Lecture Notes – Qualitative Analysis of Differential Equations

Introduction Introduction Qualitative Behavior of Differential Equations Qualitative Behavior of Differential Equations More Examples More Examples Outline Calculus for the Life Sciences Lecture Notes – Qualitative Analysis 1 Introduction of Differential Equations Bacterial Growth Logistic Growth Staph Experimental Fit Joseph M. Mahaffy, [email protected] 2 Qualitative Behavior of Differential Equations h i Example: Logistic Growth Example: Sine Function Department of Mathematics and Statistics Dynamical Systems Group Computational Sciences Research Center 3 More Examples San Diego State University Left Snail Model San Diego, CA 92182-7720 Pitchfork Bifurcation http://www-rohan.sdsu.edu/∼jmahaffy Allee Effect

Fall 2014

Lecture Notes – Qualitative Analysis of Differential Equations Lecture Notes – Qualitative Analysis of Differen Joseph M. Mahaffy, [email protected] — (1/45) Joseph M. Mahaffy, [email protected] — (2/45)

Introduction Bacterial Growth Introduction Bacterial Growth Qualitative Behavior of Diﬀerential Equations Logistic Growth Qualitative Behavior of Diﬀerential Equations Logistic Growth More Examples Staph Experimental Fit More Examples Staph Experimental Fit Introduction Growth of Bacteria

Modeling and Differential Equations Growth of Bacteria Biological Models often use differential equations, Bacterial growth usually follows a regular pattern which have no analytical solution They are inoculated into a broth Culture has a lag period (adjusting to the new growing Properties of the model and differential equation can conditions) provide some insight A period of exponential growth (Malthusian growth Qualitative analysis techniques provide simple tools for model) understanding models Cell growth slows to stationary growth (nutrients become limiting or waste products build) Analysis helps understand equilibria and behavior near the Population levels off or declines using different pathways to equilibria survive the lean times

Lecture Notes – Qualitative Analysis of Differential Equations Lecture Notes – Qualitative Analysis of Differen Joseph M. Mahaffy, [email protected] — (3/45) Joseph M. Mahaffy, [email protected] — (4/45) Introduction Bacterial Growth Introduction Bacterial Growth Qualitative Behavior of Differential Equations Logistic Growth Qualitative Behavior of Differential Equations Logistic Growth More Examples Staph Experimental Fit More Examples Staph Experimental Fit BacterialGrowthExperiments 1 BacterialGrowthExperiments 2

Bacterial Growth Experiments Bacterial Growth Experiments Graph of data and logistic growth model Staphylococcus aureus is a common pathogen that can cause food poisoning Staphylococcus aureus 0.4 Cultures of this bacterium satisfy the logistic growth Data from one experiment by Carl Gunderson (Lab of Anca ) 0.3

Segall) 460 Normal strain is grown to the optical density (OD650),

which estimates the number of bacteria 0.2 t (hr) 0 0.5 1 1.5 2 2.5

OD650 0.032 0.039 0.069 0.110 0.170 0.229 Concentration (OD 0.1 t (hr) 3 3.5 4 4.5 5

OD650 0.261 0.288 0.309 0.327 0.347 0 0 1 2 3 4 5 6 7 Time (hr)

Lecture Notes – Qualitative Analysis of Differential Equations Lecture Notes – Qualitative Analysis of Differen Joseph M. Mahaffy, [email protected] — (5/45) Joseph M. Mahaffy, [email protected] — (6/45)

Bacterial Growth Malthusian Growth Previously studied the discrete logistic growth The experiment shows the rapid initial growth of the Could only simulate, but not solve bacteria Qualitative analysis of the equilibrium gave some insight This suggests Malthusian growth to the model behavior Earlier showed the Malthusian growth model: Growth of bacteria is a more continuous process dP Their growth is better characterized by a differential = rP, P (0) = P0 equation dt continuous logistic growth Use the model to study Solution satisfies: growth behavior rt Show several methods to analyze previous data, including P (t)= P0e qualitative methods, exact solution, and parameter fitting Only later does crowding require logistic growth model

Lecture Notes – Qualitative Analysis of Differential Equations Lecture Notes – Qualitative Analysis of Differen Joseph M. Mahaffy, [email protected] — (7/45) Joseph M. Mahaffy, [email protected] — (8/45) Introduction Bacterial Growth Introduction Bacterial Growth Qualitative Behavior of Differential Equations Logistic Growth Qualitative Behavior of Differential Equations Logistic Growth More Examples Staph Experimental Fit More Examples Staph Experimental Fit MalthusianGrowth 2 Logistic Growth 1

Graph shows early data with Malthusian and logistic growth Logistic Growth models After rapid growth the experiment shows bacterial growth Best ﬁtting Malthusian growth model is slowing . t P (t)=0.0279 e0 905 This suggests Logistic growth

Malthusian Growth Earlier showed the Logistic growth model: 0.35 Malthusian Model Logistic Model dP P 0.3 = rP 1 , P (0) = P0 Data dt − M 0.25 Solution satisﬁes: 0.2 P0M 650 P (t)= −rt OD P0 + (M P0)e 0.15 −

0.1 This solution is found by: Separation of Variables and special integration techniques 0.05 Bernoulli’s method 0 0 0.5 1 1.5 2 2.5 Want easier methods to investigate qualitative behavior t Lecture Notes – Qualitative Analysis of Differential Equations Lecture Notes – Qualitative Analysis of Differen Joseph M. Mahaffy, [email protected] — (9/45) Joseph M. Mahaffy, [email protected] — (10/45)

Introduction Bacterial Growth Introduction Example: Logistic Growth Qualitative Behavior of Differential Equations Logistic Growth Qualitative Behavior of Differential Equations Example: Sine Function More Examples Staph Experimental Fit More Examples MalthusianGrowth 2 Qualitative Behavior of Differential Equations

Graph shows the Malthusian and logistic growth models Best ﬁtting logistic growth model is Qualitative Behavior of Diﬀerential Equations 0.3450 P (t)= Techniques follow analysis of discrete dynamical systems 1+12.85 e−1.24 t Find the equilibria Logistic Growth 0.4 Determine the behavior of the solutions near the equilibria

0.35 Study Autonomous Diﬀerential Equations

0.3 dy = f(y) 0.25 dt

650 0.2 OD Graph function f(y) 0.15

0.1 Find Phase Portrait and determine local behavior near

Malthusian Model equilibria 0.05 Logistic Model Data 0 0 1 2 3 4 5 6 t Lecture Notes – Qualitative Analysis of Differential Equations Lecture Notes – Qualitative Analysis of Differen Joseph M. Mahaffy, [email protected] — (11/45) Joseph M. Mahaffy, [email protected] — (12/45) Introduction Introduction Example: Logistic Growth Example: Logistic Growth Qualitative Behavior of Differential Equations Qualitative Behavior of Differential Equations Example: Sine Function Example: Sine Function More Examples More Examples Example:LogisticGrowthModel 1 Example:LogisticGrowthModel 2

Example: Logistic Growth Model For the logistic growth equation: Consider the logistic growth equation: dP P dP P = f(P )=0.05P 1 = f(P ) = 0.05P 1 dt − 2000 dt − 2000

Phase Portrait 30 The graph above of f(P ) shows f(P ) intersects the P -axis at P = 0 and P = 2000 25 These P -intercepts are where f(P )=0or

20 dP =0 15 dt f(P)

10 There is no change in the growth of the population or the population is at equilibrium 5 This is the first step in any qualitative analysis: Find all 0 0 500 1000 1500 2000 equilibria (f(P ) = 0) population (P) Lecture Notes – Qualitative Analysis of Differential Equations Lecture Notes – Qualitative Analysis of Differen Joseph M. Mahaffy, [email protected] — (13/45) Joseph M. Mahaffy, [email protected] — (14/45)

Introduction Introduction Example: Logistic Growth Example: Logistic Growth Qualitative Behavior of Diﬀerential Equations Qualitative Behavior of Diﬀerential Equations Example: Sine Function Example: Sine Function More Examples More Examples Example:LogisticGrowthModel 3 Example:LogisticGrowthModel 4

Local Analysis: Equilibria are Pe = 0 and Pe = 2000 Phase Portrait The graph of f(P ) gives more information Use the above information to draw a Phase Portrait of

To the left of Pe = 0, f(P ) < 0 the behavior of this differential equation along the P -axis dP The behavior of the differential equation is denoted by Since = f(P ) < 0, P (t) is decreasing dt arrows along the P -axis Note that this region is outside the region of biological significance When f(P ) < 0, P (t) is decreasing and we draw an arrow to the left For 0

0 When f(P ) > 0, P (t) is increasing and we draw an arrow dP to the right Since dt = f(P ) > 0, P (t) is increasing Population monotonically growing in this area Equilibria For P > 2000, f(P ) < 0 A solid dot represents an equilibrium that solutions dP approach or stable equilibrium Since dt = f(P ) < 0, P (t) is decreasing An open dot represents an equilibrium that solutions go Population monotonically decreasing in this region away from or unstable equilibrium

Lecture Notes – Qualitative Analysis of Differential Equations Lecture Notes – Qualitative Analysis of Differen Joseph M. Mahaffy, [email protected] — (15/45) Joseph M. Mahaffy, [email protected] — (16/45) Introduction Introduction Example: Logistic Growth Example: Logistic Growth Qualitative Behavior of Differential Equations Qualitative Behavior of Differential Equations Example: Sine Function Example: Sine Function More Examples More Examples Example:LogisticGrowthModel 5 Example:LogisticGrowthModel 6

Phase Portrait Diagram of Solutions for Logistic Growth Model

30 Logistic Growth Model 2500 25

20 2000 15 )

P 1500

( 10 f P(t) 5 1000 0 < > > > > <

−5 500

−500 0 500 1000 1500 2000 2500 0 population (P ) 0 50 100 150 200 t

Lecture Notes – Qualitative Analysis of Differential Equations Lecture Notes – Qualitative Analysis of Differen Joseph M. Mahaffy, [email protected] — (17/45) Joseph M. Mahaffy, [email protected] — (18/45)

Summary of Qualitative Analysis Graph shows solutions either moving away from the equilibrium Example: Sine Function at Pe = 0 or moving toward Pe = 2000 Consider the differential equation: Solutions are increasing most rapidly where f(P ) is at a dx maximum = 2 sin(πx) Phase portrait shows direction of flow of the solutions without dt solving the differential equation Find all equilibria Solutions cannot cross in the tP -plane Phase Portrait analysis Determine the stability of the equilibria Behavior of a scalar differential equation found by just Sketch the phase portrait graphing function Show typical solutions Equilibria are zeros of function Direction of flow/arrows from sign of function Stability of equilibria from whether arrows point toward or away from the equilibria Lecture Notes – Qualitative Analysis of Differential Equations Lecture Notes – Qualitative Analysis of Differen Joseph M. Mahaffy, [email protected] — (19/45) Joseph M. Mahaffy, [email protected] — (20/45) Introduction Introduction Example: Logistic Growth Example: Logistic Growth Qualitative Behavior of Differential Equations Qualitative Behavior of Differential Equations Example: Sine Function Example: Sine Function More Examples More Examples Example:SineFunction 2 Example:SineFunction 3

For the sine function below: dx Phase Portrait: Since 2 sin(πx) alternates sign between = 2 sin(πx) dt integers, the phase portrait follows below: The equilibria satisfy 2 2 sin(πxe) = 0

1 Thus, xe = n, where n is any integer ) x

The sine function passes from negative to positive through π 0 > < > < > e

x = 0, so solutions move away from this equilibrium 2 sin(

The sine function passes from positive to negative through −1 xe = 1, so solutions move toward this equilibrium

From the function behavior near equilibria −2 All equilibria of the form xe =2n (even integer) are unstable 0 1 2 3 4 5 x All equilibria of the form xe =2n + 1 (odd integer) are stable Lecture Notes – Qualitative Analysis of Differential Equations Lecture Notes – Qualitative Analysis of Differen Joseph M. Mahaffy, [email protected] — (21/45) Joseph M. Mahaffy, [email protected] — (22/45)

Introduction Introduction Left Snail Model Example: Logistic Growth Qualitative Behavior of Differential Equations Qualitative Behavior of Differential Equations Pitchfork Bifurcation Example: Sine Function More Examples More Examples Allee Effect Example:SineFunction 4 Left Snail Model: Introduction

Diagram of Solutions for Sine Model The shell of a snail exhibits chirality, left-handed (sinistral) or right-handed (dextral) coil relative to the central axis 5 The Indian conch shell, Turbinella pyrum, is primarily a right-handed gastropod [1] 4 The left-handed shells are “exceedingly rare” 3 The Indians view the rare shells as very holy x(t) The Hindu god “Vishnu, in the form of his most celebrated 2 avatar, Krishna, blows this sacred conch shell to call the army of Arjuna into battle”

1 So why does nature favor snails with one particular handedness? Gould notes that the vast majority of snails grow the dextral 0 form. 0 0.1 0.2 0.3 0.4 0.5 0.6 t [1] S. J. Gould, “Left Snails and Right Minds,” Natural History, April 1995, 10-18, and in the compilation Dinosaur in a Haystack ( 1996) Lecture Notes – Qualitative Analysis of Differential Equations Lecture Notes – Qualitative Analysis of Differen Joseph M. Mahaffy, [email protected] — (23/45) Joseph M. Mahaffy, [email protected] — (24/45) Introduction Left Snail Model Introduction Left Snail Model Qualitative Behavior of Differential Equations Pitchfork Bifurcation Qualitative Behavior of Differential Equations Pitchfork Bifurcation More Examples Allee Effect More Examples Allee Effect Left Snail Model 1 Left Snail Model 2

Clifford Henry Taubes [2] gives a simple mathematical Taubes Snail Model model to predict the bias of either the dextral or sinistral forms for a given species Let p(t) be the probability that a snail is dextral Assume that the probability of a dextral snail breeding with A model that qualitatively exhibits the behavior described a sinistral snail is proportional to the product of the on previous slide: number of dextral snails times sinistral snails Assume that two sinistral snails always produce a sinistral dp 1 = αp(1 p) p , 0 p 1, snail and two dextral snails produce a dextral snail dt − − 2 ≤ ≤ Assume that a dextral-sinistral pair produce dextral and sinistral offspring with equal probability where α is some positive constant By the first assumption, a dextral snail is twice as likely to What is the behavior of this differential equation? choose a dextral snail than a sinistral snail What does its solutions predict about the chirality of Could use real experimental verification of the assumptions populations of snails?

[2] C. H. Taubes, Modeling Diﬀerential Equations in Biology, Prentice Hall, 2001.

Lecture Notes – Qualitative Analysis of Differential Equations Lecture Notes – Qualitative Analysis of Differen Joseph M. Mahaffy, [email protected] — (25/45) Joseph M. Mahaffy, [email protected] — (26/45)

Taubes Snail Model dp 1 Phase Portrait: = αp(1 p) p This diﬀerential equation is not easy to solve exactly dt − − 2

Qualitative analysis techniques for this diﬀerential Phase Portrait for Snail Model (α = 1.5) equation are relatively easily to show why snails are likely 0.1 to be in either the dextral or sinistral forms 2)

/ 0.05 The snail model: 1 − p )( dp 1 p = f(p)= αp(1 p) p , 0 p 1, − 0

(1 < < < > > > dt − − 2 ≤ ≤ p α

1 −0.05 Equilibria are pe =0, 2 , 1 1 f(p) < 0 for 0 0 for 2

Diagram of Solutions for Snail Model Snail Model - Summary Snail Model Figures show the solutions tend toward one of the stable 1 equilibria, pe = 0 or 1 When the solution tends toward pe = 0, then the 0.8 probability of a dextral snail being found drops to zero, so the population of snails all have the sinistral form 0.6 When the solution tends toward pe = 1, then the p(t) population of snails virtually all have the dextral form 0.4 This is what is observed in nature suggesting that this 0.2 model exhibits the behavior of the evolution of snails This does not mean that the model is a good model! 0 It simply means that the model exhibits the basic behavior 0 2 4 6 8 10 t observed experimentally from the biological experiments Lecture Notes – Qualitative Analysis of Differential Equations Lecture Notes – Qualitative Analysis of Differen Joseph M. Mahaffy, [email protected] — (29/45) Joseph M. Mahaffy, [email protected] — (30/45)

Introduction Left Snail Model Introduction Left Snail Model Qualitative Behavior of Differential Equations Pitchfork Bifurcation Qualitative Behavior of Differential Equations Pitchfork Bifurcation More Examples Allee Effect More Examples Allee Effect PitchforkBifurcation 1 PitchforkBifurcation 2

Pitchfork Bifurcation For the differential equation: Bifurcations are when behavior of the differential equation dx 3 changes = αx x dt − A supercritical pitchfork bifurcation has differing numbers of equilibria as a parameter changes Find equilibria by solving Consider the differential equation: 3 2 αxe xe =0 or xe(α xe) = 0 dx 3 − − = αx x , dt − where α can be positive, negative, or zero There is always an equilbrium at xe =0 If α< 0, then xe = 0 is the only equilibrium Find all equilibria If α> 0, then there are three equilibria given by Determine the stability of those equilibria as α changes xe =0, √α For α = 1, sketch the phase portraits and show typical ± solutions± in the x and t solution space

Lecture Notes – Qualitative Analysis of Differential Equations Lecture Notes – Qualitative Analysis of Differen Joseph M. Mahaffy, [email protected] — (31/45) Joseph M. Mahaffy, [email protected] — (32/45) Introduction Left Snail Model Introduction Left Snail Model Qualitative Behavior of Differential Equations Pitchfork Bifurcation Qualitative Behavior of Differential Equations Pitchfork Bifurcation More Examples Allee Effect More Examples Allee Effect PitchforkBifurcation 3 PitchforkBifurcation 4

Phase Portrait: For α = 1, For the diﬀerential equation: − dx dx 3 = αx x3 = x x dt − dt − − The function is always decreasing, intersecting the x-axis at e stable When x = 0 is the only equilibrium, then it is xe = 0

When there are three equilibria, then xe = 0 is an 10 unstable equilibrium, while the equilibria, xe = √α, are ± both stable 5 3 x −

As the parameter α changes from negative to positive, the x − 0 > <

diﬀerential equation’s qualitative behavior changes ) = x ( f e From having a single stable equilibrium at x =0 −5 To three equilibria with xe = 0 becoming unstable and the

other two being stable −10 −2 −1 0 1 2 x

Lecture Notes – Qualitative Analysis of Differential Equations Lecture Notes – Qualitative Analysis of Differen Joseph M. Mahaffy, [email protected] — (33/45) Joseph M. Mahaffy, [email protected] — (34/45)

The function intersects the x-axis at xe = 0, 1 1 ± 4

x(t) 0 2 3 x − x 0 >< > < ) = x ( –1 f −2

−4 –2 −2 −1 0 1 2 0 0.5 1 1.5 2 x t Lecture Notes – Qualitative Analysis of Differential Equations Lecture Notes – Qualitative Analysis of Differen Joseph M. Mahaffy, [email protected] — (35/45) Joseph M. Mahaffy, [email protected] — (36/45) Introduction Left Snail Model Introduction Left Snail Model Qualitative Behavior of Differential Equations Pitchfork Bifurcation Qualitative Behavior of Differential Equations Pitchfork Bifurcation More Examples Allee Effect More Examples Allee Effect PitchforkBifurcation 7 Allee Effect 1 dx Diagram of Solutions: For α = 1 with = x x3 dt − Thick-Billed Parrot: Rhynchopsitta pachyrhycha α = + 1 2

x(t) 0

–1

–2 0 0.5 1 1.5 2 t Lecture Notes – Qualitative Analysis of Differential Equations Lecture Notes – Qualitative Analysis of Differen Joseph M. Mahaffy, [email protected] — (37/45) Joseph M. Mahaffy, [email protected] — (38/45)

Thick-Billed Parrot: Rhynchopsitta pachyrhycha Thick-Billed Parrot: Rhynchopsitta pachyrhycha A gregarious montane bird that feeds largely on conifer The populations of these birds appear to exhibit a property seeds, using its large beak to break open pine cones for the known in ecology as the Allee effect seeds These parrots congregate in large social groups for almost These birds used to fly in huge flocks in the mountainous all of their activities regions of Mexico and Southwestern U. S. The large group allows the birds many more eyes to watch Largely because of habitat loss, these birds have lost much out for predators of their original range and have dropped to only about When the population drops below a certain number, then 1500 breeding pairs in a few large colonies in the mountains these birds become easy targets for predators, primarily of Mexico hawks, which adversely affects their ability to sustain a The pressures to log their habitat puts this population at breeding colony extreme risk for extinction

Lecture Notes – Qualitative Analysis of Differential Equations Lecture Notes – Qualitative Analysis of Differen Joseph M. Mahaffy, [email protected] — (39/45) Joseph M. Mahaffy, [email protected] — (40/45) Introduction Left Snail Model Introduction Left Snail Model Qualitative Behavior of Differential Equations Pitchfork Bifurcation Qualitative Behavior of Differential Equations Pitchfork Bifurcation More Examples Allee Effect More Examples Allee Effect Allee Effect 4 Allee Effect 5

Allee Effect: Equilibria: Suppose that a population study on thick-billed parrots in Set the right side of the differential equation equal to zero: 2 a particular region finds that the population, N(t), of the Ne r a(Ne b) = 0 − − parrots satisfies the differential equation: One solution is the trivial or extinction equilibrium, dN Ne = 0 2 2 = N r a(N b) , When r a(Ne b) = 0, then dt − − − − −8 2 r r where r = 0.04, a = 10 , and b = 2200 (Ne b) = or Ne = b − a ± ra Find the equilibria for this differential equation Three distinct equilibria unless r =0 or b = r/a Determine the stability of the equilibria With the parameters r = 0.04, a = 10−8, andpb = 2200, the Draw a phase portrait for the behavior of this model equilibria are Describe what happens to various starting populations of Ne = 0 Ne = 200 4200 the parrots as predicted by this model

Lecture Notes – Qualitative Analysis of Differential Equations Lecture Notes – Qualitative Analysis of Differen Joseph M. Mahaffy, [email protected] — (41/45) Joseph M. Mahaffy, [email protected] — (42/45)

Solutions: For Phase Portrait: Graph of right hand side of diﬀerential dN equation showing equilibria and their stability = N r a(N b)2 dt − −

100 Allee Effect (zoom near origin) 1 5000

50 Allee Effect (zoom near origin) 0.5 4000 300

200 dN/dt dN/dt 0 0 > < < > 3000 > > > < 100 N(t) N(t) 0 −0.5 0 100 200 300 2000 N –100 −50 –200 1000 0 200 400 600 800 t 0 1000 2000 3000 4000 5000 N 0 0 20 40 60 80 100 t

Lecture Notes – Qualitative Analysis of Differential Equations Lecture Notes – Qualitative Analysis of Differen Joseph M. Mahaffy, [email protected] — (43/45) Joseph M. Mahaffy, [email protected] — (44/45) Introduction Left Snail Model Qualitative Behavior of Differential Equations Pitchfork Bifurcation More Examples Allee Effect Allee Effect 8

Interpretation: Model of Allee Effect From the phase portrait, the equilibria at 4200 and 0 are stable The threshold equilibrium at 200 is unstable If the population is above 200, then it goes to the carrying capacity of this region and reaches the stable population of 4200 If the population falls below 200, then this model predicts extinction, Ne =0 This agrees with the description for these social birds, which require a critical number of birds to avoid predation Below this critical number, the predation increases above reproduction, and the population of parrots goes to extinction If the parrot population is larger than 4200, then their numbers will be reduced by starvation (and predation) to the carrying capacity, Ne = 4200 Lecture Notes – Qualitative Analysis of Differential Equations Joseph M. Mahaffy, [email protected] — (45/45)