Global bifurcation structure of a limiting system to the SKT competition model with cross-diffusion (Qualitative Theory on ODEs and their applications to Mathematical Modeling)
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(2) 46 for intra-specific intra‐specific competitions while bb_{2}2 and cc_{1}1 account for inter-specific inter‐specific competicompeti‐ tions. The constants α and α represent intra-specific population pressures, also \alpha_{11} \alpha_{22} intra‐specific 11 22 and α are the coefficients of inter-specific known as self-diffusion rates, and α \alpha_{12} \alpha_{21} self‐diffusion coefficients inter‐specific 12 21 population pressures, also known as cross-diffusion cross‐diffusion rates. The effect effect of cross-diffusion cross‐diffusion affects affects the population pressure between two difdif‐ ferent kinds. It is an interesting problem to see whether this effect may give rise effect to a spatial segregation or not, and clarify its mechanism. We should remark that it is well known that the important quantities involving the constants aa_{i},i , bb_{i},i , cc_{i}(i=1,2) i (i = 1, 2) are only a1 , a2. b1 , b2. c1 .. c2. A:= \frac{a_{1} {a_{2} , B:=\frac{b_{1} {b_{2} , C:=\frac{c_{1} {c_{2}. A :=. B :=. C :=. (1.5) (1.5). It seems natural to consider the following two cases separately: the ”strong “ strong comcom‐ B<C C<B petition” case B < C and the ”weak competition” case C < B. The behavior of petition” “weak competition” . solution in case BB<C < C is very different > B. different from CC>B. We We refer refer to to [7] [7] and and [8] [8] for for further further details details of of this this model. model. A lot of research works are done by the singular perturbation method, which started started from from aa theoretical theoretical research research by by Mimura Mimura [5]. [5]. Kan-on Kan‐on [1] [1] obtained obtained some some criteria criteria on on the the stability stability of of those those non-constant non‐constant solutions solutions of of (P). (P). However, However, it it is is not easy to clarify the global structure of stationary solutions and stability of stationary solutions. N ‐dimensional case Lou Lou and and Ni Ni [2], [2], [3] [3] started started to to investigate investigate N-dimensional case and and general general diffudiffu‐ sion coefficients. \alpha_{11}=\alpha_{21}= coefficients. To investigate the cross-diffusion cross‐diffusion effects, effects, let us put α 11 = α21 = α\alpha_{22}=0 α12 /d1 . We have 22 = 0 and rr := :=\alpha_{12}/d_{1} ⎧ ut = d1 Δ[(1 + rv)u] + u(a1 − b1 u − c1 v) in Ω × (0, ∞), (1.6) ⎪ ⎪ ⎪ ⎪ ⎨ vt = d2 Δv + v(a2 − b2 u − c2 v) in Ω × (0, ∞), (1.7) N (TPr ) ∂u ∂v ⎪ = =0 on ∂Ω × (0, ∞), (1.8) ⎪ ⎪ ∂ν ⎪ ⎩ ∂ν u(x, 0) = u0 (x) ≥ 0, v(x, 0) = v0 (x) ≥ 0 in Ω, (1.9). (TP_{r}^N)\{beginary}{l u_t}=d{1\triangle[(1+rv)u] (a_{1}-b uc_{1}v)in\Omega\cros(0, \infty),(1.6 v_{t}=d2\trianglev+(a_{2}-b uc_{2}v)in\Omega\cros(0,\infty),(1. 7)\frac{ptialu}{\prtialnu}=\frac{ptialv}{\prtialnu}=0on \partilOmega\cros(0,\infty),(1.8 u(x,0)=_{}(x)\geq0,v(x )=_{0}(x)\geq0in\Omega,(1.9) \end{ary}. N where uu=u(x, = u(x, t) t) and vv=v(x, = v(x, t) t).. Then, the stationary problem of (TP (TP_{r}^{N}) r ) ⎧ d1 Δ[(1 + rv)u] + u(a1 − b1 u − c1 v) = 0 in Ω, ⎪ ⎪ ⎪ ⎪ ⎨ d2 Δv + v(a2 − b2 u − c2 v) = 0 in Ω, (SN ) ∂u ∂v r ⎪ = =0 on ∂Ω, ⎪ ⎪ ∂ν ⎪ ⎩ ∂ν u ≥ 0, v ≥ 0 in Ω,. is. (1.10) (1.11). (S_{r}^N)\{begin{ar y}{l d_{1}\triangle[(1+rv)u] (a_{1}-b u-c_{1}v)=0in\Omega,(1.0) d_{2}\trianglev+(a_{2}-b u-c_{2}v)=0in\Omega,(1. ) \frac{prtialu}{0\bulet}=\frac{prtialv}{\partil\nu}=0on \partil\Omega,(1.2) u\geq0,v\geq0in\Omega,(1.3) \end{ar y}. where uu=u(x) = u(x) and vv=v(x) = v(x)... 2. (1.12). (1.13).
(3) 47 N N They obtained limiting systems as rrarrow\infty → ∞ for (TP (TP_{r}^{N}) (S_{r}^{N}) r ) and (S r ).. One of limiting systems as rrarrow\infty → ∞ are as follows. The time-dependent limiting system is time‐dependent ⎧ τ τ τ ∂ ⎪ ⎪ dx = (1.14) a1 − b1 − c1 v dx in (0, ∞), ⎪ ⎪ ∂t v v v ⎪ Ω Ω ⎪ ⎪ ⎪ ⎨ ∂v in Ω × (0, ∞), (1.15) N (TP∞ ) ∂t = d2 Δv + v(a2 − c2 v) − b2 τ ⎪ ⎪ ∂v ⎪ ⎪ ⎪ =0 on ∂Ω × (0, ∞), (1.16) ⎪ ⎪ ⎪ ⎩ ∂ν v(0, t) = v0 (x) > 0 in Ω, (1.17). (TP_{\infty}^N)\{beginary}{l \fracptil}{\ar tin_{\Omega}frc{\tuv}dx=\int_{Omega} \frc{tau}v(_{1-b}\frac{tuv}-_1)dxin(0,o)1.4 \frac{ptilv}\ar t=d_{2}\rianglev+(_{2}-c v)b_{2}\tauin Omega\cros(0,infty)(1.5 \frac{ptilv}\ar nu}=0o\partilOmega\cros(0,infty)(1.6 v0,t)=_{}(x>0in\Omega,(1.7) \end{ary}. where vv=v(x, = v(x, t) t) and τ\tau=\tau(t) = τ (t) are unknown positive functions, and τ\tau(t)/v(x, (t)/v(x, t) t) corresponds to u(x, u(x, t). t) . The stationary limiting system is ⎧ τ τ ⎪ ⎪ − c a (1.18) 1 v dx = 0, 1 − b1 ⎪ ⎪ v ⎪ Ω v ⎪ ⎨ d2 Δv + v(a2 − c2 v) − b2 τ = 0 in Ω, (1.19) (SN ∞) ⎪ ∂v ⎪ ⎪ =0 on ∂Ω, (1.20) ⎪ ⎪ ∂ν ⎪ ⎩ v(x) > 0, in Ω, (1.21). (S_{\infty}^{N)\begin{ary}l \int_{Omega}\frc{tau}v(_{1}-b \frac{tu}v-c_{1})dx=0,(1.8) d_{2}\trianglev+(a_{2}-c v)b_{2}\tau=0in\Omega,(1.9) \frac{ptialv}{\prtial\nu}=0on\partil\Omega,(1.20) v(x>0,in\Omega,(1.2) \end{ary}. where vv=v(x) = v(x) is an unknown positive function, τ\tau is an unknown positive concon‐ stant. N \Omega := For one-dimension (0, 1),, the limiting system corresponding (TP one‐dimension Ω :=(0,1) (TP_{\infty}^{N}) ∞ ) and N are (SP (SP_{\inft∞ y}^{N)}) are ⎧
(4) 1 1 ⎪ τ τ τ ∂ ⎪ ⎪ dx = a1 − b1 − c1 v dx in(0, 1) × (0, ∞),(1.22) ⎪ ⎪ ∂t v ⎪ 0 v 0 v ⎪ ⎨ ∂v (TP1∞ ) = d2 vxx + v(a2 − c2 v) − b2 τ in (0, 1), (1.23) ⎪ ∂t ⎪ ⎪ ⎪ ⎪ vx (0, t) = 0, vx (1, t) = 0, in (0, ∞), (1.24) ⎪ ⎪ ⎩ v(x, 0) = v0 (x) > 0, in (0, 1), (1.25) and. (TP_{\infty}^1)\{beginary}{l \frac{ptial}{\prtial}(\nt_{0^1}\frac{tu}vdx)=\int_{0}^1 \frac{tu}v(a_{1-b }\frac{tu}v-c_{1)dxin(0,1)\cros(0, )x1.2 \frac{ptialv}{\prtial}=d_{2vx}+(a_{2-c }v)b_{2\tauin(0,1) .23)v_{x}(0,t)=v_{x}(1,t)=0in(,o)1.24 v(x,0)=_{}(x)>0,in(1), .25 \end{ary} ⎧ 1 τ τ ⎪ ⎪ a1 − b1 − c1 v dx = 0, ⎪ ⎪ ⎪ v ⎨ 0 v 1 (S fty}^{1} ,general)) ⎪ d2 vxx + v(a2 − c2 v) − b2 τ = 0 in (0, 1), ( S_{\in∞,general ⎪ ⎪ vx (0) = 0, vx (1) = 0, ⎪ ⎪ ⎩ v(x) > 0 in (0, 1).. \{begin{ar y}{l \int_{0}^1\frac{tu}v(a_{1}-b \frac{tu}v-c_{1}v)dx=0,(1.26) d_{2}vx+v(a_{2}-c v)-b_{2}\tau=0in(,1) (.27) v_{x}(0)=,v_{x}(1)=0,(1.28) v(x>0in(,1).(29) \end{ar y}. (1.26). (1.27) (1.28) (1.29). Lou, Lou, Ni Ni and and Yotsutani Yotsutani [4] [4] obtained obtained existence existence and and non-existence non‐existence of of non-constant non‐constant steady state solutions, the asymptotic shape of solutions, and almost clarified clarified the 1 . ). structure of solutions of (S (S_{\i∞,general nfty,genera1}^{1}) 3.
(5) 48 In what follows, we concentrate on the monotone increasing case vv_{x}(x)>0 x (x) > 0 to 1 . understand the essence of structure of (S ). (S_{\i∞,general nfty,genera1}^{1}) 1 Now, we introduce aa(S_{\i(Snfty}^{1}) ∞ ) as follows: ⎧ 1 τ τ ⎪ ⎪ − c v dx = 0, (1.30) a − b ⎪ 1 1 1 ⎪ ⎪ v ⎨ 0 v (1.31) (S1∞ ) d2 vxx + v(a2 − c2 v) − b2 τ = 0 in (0, 1), ⎪ ⎪ ⎪ (1.32) vx (0) = 0, vx (1) = 0, ⎪ ⎪ ⎩ v(x) > 0, vx (x) > 0 in (0, 1). (1.33). 2. (S_{\infty}^{1)\ begin{ar y}{l \int_{0}^1\frac{tu}v(a_{1}-b \frac{tu}v-c_{1}v)dx=0, (1.30) d_{2}vx +v(a_{2}-c v)-b_{2}\tau=0in(,1) (.31) v_{x}(0)=,v_{x}(1)=0, (1.32) v(x>0,v_{x}()>0in(,1). ( 3) \end{ar y}. Results. 1 We We first first explain explain results results in in [4] [4] for for (S (S_{\in∞ fty}^{1). }) . As As for for the the existence existence and and non-existence, non‐existence, the following theorems are obtained:. Theorem ≤ B. B. Theorem A A (Existence, (Existence, weak weak competition). competition). Suppose Suppose that that CC\leq 1 (i) ≤A τ ) of A then (i) If If BB\leq then there there exists exists aa solution solution (v, (v, \tau) of (S (S_{\in∞ fty}^{1). }) . 1 (ii) + 3C)/4 < A < B,, then for dd_{2}\in 2 ∈ (ii) If If (B (B+3C)/4<A<B then there there exists exists aa solution solution of of (S (S_{\in∞ fty}^{1). }) . for 2A−(B+C) a2 (0, (0, \frac{2A-(B+C)B−C }{B-C}\cdot\frac{a_{·2} {π\pi2^{2}).) .. 1 Figure 1: Existence and non-existence ≤ B. B. non‐existence of solutions of (S (S_{\in∞ fty}^{1)}) for CC\leq Theorem ≤ B. B. Theorem B B (Non-Existence, (Non‐Existence, weak weak competition). competition). Suppose Suppose that that CC\leq 2 1 (i) 2 > a2 /π ,, then (i) If If dd_{2}>a_{2}/\pi^{2} then there there exists exists no no solution solution of of (S (S_{\in∞ fty}^{1). }) . ∗ ∗ (ii) + 3C)/4 < A < B,, then a2a_{2})>0 ) > 0 such (ii) If If (B (B+3C)/4<A<B then there there exists exists aa dd_{2}^{*}=d_{2}^{*}(A, B, C, such 2 = d2 (A, B, C, 1 ∗ 2 that there exists no solution of (S ) for d ∈ (d , a /π ). exists of (S_{\in∞ . 2 2 fty}^{1}) d_{2}\in(d_{2}^{* 2 }, a_{2}/\pi^{2}) 1 (iii) ≤ (B + 3C)/4,, there (iii) If If AA\leq(B+3C)/4 there exists exists no no solution solution of of (S (S_{\in∞ fty}^{1). }) .. 4.
(6) 49 1 Figure 1 shows the existence and non-existence non‐existence region of solutions of (S (S_{\in∞ fty}^{1)}) in A , vertical the case CC\leq ≤ BB assured by theorems A and B. Here, horizontal axis is A, + 3C)/4 < A < (B + C)/2, axis is dd_{2}2 . For the case dd_{2}2 sufficiently sufficiently close to 00 and (B (B+3C)/4<A<(B+C)/2, 1 non‐existence of solutions of (S existence and non-existence (S_{\in∞ fty}^{1)}) are not clear. 1 Figure 2 shows the existence and non-existence non‐existence region of solutions of (S (S_{\in∞ fty}^{1)}) in the case BB<C < C assured by theorems CC and D. For the case 00<d_{2}<((B+C< d2 < ((B + C − 2A)/(C − B)) · (a2 /π 2 ) and BB<A<(B+C)/2 < A < (B + C)/2,, existence and non-existence non‐existence of 2A)/(C-B))\cdot(a_{2}/\pi^{2}) 1 solutions of (S (S_{\in∞ fty}^{1)}) also are not clear.. Theorem < C.. If Theorem C C (Existence, (Existence, strong strong competition). competition). Suppose Suppose that that BB<C If a2 B + C − 2A a2 · 2 < d2 < 2 ,, max 0, C −B π π. \max\{0, \frac{B+C-2A}{C-B}\cdot\frac{a_{2} {\pi^{2} \}<d_{2}<\frac{a_{2} {\pi^{2}. (2.1) (2.1). 1 then there exists τ ) of exists aa solution (v, of (S (v, \tau) (S_{\in∞ fty}^{1)}) .. 1 Figure 2: Existence and non-existence < C. non‐existence of solutions of (S (S_{\in∞ fty}^{1)}) for BB<C.. Theorem < C. Theorem D D (Non-Existence, (Non‐Existence, strong strong competition). competition). Suppose Suppose that that BB<C. 2 1 (i) 2 > a2 /π ,, then (i) If If dd_{2}>a_{2}/\pi^{2} then there there exists exists no no solution solution of of (S (S_{\in∞ fty}^{1)}) . ∗ ∗ a2a_{2})>0 ) > 0 such (ii) ≤A< (B + C)/2,, then (ii) If If BB\leq A<(B+C)/2 then there there exists exists aa dd_{2}^{*}=d_{2}^{*}(A, B, C, such 2 = d2 (A, B, C, 1 ∗ ) for d ∈ (0, d ]. that there exists no solution of (S 2 that there exists no solution of (S_{\in∞ fty}^{1}) for d_{2}\in(0, d_{2}^{*}]. 2 1 (iii) < B,, there (iii) If If AA<B there exists exists no no solution solution of of (S (S_{\in∞ fty}^{1)}) .. In In [9], [9], Lou, Lou, Ni Ni and and Yotsutani Yotsutani conjectured conjectured that that the the situation situation of of existence, existence, nonnon‐ existence and the uniqueness drastically changes at CC=(7/3)B = (7/3)B.. For the case BB<C\leq(7/3)B < C ≤ (7/3)B,, the uniqueness seems to hold as shown in Figures 3 and 4. Recently, we have found a mathematical proof of this case. 5.
(7) 50. :(S_{\infty}^{1}) has the unique solution. Figure 3: CC=B. = B.. 1 Figure 4: Existence and non-existence < C ≤ (7/3)B. non‐existence of solutions of (S (S_{\in∞ fty}^{1)}) for BB<C\leq(7/3)B.. 2. 4. 1 Figure 5: Existence and non-existence > (7/3)B. non‐existence of solutions of (S (S_{\in∞ fty}^{1)}) for CC>(7/3)B.. On the other hand, for the case CC>(7/3)B > (7/3)B,, the existence region becomes wider wider as as shown shown in in Figure Figure 5. 5. In In [6], [6], Mori, Mori, Suzuki Suzuki and and Yotsutani Yotsutani have have obtained obtained precise numerical results with the stability and instability for this case As explained above, existence, non-existence non‐existence and multiplicity of solutions for the case BB\leq ≤ CC are precisely understood. 6.
(8) 51 51 However, it is not clarified < B.. Therefore, we investigate this case. clarified the case CC<B Figure 6 show existence, non-existence and multiplicity of non-constant non‐existence non‐constant solutions 1 ) obtained by numerical computation. for (S (S_{\in∞ fty}^{1}). A \near ow\near ow\near ow,. , : :. (S_{\infty}^{1}) has the unique solution. (S_{\infty}^{1}). has two solutions.. Figure 6: 00<C<B. < C < B.. 3 3. Representation Representation of of solutions solutions. 1 We explain the representation of solutions of (S efficient for (S_{\in∞ fty}^{1)}) , since it is very efficient 1 ) . investigating the solution structure of (S (S_{\in∞ fty}^{1}) Let us introduce a notations. Jacobi’s elliptic function sn(x, defined by sn(x, k) k) defined z dξ (3.1) sn−1 (z, k) = (3.1) 2 1 − k ξ2 1 − ξ2 0. sn^{-1}(z, k)=\int_{0}^{z}\frac{d\xi}{\sqrt{1-k^{2}\xi^{2} \sqrt{1-\xi^{2} }. for −1 ≤ z\leq z ≤ 11.. The complete elliptic integrals of the first, -1\leq first, second and third kind are defined defined by 1 1 1 − k2ξ 2 dξ , E(k) := dξ,, (3.2) K(k) := (3.2) 1 − k2ξ 2 1 − ξ 2 1 − ξ2 0 0. K(k) := \int_{0}^{1}\frac{d\xi}{\sqrt{1-k^{2}\xi^{2} \sqrt{1-\xi^{2} }, E(k) : =\int_{0}^{1}\frac{\sqrt{1-k^{2}\xi^{2} }{\sqrt{1-\xi^{2} }d\xi. and. . 1. dξ (1 + νξ 2 ) 1 − k 2 ξ 2 1 − ξ 2. \Pi(\nu, k) :=\int_{0}^{1}\frac{d\xi}{(1+\nu\xi^{2})\sqrt{1-k^{2}\xi^{2} \sqrt {1-\xi^{2} }. Π(ν, k) :=. 0. -1<\nu for 00\leq ≤k< < ν,, respectively. k<11 and −1. 7. (3.3) (3.3).
(9) 52 1 In what follows in (S (S_{\in∞ fty}^{1), }) , we will concentrate on the case. bb_{1}=1 1 = 1 and 2 = b2 = c2 = 1.. and aa_{2}=b_{2}=c_{2}=1 1 In fact, we get from (S (S_{\in∞ fty}^{1). }) . ⎧ 1
(10) 1 A τ¯ C ⎪ ⎪ − − v¯ dx = 0, ⎪ ⎪ ⎪ ⎨ 0 v¯ B v¯ B d¯2 v¯xx + v¯(1 − v¯) − τ¯ = 0 in (0, 1), ⎪ ⎪ ⎪ (0) = 0, v ¯ (1) = 0, v ¯ ⎪ x x ⎪ ⎩ in (0, 1) v¯(x) > 0, v¯x (x) > 0. (3.4) (3.4). \{beginary}{l \int_{0}^1\frac{}overlin{}(\fracA}{B-\fracovelin{\tau} overlin{}-\fracC{B}\overlin{})dx=0,(3.5) \overlin{d}_2\overlin{}_x+\overlin{}(1-\overlin{})-\overlin{\tau}=0 in(,1) 3.6 \overlin{}_x(0)=,\overlin{}_x(1)=0,3.7) \overlin{}(x)>0,\overlin{}_x()>0in,1)(3.8 \end{ary}. (3.5). (3.6) (3.7) (3.8). by employing the following change of variables. c2 d2 b2 c2 · v, τ¯ := 2 · τ, d¯2 := .. a2 a2 a2. \overline{v}:=\frac{ _{2} {a_{2} \cdot v, \overline{\tau}:=\frac{b_{2}c_{2} {a_{2}^{2} \cdot\ au, \overline{d}_{2}:=\frac{d_{2} {a_{2}. v¯ :=. (3.9) (3.9). Thus, without lose of generality, we may consider the case bb_{1}=1 1 = 1 and aa_{2}=b_{2}= 2 = b2 = cc_{2}=1 2 = 1 . 1 Now, we introduce an auxiliary problem to investigate (S 1 = a2 = (S_{\in∞ fty}^{1)}) with bb_{1}=a_{2}= = c = 1. Let d > 0 be given. Unknowns are a function v = v(x) bb_{2}=c_{2}=1 d_{2}>0 . 2 2 2 v=v(x) and a constant constant τ\tau>0. > 0. ⎧ in (0, 1), (3.10) ⎨ d2 vxx + v(1 − v) − τ = 0 in (0, 1), (3.11) (E) (E) ⎩ v(x) > 0 in [0, 1] and vx (x) > 0 (3.12) vx (0) = 0, vx (1) = 0 and τ > 0.. \{ begin{ar ay}{l} d_{2}v_{x }+v(1-v)-\tau=0in(0,1), (3.10) v(x)>0in[0,1]andv_{x}(x)>0in(0,1), (3.1 ) v_{x}(0)=0,v_{x}(1)=0and\tau>0. (3.12) \end{ar ay}. Exact Exact solutions solutions of of (E) (E) are are given given in in the the following following proposition. proposition.. 2 Proposition 2 ∈ (0, 1/π ).. All Proposition 3.1. 3.1. (E) (E) has has aa solution solution if if and and only only if Of dd_{2}\in(0,1/\pi^{2}) All solutions solutions (v(x), τ ) of (E) are represented by (v(x), \tau) of (E) are represented by √ √ v(x; d2 ,h)=\alpha+(\beta-\alpha)sn^{2}(K(\sqrt{h})_{X}, h),, (3.13) h) = α + (β − α) sn2 (K( h)x,\sqrt{h}) v(x;d_{2}, (3.13) √ αβ + βγ + γα 1 τ (d2 , h) = = − 4 d22 (h2 − h + 1)K( h)4 ,, (3.14) (3.14) 3 4. \tau(d_{2}, h)=\frac{\alpha\beta+\beta\gamma+\gamma\alpha}{3}=\frac{1}{4}- 4d_{2}^{2}(h^{2}-h+1)K(\sqrt{h})^{4}. where √ 1 − 2d2 K( h)2 (h + 1) ,, 2 √ 1 β = + 2d2 K( h)2 (2h − 1) ,, 2 √ 1 γ = + 2d2 K( h)2 (2 − h) .. 2. \alpha=\frac{1}{2}-2d_{2}K(\sqrt{h})^{2}(h+1) \beta=\frac{1}{2}+2d_{2}K(\sqrt{h})^{2}(2h-1) \gamma=\frac{1}{2}+2d_{2}K(\sqrt{h})^{2}(2-h). α=. 8. (3.15) (3.15) (3.16) (3.16) (3.17) (3.17).
(11) 53 ˜\tilde{h} is the unique solution of Here h of an an equation equation √ 1 (h + 1)K( h)2 = 4d2. (h+1)K( \sqrt{h})^{2}=\frac{1}{4d_{2}. (3.18) (3.18). √ h) is the complete in h, h, K( complete elliptic elliptic integral of of the 1st 1st kind, and and sn(·, K(\sqrt{h}) sn(\cdot, \cdot·)) is Jacobi’s elliptic elliptic function. Now, 1 = 1 is Now, we we note note that that (1.30) (1.30) with with bb_{1}=1 is rewritten rewritten as as 1 1 τ dx + c1 2 v 0 = a1 . 1 1 dx 0 v. \tau\int_{0}^{1}\frac{1}{v^{2} dx+c_{1}. \overline{\int_{0}^{1}\frac{1}{v}dx}=a_{1}.. d2 ,c_{1}) c1 ) by Thus, let us define ˜ãl1 (h; define a function a (h;d_{2}, 1 1 dx + c1 τ 2 0 v(x; d2 , h) a ˜ãl1 (h; d , c ) := .. 1 (h;d_{2},2 c_{1}) 1 1 dx 0 v(x; d2 , h). :=\frac{\tauint_{0}^1\frac{1}v(x;d_{2},h)^{2}dx+c_{1} \int_{0}^1 \frac{1}v(x;d_{2},h)dx}. (3.19) (3.19). (3.20) (3.20). a ˜ãl1 (h; d2 ,c_{1}) c1 ) is explicitly given in the following proposition. (h;d_{2}, 2 ˜ 2 )).. It holds that Proposition 3.2. Let dd_{2}\in(0,1/\pi^{2}), ∈ (0, h(d 2 ∈ (0, 1/π ), h h\in(0, \tilde{h}(d_{2})). d2 ,c_{1}) c1 ) a ˜ãl1 (h; (h;d_{2}, αβ + βγ + γαa \alpha\beta+\beta\gamma+\gamma.
(12) = β−α √ 6αβγΠ , h α.
(13)
(14) √ √ β−α √ · (γ − α)αE( h) − αγK( h) + (αβ + βγ + γα)Π , h α √ αK( h)c1
(15) , (3.21) + (3.21) β−α √ , , h Π α. =\overline{6\alpha\beta\gam a\Pi(\frac{\beta-\alpha}{\alpha},\sqrt{h}). ( \gamma-\alpha)\alpha E(\sqrt{h})-\alpha\gamma K(\sqrt{h})+(\alpha\beta+\beta \gamma+\gamma\alpha)\Pi(\frac{\beta-\alpha}{\alpha}, \sqrt{h}). +\frac{\alphaK(\sqrt{h})c_{1}{\Pi(\frac{6-\alpha}{\alpha},\sqrt{h}). where \alpha, β \beta and where α, and γ\gamma are are defined defined by by (3.15), (3.15), (3.16) (3.16) and and (3.17) (3.17) respectively. respectively. Here, Here, K(·), and Π(·, are the complete complete elliptic elliptic integral of of the 1st, 1st, 2nd and and 3rd K(\cdot), E(·) E(\cdot) and \Pi(\cdot, \cdot·)) are kind, respectively.. 9.
(16) 54 We explain the reason that the existence and non-existence non‐existence regions change at . cc_{1}=7/3(C/B=7/3) = 7/3 (C/B = 7/3). We obtain 1 1 2 2 dd_{22}\πpi^{2} (1 − c1 ) ++ (1 ++ ccl)) ˜ã1,1,22^{\cdot}· hh^{2}+\cdots + · · · ,, + a 1) + ( (1 —cl) (l 2. (h;d_{2}, c_{1})= \frac{1}{2}. a ˜ãl1 (h; d2 , c1 ) =. h , where by by Taylor’s Taylor’s expansion expansion of of (3.21) (3.21) in in h, where 3d2 π 2 4 2 2 a ˜ã1,2 )π d − 14π (c − 1)d + (c − 1) .. (35 + 13c 1,2 := 1 1 2 1 2 64(1 − π 2 d2 )2. := \frac{3d_{2}\pi^{2} {64(1-\pi^{2}d_{2})^{2} ( 35+13c_{1})\pi^{4}d_{2}^{2}-14 \pi^{2}(c_{1}-1)d_{2}+(c_{1}-1). (3.22) (3.22). (3.23) (3.23). We ˜ã1,2. 1,2 . We 2 = d+ and − by We check check the the sign sign of of the the coefficient coefficient a We get get dd_{2}=d_{+} and dd_{-} by solving solving (35 + 13c1 )π 4 d22 − 14π 2 (c1 − 1)d2 + (c1 − 1) = 0,, (35+13c_{1})\pi^{4}d_{2}^{2}-14\pi^{2}(c_{1}-1)d_{2}+(c_{1}-1)=0. (3.24) (3.24). 7(c1 − 1) + 2 3(c1 − 1)(3c1 − 7) d+ := π 2 (35 + 13c1 ). d_{+}:= \frac{7(c_{1}-1)+2\sqrt{3(c_{1}-1)(3c_{1}-7)} {\pi^{2}(35+13c_{1})}. (3.25) (3.25). 7(c1 − 1) − 2 3(c1 − 1)(3c1 − 7) .. d− := π 2 (35 + 13c1 ). d_{-} := \frac{7(c_{1}-1)-2\sqrt{3(c_{1}-1)(3c_{1}-7)} {\pi^{2}(35+13c_{1})}. (3.26) (3.26). <<00 ≥ \geq00 ≥ \geq00 <<00 ≥ \geq00. (3.27) (3.27) (3.28) (3.28) (3.29) (3.29) (3.30) (3.30) (3.31) (3.31). where. and. Thus, a ˜ã1,2 1,2 a ˜ã1,2 1,2 a ˜ã1,2 1,2 a ˜ã1,2 1,2 a ˜ã1,2 1,2. for for for for for for for for for for. 00<c_{1}<1, < c1 < 1, 11\leq ≤ cc_{1}\leq 1 ≤ 7/3, 7/3, cc_{1}>7/3, > 7/3, 1 cc_{1}>7/3, 1 > 7/3, cc_{1}>7/3, 1 > 7/3,. 00<d_{2}<d_{+} < d2 < d+ ,, 00<d_{2}<1/\pi^{2} < d2 < 1/π 2 ,, 2 dd_{+}\leq + ≤ dd_{2}<1/\pi^{2} 2 < 1/π ,, dd_{-}<d_{2}<d_{+} − < d2 < d+ ,, 00<d_{2}\leq < d2 ≤d_{-} d− ... Therefore, the behavior of a ˜ãl1 (h, d2 ,c_{1}) c1 ) near hh=0 = 0 is drastically change at cc_{1}=1 1 = 1 (h, d_{2}, and cc_{1}=7/3. 1 = 7/3.. References [1] [1] Y. Y. Kan-on. Kan‐on. Stability Stability of of singularly singularly perturbed perturbed solutions solutions to to nonlinear nonlinear diffusion diffusion systems arising in population dynamics, Hiroshima Math. J., 23 systems arising in population dynamics, Hiroshima Math. J., 23 (1993), (1993), 509‐536. 509-536. [2] [2] Y. Y. Lou, Lou, W.-M. W.‐M. Ni, Ni, Diffusion, Diffusion, self-diffusion self‐diffusion and and cross-diffusion, cross‐diffusion, J. J. Differential Differential Equations, Equations, 131 131 (1996), (1996), no. no. 1, 1, 79-131 79‐131 10.
(17) 55 [3] [3] Y. Y. Lou, Lou, W.-M. W.‐M. Ni, Ni, Diffusion Diffusion vs vs cross-diffusion: cross‐diffusion: an an elliptic elliptic approach, approach, J. J. Differential Equations, 154 (1999), no. 1, 157-190. Differential Equations, 154 (1999), no. 1, 157‐190. [4] [4] Y. Y. Lou, Lou, W.-M. W.‐M. Ni, Ni, S. S. Yotsutani, Yotsutani, On On aa limiting limiting system system in in the the Lotka-Volterra Lotka‐Volterra competition competition with cross-diffusion. cross‐diffusion. Partial differential differential equations equations and and applicaapplica‐ tions, Discrete Contin. Dyn. Syst., 10 (2004), no. 1-2, 435-458. tions, Discrete Contin. Dyn. Syst., 10 (2004), no. 1‐2, 435‐458. [5] [5] M. M. Mimura, Mimura, Stationary Stationary pattern pattern of of some some density-dependent density‐dependent diffusion diffusion system system with with competitive competitive dynamics, dynamics, Hiroshima Hiroshima Math. Math. J., J., 11 11 (1981), (1981), 621-635. 621‐635. [6] [6] T. T. Mori, Mori, T. T. Suzuki, Suzuki, S. S. Yotsutani, Yotsutani, Numerical Numerical Approach Approach to to Existence Existence and and Stability of of Stationary Solutions to aa SKT Cross-diffusion Cross‐diffusion Equation, MatheMathe‐ matical Models and Methods in Applied Sciences, Volume No.28, Issue No.11 No. 11 (2018), 2191-2210. (2018), 2191‐2210. [7] [7] A. A. Okubo, Okubo, “Diffusion “Diffusion and and Ecological Ecological Problems: Problems: Mathematical Mathematical Models”, Models Springer-Verlag, Springer‐Verlag, Berlin/New Berlin/New York, 1980. [8] [8] N. N. Shigesada, Shigesada, K. K. Kawasaki, Kawasaki, E. E. Teramoto, Teramoto, Spatial Spatial segregation segregation of of interacting interacting species, species, J.Theor.Biol., J.Theor.Biol., 79 79 (1979), (1979), 83-99. 83‐99. [9] [9] S. S. Yotsutani, Yotsutani, Structure Structure and and stability stability of of stationary stationary solutions solutions to to aa crosscross‐ diffusion equation, RIMS kˆ o kˆ u roku, 1854 (2013), 23-32. diffusion equation, RIMS kôkûroku, 1854 (2013), 23‐32.. 11.
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