Math 345 Intro to Math Biology Lecture 22: Reaction-Diffusion Models Junping Shi College of William and Mary November 12, 2018
2D diffusion equation [Skellam, 1951] P t = D( 2 P x 2 + 2 P y 2 ) + ap, (x, y) R2. Fundamental solution: P(x, y, t) = 1 4πDt eat Note that P(x, y, t)dxdy = e at. R 2 x 2 +y 2 4Dt. Let B R be the ball with radius R in R 2 ; we define R(t) as the number such that outside of B R(t), the total population is always 1. R(t) = 4aDt (this is the front of invasion)
Spreading of muskrat According to a general opinion it was in 1905 that Prince Colloredo Mannsfeld released two males and three females in his estate near Dobris, 4O km southwest of Prague, Czechoslovakia. It was presumed that they originated from Alaska or southeast Canada). During the first ten years they spread out from Dobris in concentric circles. Up to 1913 the radius of expansion increased by between four and thirty km annually. A natural barrier was formed by the mountain chains at the borders in the north and west of Czechoslovakia. But it lasted only till 1918 that the first muskrat was trapped in Bavaria, Southern Germany.
Spreading of muskrat ad = 119777300 296433 Year 1905 1909 1911 1915 1920 1927 Area (km 2 ) 0 5400 14000 37700 79300 201600 1 4π = 32.15422940 The speed of the expansion is R (t) = 4aD = 11.34093989(km/year)
Red fire ants invasion Red imported fire ant, Solenopsis invicta Buren was accidentally introduced to the United States from South America in the 1930s, this ant has since spread to more than 128 million ha in 13 states and Puerto Rico (Callcott, 2002). S. invicta is one of the worst invasive ant pests. 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 established itself throughout the northeast US, southern Quebec, and Ontario. The insect has now spread into Michigan, Minnesota, Virginia, West Virginia, and Wisconsin. Since 1980, 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 insects in the eastern United States; it and other foliage-eating pests cause an estimated $868 million 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
Australia cane toad invasion Cane toads (Bufo marinus) are large anurans (weighing up to 2 kg) that were introduced to Australia 70 years ago to control insect pests in sugar-cane fields. But the result has been disastrous because the toads are toxic and highly invasive. Recent studies find that toads with longer legs can not only move faster and are the first to arrive in new areas, but also that those at the front have longer legs than toads in older (long-established) populations. The disaster looks set to turn into an ecological nightmare because of the negative effects invasive species can have on native ecosystems; over many generations, rates of invasion will be accelerated owing to rapid adaptive change in the invader, with continual spatial selection at the expanding 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 60 km/year in 2000s 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 first introduced into the United States in 1963, whereas bighead, silver, and black carp arrived in the 1970s. All four species escaped into the Mississippi River Basin, and all but the black carp are known to have developed self-sustaining populations. http://fw.ky.gov/fish/pages/asian-carp-information.aspx
Fisher equation Reaction-diffusion equation: combining (chemical) reaction (growth/death, genetic evolution, epidemics, action potential) and (physical) diffusion (movement, dispersal) Example 1 Genetics drifting model dp (wx wy )p + (wy wz )(1 p) = p(1 p) dt w x p 2 + 2w y p(1 p) + w z (1 p) 2 where p(t) is the fraction of one Allele at generation t, and w x, w y and w z are fitness constants. dp A simpler version: = sp(1 p) dt Now consider a species randomly dispersing in an unbounded habitat, and we look for one particular gene. Let p(x, t) be the density of the species which possesses an advantageous allele. Then (Fisher equation or diffusive logistic equation, 1937) p dt (x, t) = D 2 p (x, t) + sp(x, t)(1 p(x, t)). x2 This equation can also come from ecological models.
Diffusive epidemic model Reaction-diffusion SIR model S t = D S S xx βsi I t = D I I xx + βsi αi R t = D R R xx + αi The Black Death, also known as the Black Plague, was a devastating pandemic that first struck Europe in the mid-late 14th century (1347-1350), killing between a third and two-thirds of Europe s population. Almost simultaneous epidemics occurred across 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 75 million people. http://en.wikipedia.org/wiki/black_death
Epidemic wave S t = βsi I t = D I I xx + βsi αi R t = αi Phenomenon: The locally infected group reaches a high, then returns to zero; but nearby region will have the peak of infected people in a slightly later time; so the peak of 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 removed class is dead (thus no diffusion). Healthy foxes tend to stay in their own territory, but rabid will travel at random and attack other foxes. Thus only infected class will diffuse.
Diffusive FitzHugh-Nagumo equation ɛ v t = 2 v w + v(v a)(1 v) w, x2 t = v bw c. v(t, x) is the excitability of the system (voltage), w(t, x) is recovery variable representing the force that tends to return the resting state
Traveling wave pattern Each of biological invasion, epidemic wave and neural propagation has a spatiotemporal pattern of wave movement. In this wave, a gene, a new species, a disease, or a neural signal is propagated through the space. Reaction-diffusion equations are one of mathematical models which is capable of describing such wave propagation. Traveling wave solution: a solution with fixed spatial shape but moving when time evolves. Suppose the shape is w(x) when t = 0 and the speed of moving is c, then the 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 ± (a single 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 = 2 u(x, t) x 2 + 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 c dw dz = d2 w dz 2 + f (w), which is an ordinary differential equation. w + 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) We can do phase plane analysis!
Traveling wave solutions of Fisher s equation u(x, t) t = D 2 u(x, t) x 2 let u(x, t) = w(x ct) Dw + cw + aw(1 w) = 0, let v = w { w = v + au(x, t)(1 u(x, t)) v = c D v a w(1 w) D Equilibrium points: (0, 0) and (1, 0), we look for a solution satisfying and lim w(z) = 1 (road to the advantageous gene). z lim w(z) = 0 z
Existence of traveling waves Theorem (Fisher, 1937; Kolmogoroff-Petrovsky-Piscounoff, 1937) u(x, t) Consider = D 2 u(x, t) t x 2 + 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 lim w(z) = 0 and z lim w(z) = 1. Moreover, when the initial condition is a step function u 0(x) = 1 z when x < 0 and u 0 (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 the other 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 an example 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.
Traveling waves in epidemic models S t = D S S xx R 0 SI I t = D I I xx + R 0 SI I Looking for a traveling wave solution (S(x ct), I (x ct)) with lim S(z) = 1, lim S(z) = S 0, and lim I (z) = 0 z z z ± (S(z) is a wave front, and I (z) is a traveling pulse) There exists a traveling wave for c c min Minimal speed c min = 2 D I (1 R 1 0 ), so there is an epidemic wave when R 0 > 1 (traveling outbreak).