Cross-diffusion models in Ecology

Transcription

1 Cross-diffusion models in Ecology Gonzalo Galiano Dpt. of Mathematics -University of Oviedo (University of Oviedo) Review on cross-diffusion 1 / 42

2 Outline 1 Introduction: The SKT and BT models The BT model deduced from particle systems What about the SKT model? Models deduction from splitting and differentation State of the art 2 The BT model (University of Oviedo) Review on cross-diffusion 2 / 42

3 Outline 1 Introduction: The SKT and BT models The BT model deduced from particle systems What about the SKT model? Models deduction from splitting and differentation State of the art 2 The BT model (University of Oviedo) Review on cross-diffusion 3 / 42

4 Introduction Introduction: The SKT and BT models First ecological model of interacting populations taking into account drifts caused by the other population seems to be due to Kerner (1959): with a ij, α i, β i positive. t u 1 div ( a 11 u 1 + a 12 u 2 ) = u1 (α 1 β 1 u 2 ), t u 2 div ( a 21 u 1 + a 22 u 2 ) = u2 ( α 2 + β 2 u 1 ), Later on, Jorné (1977) produced a linear stabilty analysis of the model demonstrating that: while self-diffusion tends to damp out all spatial variations in the Lotka-Volterra system, cross-diffusion may give rise to instabilities and to non-constant stationary solutions. (University of Oviedo) Review on cross-diffusion 4 / 42

5 Introduction Introduction: The SKT and BT models First ecological model of interacting populations taking into account drifts caused by the other population seems to be due to Kerner (1959): with a ij, α i, β i positive. t u 1 div ( a 11 u 1 + a 12 u 2 ) = u1 (α 1 β 1 u 2 ), t u 2 div ( a 21 u 1 + a 22 u 2 ) = u2 ( α 2 + β 2 u 1 ), Later on, Jorné (1977) produced a linear stabilty analysis of the model demonstrating that: while self-diffusion tends to damp out all spatial variations in the Lotka-Volterra system, cross-diffusion may give rise to instabilities and to non-constant stationary solutions. (University of Oviedo) Review on cross-diffusion 4 / 42

6 Introduction Introduction: The SKT and BT models First ecological model of interacting populations taking into account drifts caused by the other population seems to be due to Kerner (1959): with a ij, α i, β i positive. t u 1 div ( a 11 u 1 + a 12 u 2 ) = u1 (α 1 β 1 u 2 ), t u 2 div ( a 21 u 1 + a 22 u 2 ) = u2 ( α 2 + β 2 u 1 ), Later on, Jorné (1977) produced a linear stabilty analysis of the model demonstrating that: while self-diffusion tends to damp out all spatial variations in the Lotka-Volterra system, cross-diffusion may give rise to instabilities and to non-constant stationary solutions. (University of Oviedo) Review on cross-diffusion 4 / 42

7 Introduction: The SKT and BT models Nonlinear models First nonlinear cross-diffusion models seem to have been introduced by Shigesada, Kawasaki and Teramoto (1979), and Busenberg and Travis (1983), from different modeling points of view. Shigesada et al. starts from a single continuity equation t u div J(u) = u(α βu), with J(u) = ((c + au)u) + bu Φ. The flow J is composed by three terms: Random dispersal, c u, Dispersal due to population pressure (avoiding over-crowding), a 2 u u, Drift directed to the minima of the environmental potential, Φ. (University of Oviedo) Review on cross-diffusion 5 / 42

8 Introduction: The SKT and BT models Nonlinear models First nonlinear cross-diffusion models seem to have been introduced by Shigesada, Kawasaki and Teramoto (1979), and Busenberg and Travis (1983), from different modeling points of view. Shigesada et al. starts from a single continuity equation t u div J(u) = u(α βu), with J(u) = ((c + au)u) + bu Φ. The flow J is composed by three terms: Random dispersal, c u, Dispersal due to population pressure (avoiding over-crowding), a 2 u u, Drift directed to the minima of the environmental potential, Φ. (University of Oviedo) Review on cross-diffusion 5 / 42

9 Introduction: The SKT and BT models Generalizing the scalar equation to two populations (SKT model) ( t u 1 div ( ) ) (c 1 + a 11 u 1 + a 12 u 2 )u 1 + b1 u 1 Φ = f 1 (u 1, u 2 ), ( t u 2 div ( ) ) (c 2 + a 21 u 1 + a 22 u 2 )u 2 + b2 u 2 Φ = f 2 (u 1, u 2 ), with competitive Lotka-Volterra source f i (u 1, u 2 ) = u i ( αi (β i1 u 1 + β i2 u 2 ) ). (University of Oviedo) Review on cross-diffusion 6 / 42

10 Nonlinear models Introduction: The SKT and BT models Busenberg and Travis generalize the one-population flow J(u) = au u in the following way: Each individual population flow J i is proportional to the gradient of the total population density, e.g. J i (u 1, u 2 ) = au i (u 1 + u 2 ). More in general, for a potential Ψ, they consider u i J i (u 1, u 2 ) = a Ψ(u 1 + u 2 ). u 1 + u 2 Math. analisys by Bertsch et al. in a series of papers starting in the 80 s. (University of Oviedo) Review on cross-diffusion 7 / 42

11 Nonlinear models Introduction: The SKT and BT models Busenberg and Travis generalize the one-population flow J(u) = au u in the following way: Each individual population flow J i is proportional to the gradient of the total population density, e.g. J i (u 1, u 2 ) = au i (u 1 + u 2 ). More in general, for a potential Ψ, they consider u i J i (u 1, u 2 ) = a Ψ(u 1 + u 2 ). u 1 + u 2 Math. analisys by Bertsch et al. in a series of papers starting in the 80 s. (University of Oviedo) Review on cross-diffusion 7 / 42

12 Nonlinear models Introduction: The SKT and BT models Busenberg and Travis generalize the one-population flow J(u) = au u in the following way: Each individual population flow J i is proportional to the gradient of the total population density, e.g. J i (u 1, u 2 ) = au i (u 1 + u 2 ). More in general, for a potential Ψ, they consider u i J i (u 1, u 2 ) = a Ψ(u 1 + u 2 ). u 1 + u 2 Math. analisys by Bertsch et al. in a series of papers starting in the 80 s. (University of Oviedo) Review on cross-diffusion 7 / 42

13 Introduction: The SKT and BT models As remarked by Gurtin and Pipkin (1984), the individual flows J i may depend, instead of in the total population density u 1 + u 2, in a general linear combination of both population densities, possibly different for each population. These weighted sums are motivated when considering a set of species with different characteristics, such as size, behavior with respect to overcrowding, etc. Assuming, in addition, environmental and random effects, J i (u 1, u 2 ) = u i (a i1 u 1 + a i2 u 2 + b i Φ) + c i u i. We refer to this model as to the BT model. (University of Oviedo) Review on cross-diffusion 8 / 42

14 Introduction: The SKT and BT models As remarked by Gurtin and Pipkin (1984), the individual flows J i may depend, instead of in the total population density u 1 + u 2, in a general linear combination of both population densities, possibly different for each population. These weighted sums are motivated when considering a set of species with different characteristics, such as size, behavior with respect to overcrowding, etc. Assuming, in addition, environmental and random effects, J i (u 1, u 2 ) = u i (a i1 u 1 + a i2 u 2 + b i Φ) + c i u i. We refer to this model as to the BT model. (University of Oviedo) Review on cross-diffusion 8 / 42

15 The BT model deduced from particle systems The BT model deduced from particle systems Two populations of particle (i = 1, 2) described by their trajectories (stochastic processes) t R + Xj i i (t) R m, j i = 1,..., N i (N 1 = N 2 = n to simplify). Lagrangian approach: specify interacting laws trajectories are solutions of dx i j (t) = F i j (X 1 1 (t),..., X 1 n (t), X 2 1 (t),..., X 2 n (t))dt + σ i ndw i j (t), X i j (0) = X i j0. Here, F i j : R 2n R m describe deterministic interactions. σ i N R are intensities of random dispersal (Brownian motions). (University of Oviedo) Review on cross-diffusion 9 / 42

16 The BT model deduced from particle systems The BT model deduced from particle systems Two populations of particle (i = 1, 2) described by their trajectories (stochastic processes) t R + Xj i i (t) R m, j i = 1,..., N i (N 1 = N 2 = n to simplify). Lagrangian approach: specify interacting laws trajectories are solutions of dx i j (t) = F i j (X 1 1 (t),..., X 1 n (t), X 2 1 (t),..., X 2 n (t))dt + σ i ndw i j (t), X i j (0) = X i j0. Here, F i j : R 2n R m describe deterministic interactions. σ i N R are intensities of random dispersal (Brownian motions). (University of Oviedo) Review on cross-diffusion 9 / 42

17 The BT model deduced from particle systems Particle state modeled as positive Radon measure ɛ X i { 1 if X i j (t) (B) = j (t) B 0 if Xj i (t) / B for all B B(R m ), Collective behavior in terms of spatial distribution t (empirical measures): u i n(t) = 1 n n j=1 ɛ X i j (t) M(Rm ), giving spatial relative frequency of i-th population, at time t. Lagrangian description: dx i j (t) = F i [u 1 n(t), u 2 n(t)](x i j (t))dt + σi ndw i j (t). (University of Oviedo) Review on cross-diffusion 10 / 42

18 The BT model deduced from particle systems Particle state modeled as positive Radon measure ɛ X i { 1 if X i j (t) (B) = j (t) B 0 if Xj i (t) / B for all B B(R m ), Collective behavior in terms of spatial distribution t (empirical measures): u i n(t) = 1 n n j=1 ɛ X i j (t) M(Rm ), giving spatial relative frequency of i-th population, at time t. Lagrangian description: dx i j (t) = F i [u 1 n(t), u 2 n(t)](x i j (t))dt + σi ndw i j (t). (University of Oviedo) Review on cross-diffusion 10 / 42

19 Deterministic interactions The BT model deduced from particle systems Force exerted on Xj i (t) due to the interaction with all the other particles: 2 n I i j = k=1 a ik n l=1 ζ ε (X i j (t) X k l (t)), with ζ ε a regularizing kernel. In terms of the empirical measure, 2 Ij i = a ik (un k (t) ζ ε )(Xj i (t)). k=1 (University of Oviedo) Review on cross-diffusion 11 / 42

20 The BT model deduced from particle systems Two kinds of deterministic interactions, F i = F i 1 + F i 2, Repulsive: F i 1[u 1 n(t), u 2 n(t)](x i j (t)) = Ii j = 2 k=1 a ik (u k n (t) ζ ε )(X i j (t)). Local attraction, independent of scale, derived from a potential For the stochastic part, we assume F 2 [u 1 n(t), u 2 n(t)](x i j (t)) = b i Φ(X i j (t)). lim n σi n = σ i 0. In some contexts, σ i n stands for the mean free path: average distance covered by a moving particle between successive collisions. Therefore, σ i = 0 must not be discarded. (University of Oviedo) Review on cross-diffusion 12 / 42

21 The BT model deduced from particle systems Two kinds of deterministic interactions, F i = F i 1 + F i 2, Repulsive: F i 1[u 1 n(t), u 2 n(t)](x i j (t)) = Ii j = 2 k=1 a ik (u k n (t) ζ ε )(X i j (t)). Local attraction, independent of scale, derived from a potential For the stochastic part, we assume F 2 [u 1 n(t), u 2 n(t)](x i j (t)) = b i Φ(X i j (t)). lim n σi n = σ i 0. In some contexts, σ i n stands for the mean free path: average distance covered by a moving particle between successive collisions. Therefore, σ i = 0 must not be discarded. (University of Oviedo) Review on cross-diffusion 12 / 42

22 The BT model deduced from particle systems The Euler description Notation: µ, g = g(s) dµ(s). Ito s formula (changing to Eulerian): for any smooth f : R m R + R un(t), i f (, t) = 1 n n j=1 f (Xj u i (t), t) = n(0), i f (, 0) 2 t a ik k=1 0 t 0 un(s), i (un k (s) ζ ε )( ) f (, s) ds t +b i un(s), i Φ f (, s) ds σi n n u i n(s), s f (, s) (σi n) 2 f (, s) n j=1 t 0 f (Xj i i (s), s) dwj (s). ds (University of Oviedo) Review on cross-diffusion 13 / 42

23 The BT model deduced from particle systems The last term is the only explicit source of stochasticity, present for any n < : M i n(f, t) = σi n n n j=1 t 0 f (Xj i i (s), s) dwj (s) Doob s inequality implies M i n(f, t) 0 as n. Thus In the limit n, the Eulerian description becomes deterministic. (University of Oviedo) Review on cross-diffusion 14 / 42

24 The BT model deduced from particle systems The last term is the only explicit source of stochasticity, present for any n < : M i n(f, t) = σi n n n j=1 t 0 f (Xj i i (s), s) dwj (s) Doob s inequality implies M i n(f, t) 0 as n. Thus In the limit n, the Eulerian description becomes deterministic. (University of Oviedo) Review on cross-diffusion 14 / 42

25 Finally... The BT model deduced from particle systems Assume un(t) i u (t) i (deterministic process) represented by a density u i : lim un(t), i f (, t) = u (t), i f (, t) = f (x, t)u i (x, t)dx. n R m Then, for n, we obtain f (x, t)u i (x, t)dx = f (x, 0)u i (x, 0)dx R m R m 2 t a ik k=1 t +b i + 0 t 0 0 R m u i (x, s) u k (x, s) f (x, s)dxds R m u i (x, s) Φ(x) f (x, s)dxds R m u i (x, s) ( s f (x, s) σ2 i f (x, s) ) dxds. (University of Oviedo) Review on cross-diffusion 15 / 42

26 Finally... The BT model deduced from particle systems Assume un(t) i u (t) i (deterministic process) represented by a density u i : lim un(t), i f (, t) = u (t), i f (, t) = f (x, t)u i (x, t)dx. n R m Then, for n, we obtain f (x, t)u i (x, t)dx = f (x, 0)u i (x, 0)dx R m R m 2 t a ik k=1 t +b i + 0 t 0 0 R m u i (x, s) u k (x, s) f (x, s)dxds R m u i (x, s) Φ(x) f (x, s)dxds R m u i (x, s) ( s f (x, s) σ2 i f (x, s) ) dxds. (University of Oviedo) Review on cross-diffusion 15 / 42

27 Finally... The BT model deduced from particle systems Which is a weak formulation of the Cauchy BT model: t u i div ( u i (a i1 u 1 + a i2 u 2 b i Φ) ) c i u i = 0 u i (, 0) = u i0, and c i = σ 2 i /2. The passing to the limit is justified: Capasso et al. (2008), Lachowicz (2011),.. (University of Oviedo) Review on cross-diffusion 16 / 42

28 What about the SKT model? What about the SKT model? Two (unsatisfactory) ways of fitting to the particle method: Diffusion coefficient: σ i = ( c i + a i1 u 1 + a i2 u 2 ) 1 2. But diffusion should decrease with concentration (sense of mean free path). Convection: ( u 1 (a 11 u 1 + a 12 u 2 ) ) ( ) = u 1 2a11 u 1 + a 12 u 2 + a12 u 2 u 1 ( (2a11 u 2 ) ) = u 1 + a 12 u1 + a 12 u 2 u 1 Not the first flow one thinks of... (University of Oviedo) Review on cross-diffusion 17 / 42

29 What about the SKT model? What about the SKT model? Two (unsatisfactory) ways of fitting to the particle method: Diffusion coefficient: σ i = ( c i + a i1 u 1 + a i2 u 2 ) 1 2. But diffusion should decrease with concentration (sense of mean free path). Convection: ( u 1 (a 11 u 1 + a 12 u 2 ) ) ( ) = u 1 2a11 u 1 + a 12 u 2 + a12 u 2 u 1 ( (2a11 u 2 ) ) = u 1 + a 12 u1 + a 12 u 2 u 1 Not the first flow one thinks of... (University of Oviedo) Review on cross-diffusion 17 / 42

30 What about the SKT model? What about the SKT model? Two (unsatisfactory) ways of fitting to the particle method: Diffusion coefficient: σ i = ( c i + a i1 u 1 + a i2 u 2 ) 1 2. But diffusion should decrease with concentration (sense of mean free path). Convection: ( u 1 (a 11 u 1 + a 12 u 2 ) ) ( ) = u 1 2a11 u 1 + a 12 u 2 + a12 u 2 u 1 ( (2a11 u 2 ) ) = u 1 + a 12 u1 + a 12 u 2 u 1 Not the first flow one thinks of... (University of Oviedo) Review on cross-diffusion 17 / 42

31 Lattice modeling What about the SKT model? Let {x j } denote a mesh of size h of a 1D interval, and consider, for i = 1, 2, d dt u i(x j ) = Rj 1 i u i(x j 1 ) + L i j+1 u i(x j+1 ) ( Rj i + L i ) j ui (x j ). (1) Dispersal rates, Rj i and L i j, may depend on concentrations. Assuming R i j = L i j = σ 0 p i (u 1 (x j ), u 2 (x j )), σ 0 > 0, Eq. (1) is a finite differences formula for (u i p i (u 1, u 2 )), when σ 0 = 1 h 2. (University of Oviedo) Review on cross-diffusion 18 / 42

32 What about the SKT model? Thus, for Constant p s, p i = c i = const linear diffusion, t u i = c i u i. Linear p s, p i = c i + a i1 u 1 + a i2 u 2 SKT model t u i = c i ( u i (c i + a i1 u 1 + a i2 u 2 ) ). BT model can, in general, not be cast in this form, since is not conservative. u i (a i1 u 1 + a i2 u 2 ) (University of Oviedo) Review on cross-diffusion 19 / 42

33 Models deduction from splitting and differentation Model deduction from differentation and splitting Species U splits into two species U 1 and U 2, but keeping most of original behavior (Sánchez-Palencia, 2011). Thus, if U satisfies U (t) = U(t) ( α βu(t) ), for t (0, t ), U(0) = U 0 > 0. (2) after splitting, (U 1, U 2 ) satisfies U i (t) = U i(t) ( α β ( U 1 (t) + U 2 (t) )), for t (t, T ), U i (t ) = U i0 > 0, with U 10 + U 20 = U(t ) Under this splitting without differentiation, U 1 + U 2 still satisfies (2) for t t. (University of Oviedo) Review on cross-diffusion 20 / 42

34 Models deduction from splitting and differentation Model deduction from differentation and splitting Species U splits into two species U 1 and U 2, but keeping most of original behavior (Sánchez-Palencia, 2011). Thus, if U satisfies U (t) = U(t) ( α βu(t) ), for t (0, t ), U(0) = U 0 > 0. (2) after splitting, (U 1, U 2 ) satisfies U i (t) = U i(t) ( α β ( U 1 (t) + U 2 (t) )), for t (t, T ), U i (t ) = U i0 > 0, with U 10 + U 20 = U(t ) Under this splitting without differentiation, U 1 + U 2 still satisfies (2) for t t. (University of Oviedo) Review on cross-diffusion 20 / 42

35 Models deduction from splitting and differentation Model deduction from differentation and splitting Species U splits into two species U 1 and U 2, but keeping most of original behavior (Sánchez-Palencia, 2011). Thus, if U satisfies U (t) = U(t) ( α βu(t) ), for t (0, t ), U(0) = U 0 > 0. (2) after splitting, (U 1, U 2 ) satisfies U i (t) = U i(t) ( α β ( U 1 (t) + U 2 (t) )), for t (t, T ), U i (t ) = U i0 > 0, with U 10 + U 20 = U(t ) Under this splitting without differentiation, U 1 + U 2 still satisfies (2) for t t. (University of Oviedo) Review on cross-diffusion 20 / 42

36 Models deduction from splitting and differentation By splitting with differentiation we mean that (U 1, U 2 ) satisfies U i (t) = U i(t) ( α i ( β i1 U 1 (t) + β i2 U 2 (t) )) Under this splitting, U 1 + U 2 does not satisfy, in general, the original problem for t t. (University of Oviedo) Review on cross-diffusion 21 / 42

37 Models deduction from splitting and differentation The splitting model in terms of PDEs: cross-diffusion We start with one species, satisfying t u div J(u) = f (u) in Q (0,t ) = (0, t ) Ω, J(u) ν = 0 on Γ (0,t ) = (0, t ) Ω, u(0, ) = u 0 0 on Ω, with J(u) = u u + uq, f (u) = u(α βu). q usually determined as q = ϕ. (University of Oviedo) Review on cross-diffusion 22 / 42

38 Models deduction from splitting and differentation After splitting, u 10 + u 20 = u(t, ). t u i div J i (u 1, u 2 ) = f i (u 1, u 2 ) in Q (t,t ), J i (u 1, u 2 ) ν = 0 on Γ (t,t ), u i (t, ) = u i0 on Ω, If differentiation only takes place through the LV term, J 1 (u 1, u 2 ) + J 2 (u 1, u 2 ) = J(u 1 + u 2 ) = (u 1 + u 2 ) (u 1 + u 2 ) + (u 1 + u 2 )q (University of Oviedo) Review on cross-diffusion 23 / 42

39 Models deduction from splitting and differentation After splitting, u 10 + u 20 = u(t, ). t u i div J i (u 1, u 2 ) = f i (u 1, u 2 ) in Q (t,t ), J i (u 1, u 2 ) ν = 0 on Γ (t,t ), u i (t, ) = u i0 on Ω, If differentiation only takes place through the LV term, J 1 (u 1, u 2 ) + J 2 (u 1, u 2 ) = J(u 1 + u 2 ) = (u 1 + u 2 ) (u 1 + u 2 ) + (u 1 + u 2 )q (University of Oviedo) Review on cross-diffusion 23 / 42

40 Models deduction from splitting and differentation The linear terms differentiate in a natural way, e.g. (u 1 + u 2 )q = u 1 q + u 2 q. The nonlinear term admits several reasonable decompositions. 1 Leading to (a special case of) the SKT model (G. 2012), u i u i + b i (u 1 u 2 ), b i 0, b 1 + b 2 = 1. 2 Leading to BT model (G. and Selgas, 2015) J i (u 1, u 2 ) = u i (u 1 + u 2 ) + u i q, (University of Oviedo) Review on cross-diffusion 24 / 42

41 Models deduction from splitting and differentation The linear terms differentiate in a natural way, e.g. (u 1 + u 2 )q = u 1 q + u 2 q. The nonlinear term admits several reasonable decompositions. 1 Leading to (a special case of) the SKT model (G. 2012), u i u i + b i (u 1 u 2 ), b i 0, b 1 + b 2 = 1. 2 Leading to BT model (G. and Selgas, 2015) J i (u 1, u 2 ) = u i (u 1 + u 2 ) + u i q, (University of Oviedo) Review on cross-diffusion 24 / 42

42 Models deduction from splitting and differentation The linear terms differentiate in a natural way, e.g. (u 1 + u 2 )q = u 1 q + u 2 q. The nonlinear term admits several reasonable decompositions. 1 Leading to (a special case of) the SKT model (G. 2012), u i u i + b i (u 1 u 2 ), b i 0, b 1 + b 2 = 1. 2 Leading to BT model (G. and Selgas, 2015) J i (u 1, u 2 ) = u i (u 1 + u 2 ) + u i q, (University of Oviedo) Review on cross-diffusion 24 / 42

43 State of the art State of the art: SKT model Existence of solutions Amann (1989) Local in time. Need L bounds to extend to T =. First existence result for full matrix: Yagi (1993) Under assumption a 12 < 8a 11, a 21 < 8a 22 i.e. diffusion matrix positive definite. 1D case without restrictions on coefficients: G., Garzón and Jüngel (2003) Extension to 3D case: Chen and Jüngel (2004) Generalizations: Desvilletes et al. (2014), Jüngel (2015), G., Jüngel and Milisic (in progress). (University of Oviedo) Review on cross-diffusion 25 / 42

44 State of the art State of the art: SKT model Pattern formation: Gambino, Lombardo and Sammartino (2012, 2013), Ruiz-Baier and Tian (2013),... Numerical discretization: G., Garzón and Jüngel (2001, 2003), Gambino, Lombardo and Sammartino (2009), Andreianov, Bendahmane, and Ruiz-Baier (2011),... Travelling wave, exact solutions: Zhao (2005), Cherniha (2008) Ecological models: G. and Velasco (2011, 2013),... Steady state... (University of Oviedo) Review on cross-diffusion 26 / 42

45 State of the art: BT model State of the art Existence of solutions A series of papers by Bertsch, Gurtin, Hilhorst, Mimura, Peletier, and others (1984, 1985, 2010, 2013, 2015) Contact-inhibition problem G. and Selgas (2014, 2015, 2016) existence for BT model, numerics G., Shmarev and Velasco (2015) Contact-inhibition problem: existence, nonuniqueness (University of Oviedo) Review on cross-diffusion 27 / 42

46 The BT model Outline 1 Introduction: The SKT and BT models The BT model deduced from particle systems What about the SKT model? Models deduction from splitting and differentation State of the art 2 The BT model (University of Oviedo) Review on cross-diffusion 28 / 42

47 The BT model The BT model For T > 0 and Ω R m, find u i : Ω (0, T ) R such that, for i = 1, 2, t u i div J i (u 1, u 2 ) = f i (u 1, u 2 ) in Q T = Ω (0, T ), J i (u 1, u 2 ) n = 0 on Γ T = Ω (0, T ), u i (, 0) = u i0 in Ω, with flow and competitive Lotka-Volterra functions given by ( J i (u 1, u 2 ) = u i ai1 u 1 + a i2 u 2 + b i q ) + c i u i, ( ) f i (u 1, u 2 ) = u i αi β i1 u 1 β i2 u 2. (University of Oviedo) Review on cross-diffusion 29 / 42

48 The BT model Assumptions 1 Ω R m (m = 1, 2 or 3) bounded, Ω Lipschitz continuous. 2 a ij, c i, α i, β ij 0 and L (Q T ), b i L (Q T ). Besides, a 0 > 0 such that 4a 11 a 22 (a 12 + a 21 ) 2 > a 0 a.e. in Q T. 3 q (L 2 (Q T )) m. 4 u i0 0, and L (Ω). (University of Oviedo) Review on cross-diffusion 30 / 42

49 The BT model Main tool for the analysis is the entropy functional E(t) = 2 i=1 Using F (u i ) = ln u i as a test function, E(t) + Q t Ω F(u i (, t)) 0, with F (s) = s(ln s 1) + 1. ( 2 (a ii u i 2 + 2c i ) u i 2 ) + (a 12 + a 21 ) u 1 u 2 i=1 and, from assumptions, = E(0) + Q t 2 i=1 ( b i q u i + f i (u 1, u 2 ) ln u i ), E(t) + a 0 Q t ( u u 2 2 ) (E(0) + C 1 ) e C 2t. (University of Oviedo) Review on cross-diffusion 31 / 42

50 The BT model Main tool for the analysis is the entropy functional E(t) = 2 i=1 Using F (u i ) = ln u i as a test function, E(t) + Q t Ω F(u i (, t)) 0, with F (s) = s(ln s 1) + 1. ( 2 (a ii u i 2 + 2c i ) u i 2 ) + (a 12 + a 21 ) u 1 u 2 i=1 and, from assumptions, = E(0) + Q t 2 i=1 ( b i q u i + f i (u 1, u 2 ) ln u i ), E(t) + a 0 Q t ( u u 2 2 ) (E(0) + C 1 ) e C 2t. (University of Oviedo) Review on cross-diffusion 31 / 42

51 The BT model Main tool for the analysis is the entropy functional E(t) = 2 i=1 Using F (u i ) = ln u i as a test function, E(t) + Q t Ω F(u i (, t)) 0, with F (s) = s(ln s 1) + 1. ( 2 (a ii u i 2 + 2c i ) u i 2 ) + (a 12 + a 21 ) u 1 u 2 i=1 and, from assumptions, = E(0) + Q t 2 i=1 ( b i q u i + f i (u 1, u 2 ) ln u i ), E(t) + a 0 Q t ( u u 2 2 ) (E(0) + C 1 ) e C 2t. (University of Oviedo) Review on cross-diffusion 31 / 42

52 The BT model The limit case 4a 11 a 22 = (a 12 + a 21 ) 2 Parabolic-hyperbolic problem Solved in (at least) two cases: General initial data, but strong assumptions on coefficients: t u i div ( au i (u 1 + u 2 ) + bqu i + c u i ) = ui (α β(u 1 + u 2 )), The key is that u = u 1 + u 2 satisfies a porous medium type PDE. Thus, if u 10 + u 20 > 0 in Ω then u is smooth. An approximation is constructred via nonlinear regularization (SKT type), adding a δ 2 ( u i (u 1 + u 2 ) ) for which u1 δ + u2 δ still satisfies a PM equation. ui δ only converges weakly* in L, but u δ strongly in H 1 (Ω) (University of Oviedo) Review on cross-diffusion 32 / 42

53 The BT model The limit case 4a 11 a 22 = (a 12 + a 21 ) 2 Parabolic-hyperbolic problem Solved in (at least) two cases: General initial data, but strong assumptions on coefficients: t u i div ( au i (u 1 + u 2 ) + bqu i + c u i ) = ui (α β(u 1 + u 2 )), The key is that u = u 1 + u 2 satisfies a porous medium type PDE. Thus, if u 10 + u 20 > 0 in Ω then u is smooth. An approximation is constructred via nonlinear regularization (SKT type), adding a δ 2 ( u i (u 1 + u 2 ) ) for which u1 δ + u2 δ still satisfies a PM equation. ui δ only converges weakly* in L, but u δ strongly in H 1 (Ω) (University of Oviedo) Review on cross-diffusion 32 / 42

54 The BT model The limit case 4a 11 a 22 = (a 12 + a 21 ) 2 Parabolic-hyperbolic problem Solved in (at least) two cases: General initial data, but strong assumptions on coefficients: t u i div ( au i (u 1 + u 2 ) + bqu i + c u i ) = ui (α β(u 1 + u 2 )), The key is that u = u 1 + u 2 satisfies a porous medium type PDE. Thus, if u 10 + u 20 > 0 in Ω then u is smooth. An approximation is constructred via nonlinear regularization (SKT type), adding a δ 2 ( u i (u 1 + u 2 ) ) for which u1 δ + u2 δ still satisfies a PM equation. ui δ only converges weakly* in L, but u δ strongly in H 1 (Ω) (University of Oviedo) Review on cross-diffusion 32 / 42

55 The BT model The limit case 4a 11 a 22 = (a 12 + a 21 ) 2 Parabolic-hyperbolic problem Solved in (at least) two cases: General initial data, but strong assumptions on coefficients: t u i div ( au i (u 1 + u 2 ) + bqu i + c u i ) = ui (α β(u 1 + u 2 )), The key is that u = u 1 + u 2 satisfies a porous medium type PDE. Thus, if u 10 + u 20 > 0 in Ω then u is smooth. An approximation is constructred via nonlinear regularization (SKT type), adding a δ 2 ( u i (u 1 + u 2 ) ) for which u1 δ + u2 δ still satisfies a PM equation. ui δ only converges weakly* in L, but u δ strongly in H 1 (Ω) (University of Oviedo) Review on cross-diffusion 32 / 42

56 The BT model Contact-inhibition problem. Initial data segregated: u 10 + u 20 > 0 and supp(u 10 ) supp(u 20 ) = Bertsch et al. introduce solving (P) B with F i given in terms of f i. u = u 1 + u 2, r = u 1 u, t u x (u( x u)) = F 1 (u, r) in Q T, t r x u x r δ xx r = F 2 (u, r) in Q T, x u = x r = 0 on Γ T, u(0, ) = u 0, r(0, ) = r0 δ in Ω, (University of Oviedo) Review on cross-diffusion 33 / 42

57 The BT model Bertsch et al. show existence in 1D and multi-dimensional (more restrictions) using parabolic regulariztion of the auxiliar problem, characteristics. Using part of their arguments, we also show existence in 1D with the direct parabolic regularization. The key is a BV (Ω) bound, maximal regularity expected. (University of Oviedo) Review on cross-diffusion 34 / 42

58 The BT model Example: an explicit solution H Heaviside and B Barenblatt. u 10 (x) = H(x x 0 )B(0, x), u 20 (x) = H(x 0 x)b(0, x). The L (0, T ; BV (Ω)) functions u 1 (t, x) = H(x η(t))b(t, x), u 2 (t, x) = H(η(t) x)b(t, x), with η(t) = x 0 (1 + t t )1/3, are a weak solution of t u i x (u i x (u 1 + u 2 )) = 0 (University of Oviedo) Review on cross-diffusion 35 / 42

59 The BT model Example: an explicit solution H Heaviside and B Barenblatt. u 10 (x) = H(x x 0 )B(0, x), u 20 (x) = H(x 0 x)b(0, x). The L (0, T ; BV (Ω)) functions u 1 (t, x) = H(x η(t))b(t, x), u 2 (t, x) = H(η(t) x)b(t, x), with η(t) = x 0 (1 + t t )1/3, are a weak solution of t u i x (u i x (u 1 + u 2 )) = 0 (University of Oviedo) Review on cross-diffusion 35 / 42

60 The BT model Heaviside-Barenblatt example 4 2 h=0.04, t= x h=0.008, t= x 4 2 h=0.04, t= x 4 2 h=0.008, t= x h=0.04, t= x h=0.008, t= x (University of Oviedo) Review on cross-diffusion 36 / 42

61 The BT model Invasion example h=0.008, t= h=0.008, t= x h=0.008, t= x x h=0.008, t= x (University of Oviedo) Review on cross-diffusion 37 / 42

62 The BT model SKT versus BT 30 u 1 30 u 1 u 2 u (University of Oviedo) Review on cross-diffusion 38 / 42

63 The BT model 0.5 u 1 u u 1 u u 1 u Figure : Transient states corresponding to: top left: b 1 = 1, b 2 = 10, top right: a 11 = a 12 = 3, a 21 = a 22 = 1, bottom: a 11 = 4, a 12 = 0, a 21 = 3.9, a 22 = 1. (University of Oviedo) Review on cross-diffusion 39 / 42

64 The BT model (University of Oviedo) Review on cross-diffusion 40 / 42

65 The BT model (University of Oviedo) Review on cross-diffusion 41 / 42

66 The BT model Thank you! (University of Oviedo) Review on cross-diffusion 42 / 42