Center for Partial Differential Equations, ECNU                                                                             No. 2013-02
                               Diffusion and Advection: some PDE models in
                                                                           Spatial Ecology
                                 Lecture notes from Program on Nonlinear Equations in
                                                                           Population Biology,
                                            Center for PDE, ECNU, mid April-June, 2013
                                                                                    Yuan Lou
                                                The Ohio State University, USA.lou@math.ohio-state.edu
                         The Center receives funding from                          Cente for PDE 
                                                                                   500 Dongchuan Road 
                                                                                   Administration Building 12th floor 
                                                                                   Minhang Campus, East China Normal University 
                                                                                   Shanghai, 200241, China  
                                                                                   Email: admin@cpde.ecnu.edu.cn 
              DIFFUSION AND ADVECTION: SOME PDE MODELS IN SPATIAL
                                             ECOLOGY
                                              Yuan Lou
                   Abstract. This series of lectures will focus on the dynamics of some reaction-diffusion
                   advection models from spatial ecology. Mathematically we are interested in the effect of
                   diffusion and advection on population dynamics in spatially heterogeneous environment.
                   Biologically we are interested in understanding the evolution of dispersal; i.e., loosely
                   speaking, to investigate what kind of dispersal strategies are optimal.
                                            LECTURE1
                                          1. Logistic model
             (1.1)                         dN =rN(1−N)
                                            dt          K
                            r : intrinsic growth rate (no unite);
                            K: carrying capacity (same unit as population size).
               Discrete-time model
                              N =N(t):populationsizeattimet,t = 0,1,2,···
                                t
               Geometric growth N  =RN
                                t+1     t
                                   R=Nt+1 = offspring ♯, ♯ : numbers
                                        Nt     parents ♯
                                               r = lnR
               General model N  =f(N)
                             t+1      t
               Beverton-Holt model
             (1.2)                         N = RNt
                                            t+1   1+R−1N
                                                      K   t
               K: population size where the parent vs offspring ratio
               1991 Mathematics Subject Classification. Primary: 35B44, 35K57; Secondary: 35B30, 35K51.
               Key words and phrases. Diffusion, Advection, Logistic, Lotka-Volterra, Spatial Ecology.
                                                  1
            2                              YUANLOU
             For geometric model,
                                            Nt = 1
                                           Nt+1  R
            What is the next level of models in terms of complexity?
                             Nt =linear function of N
                            N                     t
                              t+1
                                 =line passing through (K,1) and (0, 1 ).
                                                               R
             Then we get (1.2).
             Change (1.2),
                                             RhN
                                    N =         t  , h > 0.
                                     t+h  1+Rh−1N
                                               K   t
                     N −N 1[ RhN                ]
                       t+h   t =         t  −N
                         h      h 1+Rh−1N       t
                                  [    K   t        ]
                                1  (Rh −1)N − Rh−1N2
                              =            t   K   t
                                h      1+Rh−1N
                                      [    K   t ]
                                Rh−1 (1−Nt)N                 N
                              =             K  t  →lnRN(1− ),ash→0.
                                  h    1+Rh−1N           t   K
                                            K   t
            Remark 1.1. Two species model
                                               R1N1(t)
                               
                               N(t+1)=
                                1        1+α N (t)+β N (t)
                                               1 1     1 2
                                               R2N2(t)
                               
                               N(t+1)=                     .
                                  2       1+α N (t)+β N (t)
                                               2 1     1 2
            which is equivalent to
                                N(t)      1+α N (t)+β N (t)
                                1             1 1     1 2
                                       =
                               N(t+1)            R
                                  1                 1
                                N(t)      1+α N (t)+β N (t)
                                2             2 1     1 2
                               N(t+1) =          R         .
                                  2                 2
             Corresponding continuous-time model
                                dN
                                 1
                                     =lnR (1−C N −D N )
                                 dt       1     1 1   1 2
                                dN
                                 2
                                     =lnR (1−C N −D N ),
                                   dt      2     2 1   2 2
            where C ,D , i = 1,2 depend upon α ,β i = 1,2.
                  i  i                   i  i
                           DIFFUSION AND ADVECTION: SOME PDE MODELS IN SPATIAL ECOLOGY                       3
                                       2. Diffusion models of single species
                                                     −→                              n
                (2.1)          ut = ∇·[d(x)∇u]+ b ·∇u+uf(x,u), x ∈ Ω ⊂ R , t > 0.
                                u(x,t) : density at location x and time t.
                                                   −→
                                d(x) > 0,smooth, b = (b1,b2,··· ,bn) Holder continuous.
                  Boundary condition:
                                                          −→
                (2.2)  −→                            ∇u· n =0 on ∂Ω
                where n is the outward unit normal vector.
                  Obvious, u ≡ 0 is a steady state of (2.1)
                  Stability of u ≡ 0: It is determined by the smallest eigenvalue (denoted by σ1).
                                                   −→
                                  
                (2.3)               ∇·[d(x)∇φ]+ b ·∇φ+f(x,0)φ+σφ=0, in Ω
                                        −→
                                    ∇φ· n =0 on ∂Ω.
                Proposition 2.1.
                If σ > 0, then u ≡ 0 is locally stable;
                   1
                If σ < 0, then u ≡ 0 is unstable.
                   1
                Proof. Sub-solution: Consider σ1 < 0. Set u(x) = εφ1(x) where ε > 0, φ1 > 0 is an
                eigenfunction of σ1.
                  Recall                                        −→
                                          u =∇·[d(x)∇u]+ b ·∇u+uf(x,u).
                                           t
                  It suffices to show:
                                                               −→
                                         ut ≤ ∇·[d(x)∇u]+ b ·∇u+uf(x,u)
                                                              −→
                                      ⇔0≤ε∇·[d∇φ1]+εb ·∇φ1+εφ1f(x,εφ1)
                                      ⇔0≤−f(x,0)φ −σ φ +φ f(x,εφ )
                                                        1    1 1      1       1
                                      ⇔0≤[f(x,εφ )−f(x,0)]−σ .
                                                      1                1
                The last inequality holds for 0 < ε << 1 since σ < 0.
                                                                   1
                                                                                                            
                Remark 2.1.
                             {                      −→                               n
                               ut = ∇·[d(x)∇u]+ b ·∇u+uf(x,u), x ∈ Ω ⊂ R , t > 0,
                               u(x,0) = εφ1 (sub−solution).
                  By maximum principles, u(x,t) is increasing in t for every x ⇒ u ≡ 0 is unstable.
                  Exercise: To check, if σ < 0, u ≡ 0 is locally stable (construct super-solution).
                                           1