math 442 partial differential equations lecture 22: growth

Diffusion Minimal Patch Size Biological invasion Reaction-diffusion equation

Math 442 Partial Differential EquationsLecture 22: Growth and Spreading with Diffusion

Junping Shi

William & Mary

April 27, 2020


Diffusion

Diffusion is the spontaneous spreading of matter (particles or molecules), heat,momentum, or light. Diffusion is one type of transport phenomenon. Diffusion is themovement of particles from higher chemical potential to lower chemical potential(chemical potential can in most cases be represented by a change in concentration). Itis readily observed, for example, when dried food like spaghetti is cooked; watermolecules diffuse into the spaghetti strings, making them thicker and more flexible. Itis a physical process rather than a chemical reaction, which requires no net energyexpenditure. In cell biology, diffusion is often described as a form of passive transport,by which substances cross membranes.


Brownian motion

The concept of Brownian motion is closely related to diffusion. Brownian motion isthe random movement of microscopic particles in a gas or liquid. This motion iscaused by the collision of the microparticle with the moving atoms of the surroundingmedium.

“Having found motion in the particles of the pollen of all the living plants which I hadexamined, I was led next to inquire whether this property continued after the death ofthe plant, and for what length of time it was retained.”

Robert Brown (1828)people.sc.fsu.edu/~jburkardt/m_src/brownian_motion_simulation/brownian_motion_simulation.html

people.sc.fsu.edu/~jburkardt/m_src/brownian_motion_simulation/brownian_motion_simulation.html


Diffusion equation

u(x , t) is the population density at location x and time t:∂u

∂t(x , t) = D

∂2u

∂x2(x , t), u(x , 0) = f (x).

1 Diffusion is the spontaneous spreading of matter (particles or molecules), heat,momentum, or light. The rate of change w.r.t. time is caused by the the spatialmovement, and the probability of moving to a neighboring location is the samefor all directions.

2 It was first derived by Joseph Fourier in his treatise Theorie analytique de lachaleur, published in 1822, to describe the heat conduction. The particlediffusion equation was originally derived by Albert Einstein in 1905 in order tomodel the Brownian motion.

3 Solution when x ∈ R: u(x , t) =1

√4πDt

∫ ∞−∞

e−(x−y)2

4Dt f (y)dy . In particular

when f (x) = δ(x), u(x , t) =1

√4πDt

e−x2

4Dt .

4 Solution when x ∈ (0, L) with Dirichlet boundary condition

u(0, t) = u(L, t) = 0: u(x , t) =∞∑

k=1

ak e−D k2π2

L2 tsin

(kπx

L

)→ 0(t →∞).

5 Solution when x ∈ (0, L) with Neumann BC ux (0, t) = ux (L, t) = 0:

u(x , t) =1

L

∫ L

0f (y)dy +

∞∑k=1

ak e−D k2π2

L2 tcos

(kπx

L

)→

1

L

∫ L

0f (y)dy(t →∞).


Fourier, Brown and Einstein

(Left) Joseph Fourier (1768-1830) French mathematician and physicist best known forFourier series and their applications to problems of heat transfer and vibrations.(Middle) Robert Brown (1773-1858) Scottish botanist and palaeobotanist whosecontributions include pioneering use of the microscope; descriptions of the cell nucleusand cytoplasmic streaming; and the observation of Brownian motion.(Right) Albert Einstein (1879-1955) German-born theoretical physicist. In 1905, calledhis annus mirabilis (miracle year), he published four groundbreaking papers, whichattracted the attention of the academic world; the first outlined the theory of thephotoelectric effect, the second paper explained Brownian motion, the third paperintroduced special relativity, and the fourth mass-energy equivalence.


Growth in a bounded region

Let u(x , t) be the population density of a biological species living in a habitat (0, L).ut = Duxx + au, 0 < x < L, t > 0,

u(0, t) = u(L, t) = 0, t > 0,

u(x , 0) = f (x), 0 < x < L.

u(x , t) =∞∑

k=1

ak e−D k2π2

L2 t+atsin

(kπx

L

), ak =

∫ L

0f (x)sin

(kπx

L

)dx∫ L

0sin2

(kπx

L

)dx

.

Growth or extinction?

u(x , t) ≈ a1e(−D π2

L2 +a)tsin(πx

L

).

If a > Dπ2/L2 (or L > Daπ), then u(x , t)→∞ as t →∞ (growth).

If a < Dπ2/L2 (or L < Daπ), then u(x , t)→ 0 as t →∞ (extinction).

L = Dπ/a is called minimal patch size for the species for survival.


Habitat fragmentation

Habitat fragmentation describes the emergence of discontinuities (fragmentation) inan organism’s preferred environment (habitat), causing population fragmentation andecosystem decay. It includes (i) reduction in the total area of the habitat; (ii) decreaseof the interior:edge ratio; (iii) isolation of one habitat fragment from other areas ofhabitat; (iv) breaking up of one patch of habitat into several smaller patches; and (v)decrease in the average size of each patch of habitat.

Habitat fragmentation decreases L (area in 1D), and it may cause the size of habitatbelow the minimal patch size to induce the species extinction.


Habitat connectivity

Increasing the habitat network connectivity will increase L, and it helps animals topersist.


Diffusion equation in 2D

u(x , y , t) is the population density at location (x , y) and time t:

∂u

∂t(x , y , t) = D

(∂2u

∂x2(x , y , t) +

∂2

∂y2(x , y , t)

).

Solution with a point concentration (δ(x) initial condition at t = 0):

u(x , y , t) =1

4πDte−

x2+y2

4Dt .

(normal distribution with mean 0 and linearly increasing variance.)

[Skellam, 1951] “Random Dispersal in Theoretical Populations”. Biometrika.∂P

∂t= D

(∂2P

∂x2+∂2P

∂y2

)+ aP, (x , y) ∈ R2.

Fundamental solution: P(x , y , t) =1

4πDteat− x2+y2

4Dt . Note

∫R2

P(x , y , t)dxdy = eat .

Let BR be the ball with radius R in R2; we define R(t) as the number such thatoutside of BR(t), the total population is always 1.

R(t) =√

4aDt (this is the front of invasion)


Spreading of muskrat

It was in 1905 that Prince Colloredo Mannsfeld released 5 muskrats in his estate nearDobris, 40 km southwest of Prague. They originated from North America. During thefirst ten years they spread out from Dobris in concentric circles. Up to 1913 the radiusof expansion increased by between 4 and 13 km annually. A natural barrier was formedby the mountain chains at the borders in the north and west of Czechoslovakia. But itlasted only till 1918 that the first muskrat was trapped in Bavaria, Southern Germany.

Year 1905 1909 1911 1915 1920 1927Area (km2) 0 5400 14000 37700 79300 201600

aD =119777300

296433·

1

4π= 32.15. The speed of the expansion is

R′(t) =√

4aD = 11.34(km/year)


Red fire ants invasion

Red imported fire ant, Solenopsis invicta Buren was accidentally introduced to theUnited States from South America in the 1930s, this ant has since spread to morethan 128 million ha in 13 states and Puerto Rico (Callcott, 2002). S. invicta is one ofthe worst invasive ant pests.

http://www.ars.usda.gov/research/docs.htm?docid=9165

http://www.ars.usda.gov/research/docs.htm?docid=9165


Gypsy moss invasion

The gypsy moth (Lymantria dispar) was introduced in 1868 into the United States.The first US outbreak occurred in 1889, and by 1987, the gypsy moth had establisheditself throughout the northeast US, southern Quebec, and Ontario. The insect hasnow spread into Michigan, Minnesota, Virginia, West Virginia, and Wisconsin. Since1980, the gypsy moth has defoliated over one million acres of forest each year.According to a 2011 report, the gypsy moth is now one of the most destructive insectsin the eastern United States; it and other foliage-eating pests cause an estimated $868million in annual damages in the U.S.

Invasion speed: [Elton 1958]√

4aD = 9.8 km/year

http://en.wikipedia.org/wiki/Gypsy_moths_in_the_United_States

http://en.wikipedia.org/wiki/Gypsy_moths_in_the_United_States


Australia cane toad invasion

Cane toads (Bufo marinus) are large anurans (weighing up to 2 kg) that wereintroduced to Australia 70 years ago to control insect pests in sugar-cane fields. Butthe result has been disastrous because the toads are toxic and highly invasive. Recentstudies find that toads with longer legs can not only move faster and are the first toarrive in new areas, but also that those at the front have longer legs than toads inolder (long-established) populations. The disaster looks set to turn into an ecologicalnightmare because of the negative effects invasive species can have on nativeecosystems; over many generations, rates of invasion will be accelerated owing torapid adaptive change in the invader, with continual ‘spatial selection’ at theexpanding front favouring traits that increase the toads’ dispersal.

Invasion speed: [Shine et.al. 2006,2011] 10 km/year in 1950-60s, but close to 50− 60km/year in 2000shttp://sydney.edu.au/science/biology/shine/canetoad_research/

scientific-publications-the-cane-toad-invasion.shtml

http://www.nature.com/nature/journal/v439/n7078/pdf/439803a.pdf

http://sydney.edu.au/science/biology/shine/canetoad_research/scientific-publications-the-cane-toad-invasion.shtml

http://sydney.edu.au/science/biology/shine/canetoad_research/scientific-publications-the-cane-toad-invasion.shtml

http://www.nature.com/nature/journal/v439/n7078/pdf/439803a.pdf


Asian carp invasion

Bighead, silver, grass, and black carp are native to Asia. Grass carp were firstintroduced into the United States in 1963, whereas bighead, silver, and black carparrived in the 1970s. All four species escaped into the Mississippi River Basin, and allbut the black carp are known to have developed self-sustaining populations.

http://fw.ky.gov/Fish/Pages/Asian-Carp-Information.aspx

http://fw.ky.gov/Fish/Pages/Asian-Carp-Information.aspx


Comparison of invasion speed

Species Intrinsic Diffusion Theoretical Observedgrowth rate coefficient spread rate spread ratea (per year) D (km2/year) c (km/year) c (km/year)

Muskrat 0.2-1.1 51-230 6-32 1-25Cabbage butterfly 9-32 2.4-64 9.3-90 15-170Cereal Leaf beetle 1.6-1.9 0.4 1.6-1.7 27-90Gypsy moth 4.6 <0.34 <2.5 3-20Sea otter (North) 0.056 13.5 1.74 1.4Plague 19 25000 720 320-650Rabies 66 40-50 70 30-60

Table : Rates of spread for various species.

From [Lockwood-Hoopes-Marchetti, 2013] Invasion Ecology, John & Wiley.


Fisher-KPP equation

Reaction-diffusion equation: combining (chemical) reaction (growth/death, geneticevolution, epidemics, action potential) and (physical) diffusion (movement, dispersal)

Example 1 Genetics drifting modeldp

dt= sp(1− p), where p(t) is the fraction of one

Allele at generation t, and wx , wy and wz are fitness constants. Now consider aspecies randomly dispersing in an unbounded habitat. Let p(x , t) be the density of thespecies which possesses an advantageous allele.

∂p

dt(x , t) = D

∂2p

∂x2(x , t) + sp(x , t)(1− p(x , t)).

[Fisher, 1937] [Kolmogorov, Petrovski, Piskunov, 1937]

R. Fisher (1890-1962), A. Kolmogorov (1903-1987), I. Petrovsky (1901-1973)


Traveling wave pattern

Each of biological invasion, epidemic wave and neural propagation has aspatiotemporal pattern of wave movement. In this wave, a gene, a new species, adisease, or a neural signal is propagated through the space. Reaction-diffusionequations are one of mathematical models which is capable of describing such wavepropagation.

Traveling wave solution: a solution with fixed spatial shape but moving when timeevolves. Suppose the “shape” is w(x) when t = 0 and the speed of moving is c, thenthe solution is u(x , t) = w(x − ct), i.e. when t = 1, u(x , 1) = w(x − c), when t = 2,u(x , 2) = w(x − 2c), etc.


Types of traveling waves

Wave front: w(y) monotone function, increases from one state to another state(Fisher equation or epidemic wave S(x , t))Traveling pulse: w(y) has one peak but decays to the same state as y → ±∞ (asingle neural signal or epidemic wave I (x , t))Wave train: w(y) is periodic in y (periodic neural signal or surfing)


Finding traveling wave

∂u(x , t)

∂t= D

∂2u(x , t)

∂x2+ f (u(x , t))

Format of traveling wave solution: u(x , t) = w(x − ct)then w(z) (here we use z = x − ct) satisfies

−cdw

dz= D

d2w

dz2+ f (w), which is an ordinary differential equation.

Dw ′′ + cw ′ + f (w) = 0, a 2nd order differential equation.

Let v = w ′, then w ′ = v and v ′ = w ′′ = −cw ′ − f (w) = −cv − f (w){w ′ = v

v ′ = [−cv − f (w)]/D.

We can do phase plane analysis!


Traveling wave solutions of Fisher’s equation

∂u(x , t)

∂t= D

∂2u(x , t)

∂x2+ au(x , t)(1− u(x , t))

let u(x , t) = w(x − ct)Dw ′′ + cw ′ + aw(1− w) = 0, let v = w ′{

w ′ = v

v ′ = −c

Dv −

a

Dw(1− w)

Equilibrium points: (0, 0) and (1, 0), we look for a solution satisfying limz→−∞

w(z) = 0

and limz→∞

w(z) = 1 (road to the advantageous gene).


Existence of traveling waves

Theorem [Fisher, 1937] [Kolmogoroff-Petrovsky-Piscounoff, 1937]

Consider∂u(x , t)

∂t= D

∂2u(x , t)

∂x2+ au(x , t)(1− u(x , t)). For every c ≥

√4aD, there

is a traveling wave solution u(x , t) = w(x − ct) so that limz→−∞

w(z) = 0 and

limz→∞

w(z) = 1. Moreover, when the initial condition is a step function u0(x) = 1

when x < 0 and u0(x) = 0 when x > 0, then the solution u(t, x) tends to the

traveling wave solution with speed c =√

4aD (minimum speed).

Biological meaning: initially advantageous gene occupies one region x < 0, and theother region x > 0 is occupied by recessive gene. Now the evolution and diffusion(Fisher equation) pushes the advantageous gene into the recessive region, with a

speed√

4aD, and eventually the advantageous gene will occupy all regions. This is anexample of biological invasion. The speed of invasion is again

√4aD, which is same

the one derived by Skellam in 1951 using the linear diffusion equation.


Diffusive epidemic model

The Black Death, also known as the “Black Plague”, was a devastating pandemicthat first struck Europe in the mid-late 14th century (1347-1350), killing between athird and two-thirds of Europe’s population. Almost simultaneous epidemics occurredacross large portions of West Asia and the Middle East during the same period,indicating that the European outbreak was actually part of a multi-regional pandemic.Including Middle Eastern lands, India and China, the Black Death killed at least 75million people. http://en.wikipedia.org/wiki/Black_Death

1918-1920 Spanish pandemic (infected 500 million, death 17-50 mil),2019-2020 Coronavirus pandemic (infected > 3 million, death > 0.2 million)

http://en.wikipedia.org/wiki/Black_Death


Sub-populations

1 Susceptible population S(t): who are not yet infected

2 Infective population I (t): who are infected at time t and are able to spread thedisease by contact with susceptible

3 Removed population R(t): who have been infected and then removed from the

possibility of being infected again or spreading (Methods of removal: isolation orimmunization or recovery or death)

4 The total population N = S(t) + I (t) + R(t) is assumed to be a constant, sobirth and death are ignored.


SIR model

1 Total population is a constant N (except death from the disease) (dimension ofN, S , I ,R: M, number of people, dimension of t: T )

2 A average infective makes contact sufficient to transmit infection with βNothers per unit time (dimension of β: M−1T−1, per person per unit time)

3 A fraction α of infectives leave the infective class per unit time (dimension of α:T−1). If a patient is cured in n units of time, then α = 1/n.

S ′ = −βSI ,

I ′ = βSI − αI ,

R′ = αI .

You can drop the third equation since N = S + I + R.


Epidemic wave

St = −R0SI

It = DI Ixx + R0SI − I

Phenomenon: The locally infected group reaches a high, then returns to zero; butnearby region will have the peak of infected people in a slightly later time; so the peakof infected propagates from center to surrounding areas. That is an epidemic wave.

Here consider the case of rabies (population of foxes). Rabies is lethal so that removedclass is dead (thus no diffusion). Healthy foxes tend to stay in their own territory, butrabid will travel at random and attack other foxes. Thus only infected class will diffuse.Looking for a traveling wave solution (S(x − ct), I (x − ct)) withlim

z→∞S(z) = 1, lim

z→−∞S(z) = S∞ ≥ 0, and lim

z→±∞I (z) = 0

(S(z) is a wave front, and I (z) is a traveling pulse)

There exists a traveling wave for c ≥ cmin = 2√

DI (1− R−10 ), so there is an epidemic

wave when R0 > 1 (traveling outbreak).

math 442 partial differential equations lecture 22: growth

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