Reaction-Diffusion Equations with Applications

Reaction-Diffusion equations with applications Christina Kuttler Sommersemester 2011 Contents 1 Reaction & Diffusion 3 1.1 Introduction....................................... ..... 3 1.2 Examples for typical reactions . ...... 3 1.2.1 Population dynamics . 3 1.2.2 Predator-prey . 4 1.2.3 Competition .................................... ... 4 1.2.4 Symbiosis ....................................... 4 1.2.5 Chemical reactions . 5 1.3 Diffusionequation .................................... .... 9 1.3.1 Fick’s law / Conservation of mass . ....... 9 1.3.2 (Weak) Maximum principle . 10 1.3.3 Uniqueness . 11 1.3.4 Fundamental solution of the Diffusion equation . 12 1.3.5 Including a source . 14 1.3.6 Random motion and the Diffusion equation . 15 1.3.7 “Time”ofDiffusion ................................. 17 2 Mathematical basics / Overview 19 2.1 Some necessary basics from ODE theory . 19 2.2 Boundary conditions . 21 2.3 Separation of Variables . ...... 22 2.4 Similaritysolutions................................ ........ 24 2.5 Blowup........................................... 26 3 Comparison principles 29 4 Existence and uniqueness 31 4.1 Problemformulation ................................. ...... 31 4.2 Abstract linear problems . 33 4.3 Abstract semilinear problems . 34 4.4 Existence and uniqueness for reaction-diffusion problems . 37 4.5 Globalsolutions.................................... ...... 39 5 Stationary solutions / Stability 40 5.1 Basics ........................................... 40 5.2 Example: The spruce budworm model . 41 6 Reaction-diffusion systems 50 6.1 What can be expected from Reaction-diffusion systems? . 50 6.2 Conservativesystems................................ ....... 50 6.2.1 Lotka-Volterrasystem . ..... 50 6.3 Some more basics for Reaction-diffusion systems . ..... 52 7 Waves 57 7.1 Planewaves........................................ 57 7.1.1 Introduction ..................................... 57 7.1.2 Different types of waves . 64 7.1.3 Radially symmetric wave . ..... 64 1 7.2 Waves in Reaction-Diffusion systems . ..... 64 7.2.1 The spread of rabies . 65 7.2.2 Waves of pursuit and evasion in predator-prey systems . 67 7.2.3 Travelling fronts in the Belousov-Zhabotinskii Reaction . ......... 70 7.2.4 Waves in Excitable Media . 72 7.2.5 Travelling Wave Trains in Reaction Diffusion Systems with Oscillatory Kinetics . 79 7.3 Spiralwaves........................................ 82 8 Pattern formation 87 8.1 Turing Mechanisms . 87 8.2 Linear stability analysis for Diffusion-driven instability . ............ 89 8.3 Analysis of Pattern Initiation for the Schnakenberg system . .......... 92 8.4 Mode Selection . 97 8.4.1 Whitenoise...................................... 97 8.4.2 Travelling wave initiation of patterns . ........ 98 8.4.3 Pattern formation in growing domains . ....... 98 8.5 Nonexistence of spatial patterns in reaction diffusion systems . ....... 99 Literature 101 Index 103 2 Chapter 1 Reaction & Diffusion 1.1 Introduction References: [5] A reaction-diffusion equation comprises a reaction term and a diffusion term, i.e. the typical form is as follows: ut = D∆u + f(u) u = u(x,t) is a state variable and describes density/concentration of a substance, a population ... at position x Ω Rn at time t (Ω is a open set). ∆ denotes the Laplace operator. So the first term on the right hand∈ side⊂ describes the “diffusion”, including D as diffusion coefficient. The second term, f(u) is a smooth function f : R R and describes processes with really “change” the present u, i.e. something happens to it (birth, death,→ chemical reaction ...), not just diffuse in the space. It is also possible, that the reaction term depends not only on u, but also on the first derivative of u, i.e. u; and/or explicitly on x. Instead∇ of a scalar equation, one can also introduce systems of reaction diffusion equations, which are of the form u = D∆u + f(x,u, u), t ∇ where u(x,t) Rm. ∈ In this lecture, we will deal with such reaction-diffusion equations, from both, an analytical point of view, but also learn something about the applications of such equations. 1.2 Examples for typical reactions In this section, we consider typical reactions which may appear as “reaction” terms for the reaction- diffusion equations. When the diffusion (i.e. spatial effects) are ignored, they are ODEs, u˙ = f(u) 1.2.1 Population dynamics Often, reaction-diffusion equations are used to describe the spread of populations in space. So, we need some basics about populations dynamics. Generally, the possible stationary states (whereu ˙ = 0) and their stability are of interest, which correspond to population sizes which don’t change over time. Exponential growth f(u) = au for a = const. (the growth rate). 3 Logistic growth u f(u) = au 1 , · − K adding a carrying capacity K as limitation of growth.³ ´ Allee effect n n f(u) = au 1 1 K0 − − K µ ¶ ³ ´ The basis of this model approach is still the logistic growth, but if the population is too low, it will also die out. Such phenomena may appear due to the necessity to find a mate for reproduction, or to defence the group against predators. This leads to the additional factor n 1 . K0 − ³ ´ The above mentioned models concern the growth of single populations. Of course, also the interaction of several populations is of interest. The prototype are interactions of two populations, which can be formulated generally as 2D ODE: x˙ = f(x,y) y˙ = g(x,y) There are three typical special cases for the interaction of two populations: 1.2.2 Predator-prey A predator population y eats from a prey population x, the most famous predator prey model (Lotka Volterra) reads x˙ = ax bxy − y˙ = cxy dy − Modifications are possible and often necessary; e.g. a limited growth of prey for absent predators could be introduced, or other “functional responses” (how much of the prey is eaten by the predators respectively how the predators benefit from the prey), for more details see e.g. [9]. 1.2.3 Competition Here, two species compete for a common food source. Mathematically, one species reduces the carrying capacity of the other species and vice versa. The standard approach of Lotka-Volterra reads x + α y x˙ = a x 1 12 1 − K µ 1 ¶ y + α x y˙ = a y 1 21 2 − K µ 2 ¶ 1.2.4 Symbiosis In this case, the two species benefit from each other. In some sense, it is the opposite of the competition model above: each species increases the carrying capacity of the other species. x α y x˙ = a x 1 − 12 1 − K µ 1 ¶ y α x y˙ = a y 1 − 21 2 − K µ 2 ¶ Of course, there are many more possibilities for such models, these are just rough basic ideas! For more details of the modelling of population growth see e.g. [9]. 4 1.2.5 Chemical reactions References: [3, 6] In many applications (bio)chemical reactions are considered, here we just collect a few very basic ideas how to translate such reactions into ODEs. Law of mass action We consider a very simple irreversible reaction: A + B k C, −→ k is the so-called reaction rate. Assumption: The change of product in time corresponds to the number of collisions between molecules A and B, multiplied by the probability that indeed a reaction happens in case of collision (e.g. there is enough kinetic energy available to initialise the reaction). Let a = [A], b = [B], c = [C] be the concentrations of A, B and C. The term r1ab∆t approximates the number of collisions in ∆t. For the above mentioned probability we choose a constant r2, so the change of c in time can be described by ∆c = abk∆t, where k = r1r2, furthermore (∆t 0) → dc =c ˙(t) = kab, dt the so-called Law of mass action. (This is a mathematical model, not a “law”) Remark that reaction rates and concentrations should remain nonnegative! Reversible reactions Now we consider a reversible reaction, k+ A + B ⇋ C, k− with the reaction rates k+ and k . Assume that the split-up of C is proportional to c. The corresponding ODE system reads: − dc = k+ab k c dt − − da = k c k+ab dt − − db = k c k+ab. dt − − General reaction systems As a generalisation, we consider now r chemical reactions between s species Ui (with concentrations ui), i = 1,...s, which interact simultaneously: s s k l U j r U , j = 1, . r. ij i −→ ij i i=1 i=1 X X lij , rij are the so-called stoichiometric coefficients; they describe loss and gain of the number of molecules Ui in reaction j; kj (t) is the corresponding reaction rate (dependency on t might appear due to change of temperature etc.). The so-called rate function, s lnj gj (t,u) = kj (t) (un) n=1 Y corresponds to the speed of reaction j and can be used to formulate the ODEs for the ui, as a net result of all reactions on Ui: r u˙ = (r l )g (t,u(t)), i = 1,...,s, i ij − ij j j=1 X 5 or in matrix notation u˙ = Sg(t,u(t)), where u = (u ,...,u )T , S = (r l ) the stoichiometric matrix of size s r, g(t,u) = (g (t,u)) Rr. 1 s ij − ij × j ∈ Michaelis-Menten Literature: Murray [11], a nice short introduction can be found in Wikipedia. The enzyme kinetics was established by Leonor Michaelis and Maud Menten in 1913. Generally, en- zymes can compensate fluctuating concentrations of substrate, i.e. they adapt their activity and thereby tune a steady state. This behaviour is the common one, but of course, there are also exceptions. In opposite to the kinetics of chemical reactions, in enzyme kinetics there is the phenomenon of saturation. Even for very high concentrations of the substrate, the metabolic rate cannot be increased unlimited, there is a maximum value vmax.

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