23 moving populations and travelling waves

consider another example of reaction-diffusion equation, nonlinear: Fisher’s equation for the spread of an advantageous allele (1937) (still have diffusion, in 1D) study the dynamics using phase-plane techniques(as non-linear we dont find explicit solution) look for a travelling wave solution find a critical minimal speed for a travelling wave solution to exist (necessary condition)

WAVE OF ADVANCE OF ADVANTAGEOUS GENES By R. A. FISHER, Sc.D., F.R.S. I. THE PROBLEM OF GENE DISPERSION CONSIDER a population distributed in a linear habitat, such as a shore line, which it occupies with uniform density. If at any point of the habitat a mutation occurs, which happens to be in some degree, however slight, advantageous to survival, in the totality of its efferts, we may expect the mutant gene to increase at the expense of the allelomorph or allelomorphs previously occupying the same locus. This process will be first completed in the neighbour-hood of the occurrence of the mutation, and later, as the advantageous gene is diffused into the surrounding population, in the adjacent portions of its range. Supposing the range to be long compared with the distances separating the sites of offspring from those of their parents,** there will be, advancing from the origin, a wave of increase in the gene frequency**-

Assume that : individuals move randomly in a one-dimensional domain, with diffusion coefficient D allele A and B only; A is advantageous over B replacement of allele B by A is proportional to the product of their respective frequencies (σ Let: F(x, t) = fraction of allele A, advantageous over allele B. Hence: F(x, t) ∈ [0, 1] 1 − F(x, t) is the fraction of B If F=1 pop is only allele A If F=0 pop is only allele B σ = selection intensity, so that replacement of B by A happens at rate σF(1 − F) assumed positive we assume the advantageous gene is introduced from the left, diffuses through the population What is the equation that governs the dynamics of F?

F satisfies the equation ∂F(x, t)/∂t = D∂²F(x, t)/∂x + σF(x, t)(1 − F(x, t)) x ∈ IR σ = selection intensity, ∂F(x, t)/∂t =diffusion + replacement of B by A Hence, the reaction term is a logistic growth R(F) = σF(1 − F)

d(F)/dt = σF(1 − F) w/o diffusion solution F increases, bounded above by 1 for different initial conditions 0<F<1 has two steady states: F~ = 0: only gene B present in the population F~ = 1: only gene A present in the population σ > 0: F~ = 1 is asymptotically stable Without movement (no diffusion), each point in space where the frequency of gene A is initially positive, will tend towards replacement of all B’s by A. Every point with B only, will remain with B only. F(x,t) against x starts at 0 remains at 0 for all x starts at 1 remains at 1 for all x if some points of x only formed of allele B F=0 and some other points only A F=1 then no movements, wont be mixing the system

if we perturb the system movement and gene A will increase due to advantage finally replacing B and equilibrium If we have diffusion then part of pop with gene A will move randomly and end the regions where only B was present, fraction of A will grow and replace the whole pop With diffusion, the constant solutions 0 and 1 (homogeneous in space) are still steady states, but what happen when we have only A on one side, and only B on the other? Diffusion mixes the genes at the boundary, and the advantageous gene A overtakes the region with gene B

We look for a travelling wave solution, i.e., F(x, t) = U(x − ct) where c is the constant speed of the wave, U is the shape of the wave, with special condition (wave profile which moves at a constant speed, respectively if we move at that speed its stationary imagine fixing a t then moving that image for another time t ) (invasion from left to right so c >0) (F(x,t) plotted against space x, shifts to the right as time increases) U(x)=F(x,0) profile at t=0 which moves y=x-ct U(y) → 0 as y → +∞ (far right in space x to infinity, past time) U(y) → 1 as y → −∞ (far left in spacex to -infinity, future time) and U′(y) → 0 as y → ±∞

;like SS, if the initial conditions are exactly this travelling wave then the solution is exact after infinite time will tend to this if the initial condition is not exactly the travelling wave but is a pertubation of it then as time goes on it will smooth out and tend to this travellign wave as time goes on, it is a limiting constant

Let F(x, t) = U(x − ct), then: ∂F(x, t)/∂t = −cU′(x − ct) from dU/dt x ∂y/∂t ∂²F(x, t)/∂x²= U′′(x − ct) Y=x-ct Hence U must satisfy the second-order ODE −cU′ = DU′′ − σU(1 − U) Scalar 2nd order non linear ODE We can’t find sol so we do stability analysis in the long term To do this we transfer it into a system of first order equations

imagine you have a population in 1D space with gene B Gene A individuals enter through the left side boundary, as this is the advantageous their introduction diffuses through this boundary and replaces gene B from the left for a fixed time their distribution would decrease the further right you go, but over time this pop enters diffuses and replaces more and more until constant F=1

Introduce new var v U’ = V Now -cV = DV’ - σU(1-U) V’= -cV/D + σU(1-U)/D Standard way to transform into first order eq we have a SYSTEM OF 2 And we can look at stability

SS V’= 0 and U’= 0 V= 0 -cV/D - σU(1-U)/D = 0 (U, V) = (0, 0 ) U = 0 means F = 0 no gene A only gene B RHS of travelling wave In the past and (1, 0) U=1 F=1 No gene B in the population only gene A LHS of the travelling wave Future state

Consider phase plane for U-V SS marked We are looking for a sol that starts in gene b only and ends in gene A only But we consider the limits for y for past and future previously defined Join a U shape under the U axis From (0, 1 ) to (0, 0 ). It is below because we want derivative U’(y) to be negative V(y)<0 so negative V axis part We want (0, 1 ) to be unstable and (0, 0 ) to be stable as y tends to infinity Jacobian A*= [0 1] [{-σ +2σU}/D. -c/D] Compute at SS = (0, 1 ) [0 1] [{-σ /D. -c/D] Consider eq to find the eigenvalies We have one root which is real and positive one real and negative The SS is unstable = (0, 0 ) We want this to be stable and this have eigen values with negative real part [0 1] [-σ/D. -c/D] Trace -c/D <0 Det-c/D + σ/D Positive if c> σ Negative if c <σ Eigen values could my complex and give a spiral But for the sol to be biologically feasible we want U(y) = F(x,t) > 0 Thus we need to have real eigenvalues

We look for a travelling wave solution, i.e., F(x, t) = U(x − ct) where c is the constant speed of the wave, U is the shape of the wave, with special conditions (invasion from left to right) U(y) → 0 as y → +∞ (i.e., x → +∞ or t → −∞) U(y) → 1 as y → −∞ (i.e., x → −∞ or t → +∞) and U′(y) → 0 as y → ±∞ from F→ 1 as x → −∞ F→ 0 as x → +∞

F(x,t) against x frequency 1 LHS x=-100 decreasesto RHS curve RHS F(x,t)=0 x=100

F(x,t) against x similarly but slightly to the right as the t value is increased

For our travelling wave, we look for a solution with: (U, V ) → (1, 0) as y → −∞ (U, V ) → (0, 0) as y → +∞ (y=x-ct so as y → −∞ we consider LHS or a past time, we want F to be 1) Hence: ** (1, 0) should be unstable (limit point as y → −∞) (0, 0) should be stable (limit point as y → ∞) biologically feasible solution U ∈ [0, 1], hence (0, 0) should be a stable node (real eigenvalues of the jacobian)** and no spirals ⇒ Necessary condition: c larger than the minimal wave speed c ≥ c_min = 2√{σD}

⇒ Necessary condition: c larger than the minimal wave speed c ≥ c_min = 2√{σD}

We are looking for a solution of the form prev: 1) Make the substitution of variable F(x,t)= U(x-ct)= U(y) 2)Find the ODE system of 2 ODEs from V=U’ 3) look at equilibria (0,0) and (1,0) F=-α is not biologically feasible 4) (1,0) should be unstable (from the left ) (0,0) should be stable phase portrait curve under the axis from (1,0) and both should be nodes, real eigenvalues , otherwise spiral is not biologically feasible

SI-type model: individuals can be susceptible or become infected, but cannot recover infected foxes are disoriented: move randomly (diffusion D) susceptible healthy foxes are territorial, do not diffuse ∂S(x, t)/∂t = −rS(x, t)I(x, t) (no diffusion) ∂I(x, t)/∂t = rS(x, t)I(x, t)− aI(x, t)+ D∂²I(x, t)∂x² new infections-death+diffusion #new infections proportional to #I and #S , x ∈ IR where: a is the death rate (no recovery) r is the infection rate per unit time (contact rate of one infected (accounts for how often get in contact and probability of transmission) individual with one susceptible individual, times probability of transmission (we still keep track of susceptible location as how many susceptibles dep on how many infected) {z }

Let S0 be the initial (fully susceptible) population density. We can rescale the system for dimensionless variables (exercise): t → τ = rS₀t x → x˜ = x sqrt(rS₀/D) S(x, t) −→ S˜(˜x, τ ) = S(x, t)/S₀ ∈ [0, 1] I(x, t) −→ ˜I(˜x, τ ) = I(x, t)/S₀ ∈ [0, 1] and further we define m :=a/rS₀ (note that 1/m = R₀ is the classical basic reproductive number without diffusion, see Chapter 7- exp #new infections one individual is expected to produce) The dimensionless system reads ∂S˜/∂τ = −S˜˜I ∂ ˜I/∂τ = S˜˜I − m˜I + ∂ ²˜I/∂x˜² , x˜ ∈ IR with S˜ = S˜(˜x, τ ) and ˜I = ˜I(˜x, τ )

the population is initially completely susceptible, S˜ = S/S₀ = 1 infection is seeded to the left of the domain and spreads towards the right as time increases when the infection is introduced in a susceptible population, the epidemic grows until depletion of susceptibles occurs, the n decreases leaving the susceptibles at a lower post-epidemic level S˜ = S˜_f < 1 diagram: susceptibles against x: from thr LHS =above to 0, increases and then hits 1 constant infected against x: at 0 from LHS, increases then decreases back to 0 for later times this is the same but translated to the right

We show that: the epidemic persists only if m < 1 (i.e., R₀ > 1), otherwise it dies out if m < 1, the final size is S˜_f = 1- (S_f /S₀) satisfies S˜_f < m and can be found as the solution of the final size equation S˜_f = 1 +(a/rS₀)ln S˜_f (compare with final size in the standard SIR model without diffusion from Chapter 7) We are looking for a travelling wave that moves at constant speed from L to R with speed above: if a travelling wave exists, it must satisfy c ≥ 2√(1 − m) = 2sqrt(1 −a/(rS₀) =: cmin where cmin is the minimal wave speed speed must be larger than this value (no parameter D, epidemological parameters are the only ones playing a part in epidemic number final number on the c_min)

We are looking for a travelling wave that moves at constant speed from L to R 1) introduce 2 functs U AND V ** We look for a travelling wave solution of the form S˜(˜x, τ ) = U(˜x − cτ ), (profile of susceptibles) ˜I(˜x, τ ) = V (˜x − cτ ) (profile of infected)** 2)Plot against x: Susceptibles S~ to the right of the domain are at 1 (initially) and the after they decrease on the LHS of the domain towards S_f V is 0 before and after the epidemic LHS and RHS so I ** so that, as x˜ → +∞ and τ → −∞ we have S˜ → 1,˜I → 0 (before epidemic wave)** ** and, as x˜ → −∞ and τ → +∞ we have S˜ → S˜_f ,˜I → 0 (after epidemic wave)** We are looking for a travelling wave that moves at constant speed from L to R 3) For S˜(˜x, τ ) = U(˜x − cτ ), (profile of susceptible) we want that as y˜ → −∞ U→ S˜_f , (final susceptibles left) as x → −∞ S~→ S˜_f , (for fixed time as y tends to -infinity so does x~) (for a fixed x~ and look at tau as y tends to -infinity we have tau to infinity :future time so U and S~ tend to S~_f final #susceptibles) for past time tau tends to neg infinity so y tends to +infinity or x tends to +infinity,