Slowing Allee effect vs. accelerating heavy tails in monostable r. monostable reaction diffusion equations

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1 Slowing Allee effect vs. accelerating heavy tails in monostable reaction diffusion equations Université de Montpellier NIMS, KAIST, july 205

2 Contents Some population dynamics models Propagation in t u = xx u + ru β ( u) Propagation in t u = J u u + ru β ( u)

3 Contents Some population dynamics models Propagation in t u = xx u + ru β ( u) Propagation in t u = J u u + ru β ( u)

4 Invasion waves in the life sciences Chemistry: Belousov-Zhabotinsky reaction 950 (movie)... Physics: propagation of flames (movie)... Biology: propagation of a front of calcium during fecundation of sea urchin eggs (movie), ecology, population dynamics, epidemiology, evolutionary genetics, invasion phenomena...

5 Propagation of plague in Europe

6 Bacteria invading regions heterogeneous in antibiotics Hermsen et al., PNAS, 202.

7 Reaction diffusion equations u(t, x): density of individuals at time t 0 and space position (or phenotypic trait) x R N. t u = u + f (u).

8 Fisher-KPP nonlinearity f u Growth rate is maximal at low density. Ex: f (u) = ru( u), r > 0.

9 Monostable nonlinearity (non degenerate case) f (u) u Weak Allee effect: due to lack of genetic diversity or the difficulty to find mates, the KPP assumption may be unrealistic.

10 Monostable nonlinearity (degenerate case) f (u) u Weak Allee effect: due to lack of genetic diversity or the difficulty to find mates, the KPP assumption may be unrealistic. Ex: f (u) = ru β ( u), r > 0, β >.

11 Bistable nonlinearity f u Strong Allee effect: due to lack of genetic diversity or the difficulty to find mates, the KPP assumption may be unrealistic. Ex: f (u) = ru(u θ)( u), r > 0.

12 Nonlocal dispersal (seeds) t u = J u u + f (u). J 0 is an even kernel of total mass R J(z) dz =. J(x y) is the probability of jumping from y to x. R J(x y)u(t, y) dy = J u(t, x) is the rate at which individuals arrive at x from all other positions. R J(y x)u(t, x) dy = u(t, x) is the rate at which individuals leave x to reach any other positions.

13 Anomalous diffusion t u = ( ) α u + f (u), 0 < α <. Brownian process normal diffusion Lévy process (or jump process) anomalous diffusion ( ) α, 0 < α <. Super diffusion is expected to accelerate the front! Definition of the fractional Laplacian by Fourier transform on the Schwartz space: ( ) α u = ξ 2α û.

14 Contents Some population dynamics models Propagation in t u = xx u + ru β ( u) Propagation in t u = J u u + ru β ( u)

15 Propagation in t u = xx u + ru( u) Initial data is front-like: 0 u 0, lim inf x u 0 (x) > 0, for some 0 < λ +. u 0 (x) e λx as x +. How does the stable state invade the unstable 0? Definition: we say that c R is the spreading speed if min u(t, x) as t if v < c, x vt max u(t, x) 0 as t if v > c. x vt

16 f u u The minimal speed c Fundamental solution of the linearized equation t u xx u = ru is u(t, x) = e x 2 rt e 4t, 4πt whose level sets follow x c t with c = 2 r. For compactly supported data, the spreading speed exists and is linearly determined: this is c.

17 f u u Spreading speed is selected by the exponential tail When u 0 (x) e λx, spreading speed is c λ c given by { λ + r c λ = λ > c if 0 < λ < r c = 2 r if r λ +. Fisher, Kolmogorov-Petrovsky-Piskunov, Mc Kean, Hadeler-Rothe, Aronson-Weinberger, Berestycki-Hamel, Garnier-Giletti-Nadin...etc..etc...

18 f u u Propagation in t u = J u u + ru( u) When J is exponentially bounded, the same qualitative results hold for t u = J u u + ru( u), supplemented with compactly supported initial data. Weinberger, Coville... Rk: the tail is induced by the equation itself, not by the initial data...

19 Contents Some population dynamics models Propagation in t u = xx u + ru β ( u) Propagation in t u = J u u + ru β ( u)

20 Reid s paradox of rapid plant migration Reid (899): In view of the observed average seed dispersal distance, the oak, to gain its present most northerly position in north Britain after being driven out by the cold probably had to travel fully six hundred miles and this, without external aid, would take something like a million years, whereas it actually took ten thousand years (Holocene period, warm period after glaciations period)... Why such an acceleration? Clark et al (998): rare but long range dispersal events are allowed. Heavy tails for the dispersal kernel J(x). Roques-Hamel-Fayard-Fady-Klein (200): possible remaining northern refuges. Heavy tails for the initial data u 0 (x).

21 Heavy tails Definition: A function ϕ is said to have a heavy tail at + if lim x + ϕ(x)eεx = +, ε > 0. Typical examples of heavy tails are: (ln x) b, b > 0 x α, α > 0 e axb, a > 0, 0 < b < e ax/ ln x, a > 0 VERY HEAVY. ALGEBRAIC. LIGHT AMONG HEAVY ONES. LIGHT AMONG HEAVY ONES.

22 f u u Acceleration in t u = xx u + ru( u) Initial data is front-like: 0 u 0, lim inf x u 0 (x) > 0, HEAVY TAIL: lim u 0(x)e εx = +, ε > 0. x + Theorem (Hamel-Roques 200) Acceleration occurs. More precisely, the level sets of the solution u(t, ) travel exponentially fast as t. Rk: in particular, spreading speed does not exist for heavy tails.

23 f u u Acceleration in t u = J u u + ru( u) Initial data is front-like: 0 u 0, lim inf x u 0 (x) > 0, and left compactly supported, that is u 0 (x) = 0 for x M. The dispersal kernel J has HEAVY TAILS. Theorem (Garnier 20) Acceleration occurs. More precisely, the level sets of the solution u(t, ) travel exponentially fast as t. Rk: in particular, spreading speed does not exist for heavy tails.

24 f u u Acceleration in t u = ( xx ) α u + ru( u) Fundamental solution of the linearized equation is t u + ( xx ) α u = ru u(t, x) = e rt K(t, x) = e rt F (e t 2α )(x), r +2α e whose level sets follow x t +2α t (actually not correct, because of jumps). Acceleration due to the algebraic tail of the fractional Laplacian. Theorem (Cabré-Roquejoffre 203) Acceleration occurs. More precisely, the level sets of the solution u(t, ) travel exponentially fast as t.

25 Propagation in t u = xx u + ru β ( u) Propagation in t u = J u u + ru β ( u) Contents Some population dynamics models Propagation in t u = xx u + ru β ( u) Propagation in t u = J u u + ru β ( u)

26 Propagation in t u = xx u + ru β ( u) Propagation in t u = J u u + ru β ( u) Contents Some population dynamics models Propagation in t u = xx u + ru β ( u) Propagation in t u = J u u + ru β ( u)

27 Propagation in t u = xx u + ru β ( u) Propagation in t u = J u u + ru β ( u) Assumptions for t u = xx u + ru β ( u) Initial data is front-like: 0 u 0, lim inf x u 0 (x) > 0, for some α > 0. u 0 (x) as x +. x α How does the state invade the state 0? By the way, propagation does occur...(zlatos 2005, possible quenching...) Slowing β-allee effect vs α-algebraic tails.

28 f (u) u Some population dynamics models Propagation in t u = xx u + ru β ( u) Propagation in t u = J u u + ru β ( u) Algebraic tails: cancelling acceleration by Allee effect Theorem (A. 205) β + α = acceleration is blocked. Proof: very simple! In sharp contrast with the KPP situation, where heavy tails always lead to acceleration.

29 f (u) u Some population dynamics models Propagation in t u = xx u + ru β ( u) Propagation in t u = J u u + ru β ( u) Heavy tails lighter than algebraic Corollary (A. 205) For heavy tails lighter than algebraic any small Allee effect blocks acceleration. Typical example of such tails are or u 0 (x) e ax/(ln x) as x +, a > 0 u 0 (x) e axb as x +, a > 0, 0 < b <. In contrast with the KPP situation, for which such tails always lead to acceleration.

30 f (u) u Some population dynamics models Propagation in t u = xx u + ru β ( u) Propagation in t u = J u u + ru β ( u) Algebraic tails: acceleration despite Allee effect Assume β < + α. The solution of the ODE (where x serves as a parameter) is w(t, x) = ( d dt w(t, x) = rw β (t, x), u 0 (x) β r(β )t so level sets should follow i.e. acceleration. ) β w(0, x) = u 0 (x) x(t) (r(β )t) ( x α(β ) r(β )t ) β α(β ),

31 f (u) u Some population dynamics models Propagation in t u = xx u + ru β ( u) Propagation in t u = J u u + ru β ( u) Algebraic tails: acceleration despite Allee effect Theorem (A. 205) β < + α = acceleration still occurs. We can actually sandwich the acellerating level sets: for any 0 < λ <, any ε > 0, we have, for t >>, {x : u(t, x) = λ} ((r ε)(β )t) α(β ), ((r + ε)(β )t) In contrast with the KPP situation, the level sets travel polynomially fast. α(β ) ). Proof: construction of a subsolution which has the form of a small bump and travels to the right by accelerating. A perturbation of the solution of d dt w(t, x) = rw β (t, x), w(0, x) = u 0 (x) is used.

32 f (u) u Some population dynamics models Propagation in t u = xx u + ru β ( u) Propagation in t u = J u u + ru β ( u) Heavy tails heavier than algebraic Corollary (A. 205) For heavy tails heavier than algebraic, acceleration occurs whatever the strength of the Allee effect. Typical example of such tails are u 0 (x) as x +, b > 0. (ln x) b

33 f (u) u Some population dynamics models Propagation in t u = xx u + ru β ( u) Propagation in t u = J u u + ru β ( u) β-allee effect vs. α-algebraic tail β No acceleration Acceleration β = + α α

34 Propagation in t u = xx u + ru β ( u) Propagation in t u = J u u + ru β ( u) Contents Some population dynamics models Propagation in t u = xx u + ru β ( u) Propagation in t u = J u u + ru β ( u)

35 Propagation in t u = xx u + ru β ( u) Propagation in t u = J u u + ru β ( u) Assumptions for t u = J u u + ru β ( u) Initial data is front-like: 0 u 0, lim inf x u 0 (x) > 0, and left compactly supported, that is u 0 (x) = 0 for x M. The dispersal kernel has algebraic tails: for some α > 2. J(x) as x. x α How does the state invade the state 0? Slowing β-allee effect vs α-algebraic dispersal tails.

36 f (u) u Some population dynamics models Propagation in t u = xx u + ru β ( u) Propagation in t u = J u u + ru β ( u) Ongoing results Theorem (A. and Coville 205) β + α 2 = acceleration is blocked. Proof: construct an algebraic super solution traveling at constant speed as in the local case. This is more technical but the hyperbola β = + α 2 arises rather naturally...

37 f (u) u Some population dynamics models Propagation in t u = xx u + ru β ( u) Propagation in t u = J u u + ru β ( u) Ongoing results Theorem (A. and Coville 205) β < + α = acceleration still occurs. Proof: first use J to create an algebraic tail at time t = : u 0 (x) = 0 in + = u(, x) as x +, x α then use the underlying degenerate ODE to construct a small bump as an accelerating subsolution...

38 f (u) u Some population dynamics models Propagation in t u = xx u + ru β ( u) Propagation in t u = J u u + ru β ( u) Ongoing results Theorem (A. and Coville 205) ) β + α 2 = acceleration is blocked. 2) β < + α = acceleration still occurs. Right now, something remains unclear...

39 f u f (u) u u Some population dynamics models Propagation in t u = xx u + ru β ( u) Propagation in t u = J u u + ru β ( u) Conclusion KPP situation: Heavy tails acceleration. Degenerate weak Allee effect: Algebraic tails may lead or not to acceleration. In the latter case, level sets are sandwiched and travel polynomially fast. Very heavy tails : acceleration whatever the Allee effect. Light heavy tails : any small Allee effect cancels acceleration.