Part 2. B2 Revision cards
B2 Revision Questions Part 2 Question 1 Name 2 adaptations of predators Answer 1 Hunting skills, eyes on front of head to judge distances, sharp claws and teeth. Question 2 Name 2 adaptations of prey Answer
More informationCompartmental Analysis
Compartmental Analysis Math 366 - Differential Equations Material Covering Lab 3 We now learn how to model some physical phonomena through DE. General steps for modeling (you are encouraged to find your
More information1 (t + 4)(t 1) dt. Solution: The denominator of the integrand is already factored with the factors being distinct, so 1 (t + 4)(t 1) = A
Calculus Topic: Integration of Rational Functions Section 8. # 0: Evaluate the integral (t + )(t ) Solution: The denominator of the integrand is already factored with the factors being distinct, so (t
More informationDirection fields of differential equations...with SAGE
Direction fields of differential equations...with SAGE Many differential equations cannot be solved conveniently by analytical methods, so it is important to consider what qualitative information can be
More informationHomework #4 Solutions
MAT 303 Spring 03 Problems Section.: 0,, Section.:, 6,, Section.3:,, 0,, 30 Homework # Solutions..0. Suppose that the fish population P(t) in a lake is attacked by a disease at time t = 0, with the result
More information1.2. Introduction to Modeling
G. NAGY ODE August 30, 2018 1 Section Objective(s): Population Models Unlimited Resources Limited Resources Interacting Species 1.2. Introduction to Modeling 1.2.1. Population Model with Unlimited Resources.
More informationDifferential Equations ( DEs) MA 137 Calculus 1 with Life Science Application A First Look at Differential Equations (Section 4.1.
/7 Differential Equations DEs) MA 37 Calculus with Life Science Application A First Look at Differential Equations Section 4..2) Alberto Corso alberto.corso@uky.edu Department of Mathematics University
More informationPOPULATION DYNAMICS: TWO SPECIES MODELS; Susceptible Infected Recovered (SIR) MODEL. If they co-exist in the same environment:
POPULATION DYNAMICS: TWO SPECIES MODELS; Susceptible Infected Recovered (SIR) MODEL Next logical step: consider dynamics of more than one species. We start with models of 2 interacting species. We consider,
More informationMath 266: Autonomous equation and population dynamics
Math 266: Autonomous equation and population namics Long Jin Purdue, Spring 2018 Autonomous equation An autonomous equation is a differential equation which only involves the unknown function y and its
More informationLesson 9: Predator-Prey and ode45
Lesson 9: Predator-Prey and ode45 9.1 Applied Problem. In this lesson we will allow for more than one population where they depend on each other. One population could be the predator such as a fox, and
More informationCell Division and Reproduction
Cell Division and Reproduction What do you think this picture shows? If you guessed that it s a picture of two cells, you are right. In fact, the picture shows human cancer cells, and they are nearing
More informationFirst Order Differential Equations
2 First Order Differential Equations We have seen ho to solve simple first order differential equations using Simulin. In particular e have solved initial value problems for the equations dy dt dy dt dx
More informationDerivation of Itô SDE and Relationship to ODE and CTMC Models
Derivation of Itô SDE and Relationship to ODE and CTMC Models Biomathematics II April 23, 2015 Linda J. S. Allen Texas Tech University TTU 1 Euler-Maruyama Method for Numerical Solution of an Itô SDE dx(t)
More informationAPPM 2360 Lab #3: The Predator Prey Model
APPM 2360 Lab #3: The Predator Prey Model 1 Instructions Labs may be done in groups of 3 or less. One report must be turned in for each group and must be in PDF format. Labs must include each student s:
More informationOn the Interpretation of Delays in Delay Stochastic Simulation of Biological Systems
On the Interpretation of Delays in Delay Stochastic Simulation of Biological Systems Roberto Barbuti Giulio Caravagna Andrea Maggiolo-Schettini Paolo Milazzo Dipartimento di Informatica, Università di
More informationAnalysis of bacterial population growth using extended logistic Growth model with distributed delay. Abstract INTRODUCTION
Analysis of bacterial population growth using extended logistic Growth model with distributed delay Tahani Ali Omer Department of Mathematics and Statistics University of Missouri-ansas City ansas City,
More informationsections June 11, 2009
sections 3.2-3.5 June 11, 2009 Population growth/decay When we model population growth, the simplest model is the exponential (or Malthusian) model. Basic ideas: P = P(t) = population size as a function
More informationEvolution & Natural Selection
Evolution & Natural Selection Learning Objectives Know what biological evolution is and understand the driving force behind biological evolution. know the major mechanisms that change allele frequencies
More informationWhy This Class? James K. Peterson. August 22, Department of Biological Sciences and Department of Mathematical Sciences Clemson University
Why This Class? James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University August 22, 2013 Outline 1 Our Point of View Mathematics, Science and Computer
More informationToday. Introduction to Differential Equations. Linear DE ( y = ky ) Nonlinear DE (e.g. y = y (1-y) ) Qualitative analysis (phase line)
Today Introduction to Differential Equations Linear DE ( y = ky ) Nonlinear DE (e.g. y = y (1-y) ) Qualitative analysis (phase line) Differential equations (DE) Carbon dating: The amount of Carbon-14 in
More informationMath 310: Applied Differential Equations Homework 2 Prof. Ricciardi October 8, DUE: October 25, 2010
Math 310: Applied Differential Equations Homework 2 Prof. Ricciardi October 8, 2010 DUE: October 25, 2010 1. Complete Laboratory 5, numbers 4 and 7 only. 2. Find a synchronous solution of the form A cos(ωt)+b
More informationFinal Project Descriptions Introduction to Mathematical Biology Professor: Paul J. Atzberger. Project I: Predator-Prey Equations
Final Project Descriptions Introduction to Mathematical Biology Professor: Paul J. Atzberger Project I: Predator-Prey Equations The Lotka-Volterra Predator-Prey Model is given by: du dv = αu βuv = ρβuv
More informationSpring /30/2013
MA 138 - Calculus 2 for the Life Sciences FINAL EXAM Spring 2013 4/30/2013 Name: Sect. #: Answer all of the following questions. Use the backs of the question papers for scratch paper. No books or notes
More informationModels of Infectious Disease Formal Demography Stanford Summer Short Course James Holland Jones, Instructor. August 15, 2005
Models of Infectious Disease Formal Demography Stanford Summer Short Course James Holland Jones, Instructor August 15, 2005 1 Outline 1. Compartmental Thinking 2. Simple Epidemic (a) Epidemic Curve 1:
More informationGlobal Stability Analysis on a Predator-Prey Model with Omnivores
Applied Mathematical Sciences, Vol. 9, 215, no. 36, 1771-1782 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.215.512 Global Stability Analysis on a Predator-Prey Model with Omnivores Puji Andayani
More informationWhat is essential difference between snake behind glass versus a wild animal?
What is essential difference between snake behind glass versus a wild animal? intact cells physiological properties genetics some extent behavior Caged animal is out of context Removed from natural surroundings
More informationAnimal Population Dynamics
Animal Population Dynamics Jennifer Gervais Weniger 449 737-6122 jennifer.gervais@oregonstate.edu Syllabus Course webpage http://oregonstate.edu/~gervaisj/ There is something fascinating about science.
More informationDifference Equations
Difference Equations Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4500, Spring 2017 M. Macauley (Clemson) Difference equations Math
More informationMITOCW MITRES18_005S10_DiffEqnsGrowth_300k_512kb-mp4
MITOCW MITRES18_005S10_DiffEqnsGrowth_300k_512kb-mp4 GILBERT STRANG: OK, today is about differential equations. That's where calculus really is applied. And these will be equations that describe growth.
More informationSIMPLE MATHEMATICAL MODELS WITH EXCEL
SIMPLE MATHEMATICAL MODELS WITH EXCEL Traugott H. Schelker Abstract Maybe twenty or thirty years ago only physicists and engineers had any use for mathematical models. Today the situation has totally changed.
More informationNumerical computation and series solution for mathematical model of HIV/AIDS
Journal of Applied Mathematics & Bioinformatics, vol.3, no.4, 3, 69-84 ISSN: 79-66 (print, 79-6939 (online Scienpress Ltd, 3 Numerical computation and series solution for mathematical model of HIV/AIDS
More informationA Stochastic Viral Infection Model with General Functional Response
Nonlinear Analysis and Differential Equations, Vol. 4, 16, no. 9, 435-445 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.1988/nade.16.664 A Stochastic Viral Infection Model with General Functional Response
More informationOn the Interpretation of Delays in Delay Stochastic Simulation of Biological Systems
On the Interpretation of Delays in Delay Stochastic Simulation of Biological Systems Roberto Barbuti Giulio Caravagna Andrea Maggiolo-Schettini Paolo Milazzo Dipartimento di Informatica, Università di
More informationSensitivity and Stability Analysis of Hepatitis B Virus Model with Non-Cytolytic Cure Process and Logistic Hepatocyte Growth
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 3 2016), pp. 2297 2312 Research India Publications http://www.ripublication.com/gjpam.htm Sensitivity and Stability Analysis
More informationResearch Article On the Stability Property of the Infection-Free Equilibrium of a Viral Infection Model
Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume, Article ID 644, 9 pages doi:.55//644 Research Article On the Stability Property of the Infection-Free Equilibrium of a Viral
More informationFind the orthogonal trajectories for the family of curves. 9. The family of parabolas symmetric with respect to the x-axis and vertex at the origin.
Exercises 2.4.1 Find the orthogonal trajectories for the family of curves. 1. y = Cx 3. 2. x = Cy 4. 3. y = Cx 2 + 2. 4. y 2 = 2(C x). 5. y = C cos x 6. y = Ce x 7. y = ln(cx) 8. (x + y) 2 = Cx 2 Find
More informationChapters AP Biology Objectives. Objectives: You should know...
Objectives: You should know... Notes 1. Scientific evidence supports the idea that evolution has occurred in all species. 2. Scientific evidence supports the idea that evolution continues to occur. 3.
More informationA model for transfer phenomena in structured populations
A model for transfer phenomena in structured populations Peter Hinow 1, Pierre Magal 2, Glenn F. Webb 3 1 Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455,
More informationAgent-based models applied to Biology and Economics.
Agent-based models applied to Biology and Economics. Gur Yaari Phd student, under the supervision of Prof. Sorin Solomon Racah Institute of Physics in the Hebrew University of Jerusalem, Israel Multi-Agent
More informationLOCAL AND GLOBAL STABILITY OF IMPULSIVE PEST MANAGEMENT MODEL WITH BIOLOGICAL HYBRID CONTROL
International J. of Math. Sci. & Engg. Appls. (IJMSEA) ISSN 0973-9424, Vol. 11 No. II (August, 2017), pp. 129-141 LOCAL AND GLOBAL STABILITY OF IMPULSIVE PEST MANAGEMENT MODEL WITH BIOLOGICAL HYBRID CONTROL
More informationAN EXTENDED ROSENZWEIG-MACARTHUR MODEL OF A TRITROPHIC FOOD CHAIN. Nicole Rocco
AN EXTENDED ROSENZWEIG-MACARTHUR MODEL OF A TRITROPHIC FOOD CHAIN Nicole Rocco A Thesis Submitted to the University of North Carolina Wilmington in Partial Fulfillment of the Requirements for the Degree
More informationThree Disguises of 1 x = e λx
Three Disguises of 1 x = e λx Chathuri Karunarathna Mudiyanselage Rabi K.C. Winfried Just Department of Mathematics, Ohio University Mathematical Biology and Dynamical Systems Seminar Ohio University November
More informationScience One Math. October 17, 2017
Science One Math October 17, 2017 Today A few more examples of related rates problems A general discussion about mathematical modelling A simple growth model Related Rates Problems Problems where two or
More informationObjective. Single population growth models
Objective Single population growth models We are given a table with the population census at different time intervals between a date a and a date b, and want to get an expression for the population. This
More informationModeling with differential equations
Mathematical Modeling Lia Vas Modeling with differential equations When trying to predict the future value, one follows the following basic idea. Future value = present value + change. From this idea,
More informationMA 138 Calculus 2 for the Life Sciences Spring 2016 Final Exam May 4, Exam Scores. Question Score Total
MA 138 Calculus 2 for the Life Sciences Spring 2016 Final Exam May 4, 2016 Exam Scores Question Score Total 1 10 Name: Section: Last 4 digits of student ID #: No books or notes may be used. Turn off all
More informationStability Analysis And Maximum Profit Of Logistic Population Model With Time Delay And Constant Effort Of Harvesting
Jurnal Vol 3, Matematika, No, 9-8, Juli Statistika 006 & Komputasi Vol No Juli 006 9-8, Juli 006 9 Stability Analysis And Maximum Profit Of Logistic Population Model With Time Delay And Constant Effort
More informationIntroduction to some topics in Mathematical Oncology
Introduction to some topics in Mathematical Oncology Franco Flandoli, University of Pisa y, Finance and Physics, Berlin 2014 The field received considerable attentions in the past 10 years One of the plenary
More informationDynamical Systems in Biology
Dynamical Systems in Biology Hal Smith A R I Z O N A S T A T E U N I V E R S I T Y H.L. Smith (ASU) Dynamical Systems in Biology ASU, July 5, 2012 1 / 31 Outline 1 What s special about dynamical systems
More informationStochastic Viral Dynamics with Beddington-DeAngelis Functional Response
Stochastic Viral Dynamics with Beddington-DeAngelis Functional Response Junyi Tu, Yuncheng You University of South Florida, USA you@mail.usf.edu IMA Workshop in Memory of George R. Sell June 016 Outline
More informationMark Howell Gonzaga High School, Washington, D.C.
Be Prepared for the Calculus Exam Mark Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice exam contributors: Benita Albert Oak Ridge High School,
More informationOn a non-autonomous stochastic Lotka-Volterra competitive system
Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 7), 399 38 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa On a non-autonomous stochastic Lotka-Volterra
More informationMODELLING TUMOUR-IMMUNITY INTERACTIONS WITH DIFFERENT STIMULATION FUNCTIONS
Int. J. Appl. Math. Comput. Sci., 2003, Vol. 13, No. 3, 307 315 MODELLING TUMOUR-IMMUNITY INTERACTIONS WITH DIFFERENT STIMULATION FUNCTIONS PETAR ZHIVKOV, JACEK WANIEWSKI Institute of Applied Mathematics
More informationOutline. Calculus for the Life Sciences. What is a Differential Equation? Introduction. Lecture Notes Introduction to Differential Equa
Outline Calculus for the Life Sciences Lecture Notes to Differential Equations Joseph M. Mahaffy, jmahaffy@mail.sdsu.edu 1 Department of Mathematics and Statistics Dynamical Systems Group Computational
More informationSelection 10: Theory of Natural Selection
Selection 10: Theory of Natural Selection Darwin began his voyage thinking that species could not change His experience during the five-year journey altered his thinking Variation of similar species among
More information1.1 Motivation: Why study differential equations?
Chapter 1 Introduction Contents 1.1 Motivation: Why stu differential equations?....... 1 1.2 Basics............................... 2 1.3 Growth and decay........................ 3 1.4 Introduction to Ordinary
More informationSMA 208: Ordinary differential equations I
SMA 208: Ordinary differential equations I Modeling with First Order differential equations Lecturer: Dr. Philip Ngare (Contacts: pngare@uonbi.ac.ke, Tue 12-2 PM) School of Mathematics, University of Nairobi
More informationJournal of the Vol. 36, pp , 2017 Nigerian Mathematical Society c Nigerian Mathematical Society
Journal of the Vol. 36, pp. 47-54, 2017 Nigerian Mathematical Society c Nigerian Mathematical Society A CLASS OF GENERALIZATIONS OF THE LOTKA-VOLTERRA PREDATOR-PREY EQUATIONS HAVING EXACTLY SOLUBLE SOLUTIONS
More informationLinear Variable coefficient equations (Sect. 2.1) Review: Linear constant coefficient equations
Linear Variable coefficient equations (Sect. 2.1) Review: Linear constant coefficient equations. The Initial Value Problem. Linear variable coefficients equations. The Bernoulli equation: A nonlinear equation.
More informationMathematical Models of Biological Systems
Mathematical Models of Biological Systems CH924 November 25, 2012 Course outline Syllabus: Week 6: Stability of first-order autonomous differential equation, genetic switch/clock, introduction to enzyme
More informationDIFFERENCE AND DIFFERENTIAL EQUATIONS IN MATHEMATICAL MODELLING. J. Banasiak
DIFFERENCE AND DIFFERENTIAL EQUATIONS IN MATHEMATICAL MODELLING J. Banasiak ii Contents Basic ideas about mathematical modelling. Introduction: what is mathematical modelling?.........2 Simple difference
More informationMathematical Epidemiology Lecture 1. Matylda Jabłońska-Sabuka
Lecture 1 Lappeenranta University of Technology Wrocław, Fall 2013 What is? Basic terminology Epidemiology is the subject that studies the spread of diseases in populations, and primarily the human populations.
More informationSolutions to Homework 1, Introduction to Differential Equations, 3450: , Dr. Montero, Spring y(x) = ce 2x + e x
Solutions to Homewor 1, Introduction to Differential Equations, 3450:335-003, Dr. Montero, Spring 2009 problem 2. The problem says that the function yx = ce 2x + e x solves the ODE y + 2y = e x, and ass
More informationSolving Differential Equations with Simulink
Solving Differential Equation with Simulink Dr. R. L. Herman UNC Wilmington, Wilmington, NC March, 206 Solving ODE with Simulink, ICTCM 206 R. L. Herman Mar, 206 /9 Outline Simulink 2 Solution of ODE 3
More informationFish Population Modeling
Fish Population Modeling Thomas Wood 5/11/9 Logistic Model of a Fishing Season A fish population can be modelled using the Logistic Model, P = kp(1 P/a) P is the population at time t k is a growth constant
More informationLecture 20/Lab 21: Systems of Nonlinear ODEs
Lecture 20/Lab 21: Systems of Nonlinear ODEs MAR514 Geoffrey Cowles Department of Fisheries Oceanography School for Marine Science and Technology University of Massachusetts-Dartmouth Coupled ODEs: Species
More informationGetting Started With The Predator - Prey Model: Nullclines
Getting Started With The Predator - Prey Model: Nullclines James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 28, 2013 Outline The Predator
More informationA NUMERICAL STUDY ON PREDATOR PREY MODEL
International Conference Mathematical and Computational Biology 2011 International Journal of Modern Physics: Conference Series Vol. 9 (2012) 347 353 World Scientific Publishing Company DOI: 10.1142/S2010194512005417
More informationMathematical and Computational Methods for the Life Sciences
Mathematical and Computational Methods for the Life Sciences Preliminary Lecture Notes Adolfo J. Rumbos November 24, 2007 2 Contents 1 Preface 5 2 Introduction: An Example from Microbial Genetics 7 I Deterministic
More informationM340 HW 2 SOLUTIONS. 1. For the equation y = f(y), where f(y) is given in the following plot:
M340 HW SOLUTIONS 1. For the equation y = f(y), where f(y) is given in the following plot: (a) What are the critical points? (b) Are they stable or unstable? (c) Sketch the solutions in the ty plane. (d)
More information2D-Volterra-Lotka Modeling For 2 Species
Majalat Al-Ulum Al-Insaniya wat - Tatbiqiya 2D-Volterra-Lotka Modeling For 2 Species Alhashmi Darah 1 University of Almergeb Department of Mathematics Faculty of Science Zliten Libya. Abstract The purpose
More information