Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 86, pp. 1–18. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu DYNAMICS OF A DIFFUSIVE COMPETITIVE MODEL ON A PERIODICALLY EVOLVING DOMAIN JIAZHEN ZHU, JIAZHENG ZHOU, ZHIGUI LIN Abstract. This article concerns a two-species competitive model with diffu- sive terms in a periodically evolving domain and study the impact of the spatial periodic evolution on the dynamics of the model. The Lagrangian transfor- mation approach is adopted to convert the model from a changing domain to a fixed domain with the assumption that the evolution of habitat is uniform and isotropic. The ecological reproduction indexes of the linearized model are given as thresholds to reveal the dynamic behavior of the competitive model. Our theoretical results show that a lager evolving rate benefits the persistence of competitive populations for both sides in the long run. Numerical exper- iments illustrate that two competitive species, one of which survive and the other vanish in a fixed domain, both survive in a domain with a large evolving rate, and both vanish in a domain with a small evolving rate. 1. Introduction and model formulation A considerable number of models have been introduced in population ecology. Lotka-Volterra model, a typical population model, was proposed and studied to investigate the behavior of two species that compete with each other for more survival resources [20]. To understand the possible influence of spatial diffusion which caused by the random movement of individuals within a species, we consider the classic Lotka-Volterra competitive model with diffusive terms d1δu1 and d2δu2 as follows: u1t − d1∆u1 = u1(a1 − c1u1 − b1u2), x ∈ Ω, t > 0, u2t − d2∆u2 = u2(a2 − b2u1 − c2u2), x ∈ Ω, t > 0, (1.1) where Ω ⊆ Rn is a non-empty smooth open set, ui(x, t)(i = 1, 2) represents the density of the i-th competitive species depending on location x and time t, the positive constant di(i = 1, 2) is the free-diffusion coefficient of ui, and the posi- tive constants ai, bi and ci(i = 1, 2) denote the intrinsic population growth rate, interspecific competition factor and intraspecific competition factor, respectively. 2010 Mathematics Subject Classification. 35K57, 35K55, 92D25. Key words and phrases. Competitive model; diffusion; evolving domain; ecological reproduction indexes. c©2020 Texas State University. Submitted May 9, 2019. Published August 1, 2020. 1 2 J. ZHU, J. ZHOU, Z. LIN EJDE-2020/86 Assume that there is no species across the boundary, the authors in [3, 21] studied the reaction-diffusive problem u1t − d1∆u1 = u1(a1 − c1u1 − b1u2), x ∈ Ω, t > 0, u2t − d2∆u2 = u2(a2 − b2u1 − c2u2), x ∈ Ω, t > 0, ∂u1(x, t) ∂η = ∂u2(x, t) ∂η = 0, x ∈ ∂Ω, t > 0, u1(x, 0) = u1,0(x), u2(x, 0) = u2,0(x), x ∈ Ω, (1.2) where η is the unit outer normal vector of ∂Ω. Clearly, the corresponding steady- state problem of (1.2) admits the trivial solution U0 = (0, 0) and the semi-trivial solutions U1 = (a1c1 , 0) and U2 = (0, a2c2 ). In particular, the steady-state problem admits the unique positive solution U∗ = (a1c2−a2b1c1c2−b1b2 , a2c1−a1b2 c1c2−b1b2 ) when c1 b2 > a1 a2 > b1 c2 or c1 b2 < a1 a2 < b1 c2 . Further theoretical results for stability have been achieved in [21] as follows: (i) the trivial solution U0 = (0, 0) is always unstable; (ii) U∗ is globally asymptotically stable when c1 b2 > a1 a2 > b1 c2 (weak competition); (iii) U1 is globally asymptotically stable when a1 a2 > max{ c1b2 , b1 c2 }; (iv) U2 is globally asymptotically stable when a1 a2 < min{ c1b2 , b1 c2 }; (v) U1, as well as U2, is locally asymptotically stable and U∗ is unstable when c1 b2 < a1 a2 < b1 c2 (strong competition). Most reaction-diffusion problems describing ecologic models are studied in fixed domains. However, it is common in nature that the habitats in which species live are changeable. Sometimes, boundaries of shifting habitats are unknown owing to the activities of species. For examples, the spreading of invasive species like muskrats in Europe in the early 1900s [25], Asian carps in the Illinois River since the early 1990s [11], cane toad (Bufo marinus) in tropical Australia introduced in 1935 [24] and the transmission of disease like West Nile virus [13]. Models with such unknown moving boundaries are characterized by free boundary problems and studied as a brunch of model analysis [8]. Mathematically, the free boundary induces more difficulties but it better characterizes the spreading of invasive species [6, 7, 15], and the transformation of disease [2, 9, 17]. Sometimes, habitat spaces could change following certain known pattern due to objective factors like climate change and seasonal succession. Usually, leaves keep growing before falling and the water storage of lakes annually shifts. For example, the data in [14] give that, in 2009, the wetland vegetation area of Poyang Lake was about 20.8 km2 in February and up to about 1048.9 km2 in May. Figure 1 (a) are the monthly distributions of grassland in Poyang Lake in 2009 from January to December, and Figure 1 (b) is the monthly variation curve of vegetation area [14]. Figure 1 indicates that the Poyang Lake in China is an evolving domain since the water area of the Lake changes from smaller in winter to larger in summer. Problems with such known boundaries are characterized as growing domain [4, 18] or evolving domain [12, 19, 26], and have been studied extensively. In this article, we study the Lotka-Volterra competitive model in a periodic evolving domain which refers to a domain evolving with known periodicity. We assume the domain in model (1.1) is changing with t, that is Ω = Ω(t) ⊆ Rn is time-varying and its boundary ∂Ω(t) is evolving. According to the principle of mass conservation and Reynolds transport theorem [1], model (1.1) can be converted to EJDE-2020/?? DYNAMICS OF A DIFFUSIVE COMPETITIVE MODEL 3 (a) (b) Figure 1. (a) are the monthly distribution of grassland and water area in Poyang Lake in 2009 from January to December. (b) is the monthly variation curve of vegetation area which together show the monthly area changes in Poyang Lake[14]. the following problem in a evolving domain Ω(t) with Dirichlet boundary condition which implies that there is no species on the boundary, u1t − d1∆u1 + a · ∇u1 + u1∇ · a = u1(a1 − c1u1 − b1u2), x ∈ Ω(t), t > 0, u2t − d2∆u2 + a · ∇u2 + u2∇ · a = u2(a2 − b2u1 − c2u2), x ∈ Ω(t), t > 0, u1(x(t), t) = u2(x(t), t) = 0, x ∈ ∂Ω(t), t > 0, u1(x(0), 0) = u1,0(x(0)), u2(x(0), 0) = u2,0(x(0)), x(0) ∈ Ω(0), (1.3) where a denotes the spacial flow velocity caused by the change of domain, u1 · ∇a and u2 · ∇a are called dilution terms, a · ∇u1 and a · ∇u2 are called advection terms. x = x(t) within Ω(t) is the function of t, ai = ai(t), bi = bi(t) and ci = ci(t) (i = 1, 2) are all positive and T -periodic. Assume the evolution of Ω(t) is uniform and isotropic, that is, x(t) = ρ(t)y, y ∈ Ω(0), (1.4) where ρ(t) = ρ(t+ T ) is a T -periodic function with ρ(0) = 1. Thus, u1 and u2 can be mapped as a new function with the definition u1(x, t) = v1(y, t), u2(x, t) = v2(y, t) (1.5) followed with v1t = ∂u1 ∂t + a · ∇u1, v2t = ∂u2 ∂t + a · ∇u2, ∇a = nρ̇(t) ρ(t) , ∆u1 = 1 ρ2(t) ∆v1, ∆u2 = 1 ρ2(t) ∆v2, 4 J. ZHU, J. ZHOU, Z. LIN EJDE-2020/86 where n is the dimension of the space Ω. Therefore, (1.3) is converted to the problem in a fixed domain v1t − d1 ρ2(t) ∆v1 = −nρ̇(t) ρ(t) v1 + v1(a1 − c1v1 − b1v2), y ∈ Ω(0), t > 0, v2t − d2 ρ2(t) ∆v2 = −nρ̇(t) ρ(t) v2 + v2(a2 − b2v1 − c2v2), y ∈ Ω(0), t > 0, v1(y, t) = v2(y, t) = 0, y ∈ ∂Ω(0), t > 0, v1(y, 0) = v1,0(y), v2(y, 0) = v2,0(y), y ∈ Ω(0), (1.6) the dynamics of which is related to its corresponding periodic problem V1t − d1 ρ2(t) ∆V1 = −nρ̇(t) ρ(t) V1 + V1(a1 − c1V1 − b1V2), y ∈ Ω(0), t > 0, V2t − d2 ρ2(t) ∆V2 = −nρ̇(t) ρ(t) V2 + V2(a2 − b2V1 − c2V2), y ∈ Ω(0), t > 0, V1(y, t) = V2(y, t) = 0, y ∈ ∂Ω(0), t > 0, V1(y, 0) = V1(y, T ), V2(y, 0) = V2(y, T ), y ∈ Ω(0). (1.7) In the rest of this article, we investigate the asymptotic behavior of the initial and boundary value problem (1.6) in related to the T -periodic solution of problem (1.7). In Section 2, we first present the ecological reproduction indexes of problem (1.7) as thresholds based on the principal eigenvalues of its linearized problem, and then deliver the existence of periodic solution. In Section 3, we analyze the stability of the solution to the initial and boundary value problem. In Section 4, we discuss the impact of the evolving domain on the persistence of two competitive species. In Section 5, we give some numerical simulations and ecological explanations in support of the theoretical results achieved in Section 4. 2. Ecological reproduction index In this section, we determine the existence of the solution to problem (1.7). After linearizing problem (1.7) around (0, 0), we have its eigenvalue problem as follows: φ1t − d1 ρ2(t) ∆φ1 = (a1 − nρ̇(t) ρ(t) )φ1 + λ1φ1, y ∈ Ω(0), t > 0, φ2t − d2 ρ2(t) ∆φ2 = (a2 − nρ̇(t) ρ(t) )φ2 + λ2φ2, y ∈ Ω(0), t > 0, φ1(y, t) = φ2(y, t) = 0, y ∈ ∂Ω(0), t > 0, φ1(y, 0) = φ1(y, T ), φ2(y, 0) = φ2(y, T ), y ∈ Ω(0), (2.1) and denote λ4i (i = 1, 2) the principal eigenvalue of (2.1), and φ4i the corresponding eigenfunctions with 0 ≤ φ4i ≤ 1. Furthermore, by variation method, we can give the explicit expression of principal eigenvalues as λ4i = 1 T ∫ T 0 diλ0 ρ2(t) dt− 1 T ∫ T 0 ai(t)dt (i = 1, 2), where λ0 is the principal eigenvalue of −∆φ = λφ, y ∈ Ω(0), φ(y) = 0, y ∈ ∂Ω(0). (2.2) EJDE-2020/?? DYNAMICS OF A DIFFUSIVE COMPETITIVE MODEL 5 Using the next generation operator as in [16, 27], we can define the ecological reproduction index Ri(i = 1, 2). Moreover, it follows from Lemma 13.1.1 in [27] that R1 and R2 are the principal eigenvalues of the problems ϕ1t − d1 ρ2(t) ∆ϕ1 = ( a1 R1 − nρ̇(t) ρ(t) )ϕ1, y ∈ Ω(0), t > 0, ϕ2t − d2 ρ2(t) ∆ϕ2 = ( a2 R2 − nρ̇(t) ρ(t) )ϕ2, y ∈ Ω(0), t > 0, ϕ1(y, t) = ϕ2(y, t) = 0, y ∈ ∂Ω(0), t > 0, ϕ1(y, 0) = φ1(y, T ), ϕ2(y, 0) = φ2(y, T ), y ∈ Ω(0). (2.3) In the study of epidemic models, Ri is called basic reproduction number [5, 16] and usually given as threshold. Similarly, the variation method gives Ri = ∫ T 0 ai(t)dt diλ0 ∫ T 0 1 ρ2(t)dt (i = 1, 2). (2.4) It can be verified that sgn(1−Ri) = sgn(λi) (i = 1, 2). (2.5) Similar results for general systems hold as well. For more details, see [16] and the references therein. To derive the existence of the solution to (1.7), we give the definition of upper and lower solutions. Definition 2.1. (Ṽ1, Ṽ2) and (V̂1, V̂2) is a pair of coupled upper and lower solutions of the problem (1.7), if V̂1t − d1 ρ2(t) ∆V̂1 ≤ − nρ̇(t) ρ(t) V̂1 + V̂1(a1 − c1V̂1 − b1Ṽ2), y ∈ Ω(0), t > 0, Ṽ1t − d1 ρ2(t) ∆Ṽ1 ≥ − nρ̇(t) ρ(t) Ṽ1 + Ṽ1(a1 − c1Ṽ1 − b1V̂2), y ∈ Ω(0), t > 0, V̂2t − d2 ρ2(t) ∆V̂2 ≤ − nρ̇(t) ρ(t) V̂2 + V̂2(a2 − b2Ṽ1 − c2V̂2), y ∈ Ω(0), t > 0, Ṽ2t − d2 ρ2(t) ∆Ṽ2 ≥ − nρ̇(t) ρ(t) Ṽ2 + Ṽ2(a2 − b2V̂1 − c2Ṽ2), y ∈ Ω(0), t > 0, Ṽ1(y, t) ≥ V̂1(y, t) = 0, Ṽ2(y, t) ≥ V̂2(y, t) = 0, y ∈ ∂Ω(0), t ≥ 0, V̂1(y, 0) ≤ V̂1(y, T ), V̂2(y, 0) ≤ V̂2(y, T ), y ∈ Ω(0), Ṽ1(y, 0) ≥ Ṽ1(y, T ), Ṽ2(y, 0) ≥ Ṽ2(y, T ), y ∈ Ω(0). (2.6) Let S0 := {(V1, V2) : (V̂1, V̂2) ≤ (V1, V2) ≤ (Ṽ1, Ṽ2), (y, t) ∈ Ω(0) × [0, T ]} and denote f1(V1, V2) = V1(a1 − c1V1 − b1V2)− n ˙ρ(t) ρ(t) V1, f2(V1, V2) = V2(a2 − b2V1 − c2V2)− n ˙ρ(t) ρ(t) V2. 6 J. ZHU, J. ZHOU, Z. LIN EJDE-2020/86 Then, for any (V1, V2), (Z1, Z2) ∈ S0, |f1(V1, V2)− f1(Z1, Z2)| ≤ [aM1 + (bM1 + 2cM1 ) aM1 cm1 + bM1 aM2 cm2 + nρ̇M ρm ](|V1 − Z1|+ |V2 − Z2|), |f2(V1, V2)− f2(Z1, Z2)| ≤ [aM2 + (bM2 + 2cM2 ) aM2 cm2 + bM2 aM1 cm1 + nρ̇M ρm ](|V1 − Z1|+ |V2 − Z2|), where fM = max[0,T ] f(t) and fm = min[0,T ] f(t). We find that f1 and f2 satisfy the Lipschitz condition with Lipschitz coefficients k1 = aM1 + (bM1 + 2cM1 ) aM1 cm1 + bM1 aM2 cm2 + nρ̇M ρm , (2.7) k2 = aM2 + (bM2 + 2cM2 ) aM2 cm2 + bM2 aM1 cm1 + nρ̇M ρm . (2.8) Based on the upper and lower solutions technique developed by Pao [22], we have the following result about the existence of the solution. Lemma 2.2. If (Ṽ1, Ṽ2), (V̂1, V̂2) is a pair of coupled upper and lower solutions of (1.7), then (1.7) admits at least one periodic solution (V1, V2) ∈ S0. Now we present the existence of the periodic solution to (1.7). Theorem 2.3. Denote M1 = ( 1 c1 (a1− nρ̇ ρ ))M and M2 = ( 1 c2 (a2− nρ̇ ρ ))M . Then we have the following assertions: (i) if R1 ≤ 1 and R2 ≤ 1, (1.7) admits only trivial solution (0, 0); (ii) if R1 > 1 and R2 ≤ 1, (1.7) admits a semi-trivial periodic solution (V 41 , 0); (iii) if R1 ≤ 1 and R2 > 1, (1.7) admits a semi-trivial periodic solution (0, V 42 ); (iv) if R1 > 1 and R2 > 1, together with (a1b1 )m(1 − 1 R1 ) > M2 and (a2b2 )m(1 − 1 R2 ) > M1, (1.7) admits a positive periodic solution (V ∗1 , V ∗ 2 ). Proof. (i) Let (V1, V2) be the nonnegative solution of (1.7), we claim that V1 ≡ 0 and V2 ≡ 0 in Ω(0). In fact, assume that V1 satisfies V1t − d1 ρ2(t) ∆V1 + nρ̇(t) ρ(t) V1 − a1V1 = −(b1V2 + c1V1)V1, y ∈ Ω(0), t > 0, and V1 ≥ 0(6≡ 0) by contradiction. Recalling that φ1t − d1 ρ2(t) ∆φ1 + nρ̇(t) ρ(t) φ1 − a1φ1 = λ41 φ1, y ∈ Ω(0), t > 0, we have λ41 < 0 according to the monotonicity of eigenvalues revealed in [23, Proposition 5.2]. It follows from (2.3) that λ41 < 0 implies R1 > 1, which leads a contradiction to the condition. Therefore, V1 ≡ 0 in Ω(0). Similarly, V2 ≡ 0. Thus, (0, 0) is the only nonnegative solution to (1.7). EJDE-2020/?? DYNAMICS OF A DIFFUSIVE COMPETITIVE MODEL 7 (ii) If R1 > 1 and R2 ≤ 1, consider semi-trivial solution (V 41 , 0) and V 41 satisfies V 41,t − d1 ρ2(t) ∆V 41 = −nρ̇(t) ρ(t) V 41 + V 41 (a1 − c1V 41 ), y ∈ Ω(0), t > 0, V 41 = 0, y ∈ ∂Ω(0), t ≥ 0, V 41 (y, 0) = V 41 (y, T ), y ∈ Ω(0). (2.9) It can be verified that M1 and δϕ1 is a pair of ordered upper and lower solutions of problem (2.9) for any positive constant δ < −λ41 . Furthermore, according to [10, Theorem 27.1] for the uniqueness of the solution to a problem with concave non- linearities, the positive solution V 41 is unique as a1− c1V 41 is monotone decreasing in terms of V 41 . Thus, (V 41 , 0) is the unique periodic solution of (1.7). (iii) The proof is similar to that of (ii). (iv) According to Lemma 2.2, equation (1.7) admits at least one periodic solution (V1, V2) if we can verify that (M1,M2) and (εϕ1, εϕ2) is a pair of coupled upper and lower solutions of (1.7) with positive constant ε to be determined. In fact, the choose of M1 and M2 implies that (M1,M2) is an upper solution of (1.7) as long as (εϕ1, εϕ2) is nonnegative. Clearly, the condition (a1b1 )m(1− 1 R1 ) > M2 and (a2b2 )m(1− 1 R2 ) > M1 implies that there exists a constant ε0 = min { 1 cM1 (a1(1− 1 R1 )− b1M2), 1 cM2 (a2(1− 1 R2 )− b2M1) } > 0, then for any 0 < ε < ε0, (εϕ1, εϕ2) is the lower solution of (1.7) with (M1,M2) the upper solution. Thus, (M1,M2) and (εϕ1, εϕ2) is a pair of coupled upper and lower solutions of (1.7) and the proof is complete. � 3. Dynamics of periodic solutions In this section, we discuss the stability of the solution to problem (1.6) which is related to the solution of the periodic problem (1.7). Firstly, we convert the reaction functions in problem (1.6) to be quasimonotone nondecreasing. Let M = max{M2, supy∈Ω(0) v2,0(y)}, v3 = M − v2. Then (1.6) becomes v1t − d1 ρ2(t) ∆v1 = f1(v1,M − v3), y ∈ Ω(0), t > 0, v3t − d2 ρ2(t) ∆v3 = −f2(v1,M − v3), y ∈ Ω(0), t > 0, v1(y, t) = M − v3(y, t) = 0, y ∈ ∂Ω(0), t > 0, v1(y, 0) = v1,0(y), y ∈ Ω(0), v3(y, 0) = v3,0(y) := M − v2,0(y), y ∈ Ω(0). (3.1) The corresponding periodic problem of (3.1) becomes V1t − d1 ρ2(t) ∆V1 = f1(V1,M − V3), y ∈ Ω(0), t > 0, V3t − d2 ρ2(t) ∆V3 = −f2(V1,M − V3), y ∈ Ω(0), t > 0, V1(y, t) = M − V3(y, t) = 0, y ∈ ∂Ω(0), t > 0, V1(y, 0) = V1(y, T ), V3(y, 0) = V3(y, T ), y ∈ Ω(0), (3.2) 8 J. ZHU, J. ZHOU, Z. LIN EJDE-2020/86 where f1(V1,M − V3) = −nρ̇(t) ρ(t) V1 + V1(a1 − b1(M − V3)− c1V1), −f2(V1,M − V3) = nρ̇(t) ρ(t) (M − V3)− (M − V3)(a2 − b2V1 − c2(M − V3))) are quasimonotone nondecreasing reaction functions for (V1, V3) ∈ S1, where S1 := {(V,Z) : (V̂1,M − Ṽ2) ≤ (V,Z) ≤ (Ṽ1,M − V̂2), (y, t) ∈ Ω(0)× [0, T ]}. We claim that (Ṽ1,M − V̂2) and (V̂1,M − Ṽ2) is a pair of ordered upper and lower solutions of (3.2) if (Ṽ1, Ṽ2) and (V̂1, V̂2) is a pair of coupled nonnegative upper and lower solutions of (1.7). And sequences {(V (m) 1 , V (m) 3 )} and {(V (m) 1 , V (m) 3 )} can be obtained by taking V (0) 1 = Ṽ1, V (0) 3 = M − V̂2, V (0) 1 = V̂1 and V (0) 3 = M − Ṽ2 as initial iterations and solving the linear periodic problem V (n) 1t − d1 ρ2(t) ∆V (n) 1 + k1V (n) 1 = F1(t, V (n−1) 1 , V (n−1) 3 ), y ∈ Ω(0), t > 0, V (n) 3t − d2 ρ2(t) ∆V (n) 3 + k2V (n) 3 = F2(t, V (n−1) 1 , V (n−1) 3 ), y ∈ Ω(0), t > 0, V (n) 1t − d1 ρ2(t) ∆V (n) 1 + k1V (n) 1 = F1(t, V (n−1) 1 , V (n−1) 3 ), y ∈ Ω(0), t > 0, V (n) 3t − d2 ρ2(t) ∆V (n) 3 + k2V (n) 3 = F2(t, V (n−1) 1 , V (n−1) 3 ), y ∈ Ω(0), t > 0, V (n) 1 = V (n) 1 = 0, V (n) 3 = V (n) 3 = M, y ∈ ∂Ω(0), t > 0, V (n) 1 (y, 0) = V (n−1) 1 (y, T ), V (n) 3 (y, 0) = V (n−1) 3 (y, T ), y ∈ Ω(0), V (n) 1 (y, 0) = V (n−1) 1 (y, T ), V (n) 3 (y, 0) = V (n−1) 3 (y, T ), y ∈ Ω(0), (3.3) where k1 and k2 are Lipschitz coefficients given in (2.7) and (2.8), F1(t, V1, V3) = k1V1 + f1(t, V1,M − V3), F2(t, V1, V3) = k2V3 − f2(t, V1,M − V3). Similarly, the sequences {(v(m) 1 , v (m) 3 )} and {(v(m) 1 , v (m) 3 )} can be obtained by taking v (0) 1 = ṽ1, v (0) 3 = M − v̂2, v (0) 1 = v̂1 and v (0) 3 = M − ṽ2 as initial iterations and solving the linear initial and boundary value problem v (n) 1t − d1 ρ2(t) ∆v (n) 1 + k1v (n) 1 = F1(t, v (n−1) 1 , v (n−1) 3 ), y ∈ Ω(0), t > 0, v (n) 3t − d2 ρ2(t) ∆v (n) 3 + k2v (n) 3 = F2(t, v (n−1) 1 , v (n−1) 3 ), y ∈ Ω(0), t > 0, v (n) 1t − d1 ρ2(t) ∆v (n) 1 + k1v (n) 1 = F1(t, v (n−1) 1 , v (n−1) 3 ), y ∈ Ω(0), t > 0, v (n) 3t − d2 ρ2(t) ∆v (n) 3 + k2v (n) 3 = F2(t, v (n−1) 1 , v (n−1) 3 ), y ∈ Ω(0), t > 0, v (n) 1 = v (n) 1 = 0, v (n) 3 = v (n) 3 = M, y ∈ ∂Ω(0), t > 0, v (n) 1 (y, 0) = v (n) 1 (y, 0) = v1,0(y), y ∈ Ω(0), v (n) 3 (y, 0) = v (n) 3 (y, 0) = v3,0(y), y ∈ Ω(0), (3.4) EJDE-2020/?? DYNAMICS OF A DIFFUSIVE COMPETITIVE MODEL 9 where (v1,0(y), v3,0(y)) ∈ S1. Next, we present two propositions about the sequences {(V (m) 1 , V (m) 3 )}, {(V (m) 1 , V (m) 3 )}, {(v(m) 1 , v (m) 3 )}, {(v(m) 1 , v (m) 3 )} according to Pao’s work in [22]. Proposition 3.1. (i) The sequence {(V (m) 1 , V (m) 3 )} decreases and converges mono- tonically to (V 1, V 3) which is a maximal T -periodic solution of (3.2), and the se- quence {(V (m) 1 , V (m) 3 )} increases and converges monotonically to (V 1, V 3) which is a minimal T -periodic solution of (3.2); that is, (V̂1,M − Ṽ2) ≤ (V (m) 1 , V (m) 3 ) ≤ (V (m+1) 1 , V (m+1) 3 ) ≤ (V 1, V 3) ≤ (V 1, V 3) ≤ (V (m+1) 1 , V (m+1) 3 ) ≤ (V (m) 1 , V (m) 3 ) ≤ (Ṽ1,M − V̂2). (ii) (V 1, V 3) = (V 1, V 3) when V 1(y, 0) = V 1(y, 0) and V 3(y, 0) = V 3(y, 0) which implies that (3.2) admits a unique periodic solution (V1, V3) = (V 1, V 3) = (V 1, V 3). Proposition 3.2. Both {(v(m) 1 , v (m) 3 )} and {(v(m) 1 , v (m) 3 )} converge to (v1, v3), the unique solution of (3.1) satisfying (V̂1,M − Ṽ2) ≤ (v (m) 1 , v (m) 3 ) ≤ (v (m+1) 1 , v (m+1) 3 ) ≤ (v1, v3) ≤ (v (m+1) 1 , v (m+1) 3 ) ≤ (v (m) 1 , v (m) 3 ) ≤ (Ṽ1,M − V̂2). Based on Propositions 3.1 and 3.2, we have the following lemma. A detailed proof for more general parabolic systems can be found in [22]. Lemma 3.3. Let η = (v1,0(y), v3,0(y)) and for any m and m′, if (V (m′) 1 , V (m′) 3 )(y, 0) ≤ η(y) ≤ (V (m) 1 , V (m) 3 )(y, 0), then we have (i) (V (m) 1 , V (m) 3 ) and (V (m′) 1 , V (m′) 3 ) is a pair of ordered upper and lower solu- tions of problem (3.1); (ii) the solution of (3.1) denoted by v(y, t;η) satisfies (V (m) 1 , V (m) 3 )(y, t) ≤ v(y, t+mT ;η) ≤ (V (m) 1 , V (m) 3 )(y, t) with (V 1, V 3)(y, t) ≤ lim inf m→+∞ v(y, t+mT ;η) ≤ lim sup m→+∞ v(y, t+mT ;η) ≤ (V 1, V 3)(y, t). (3.5) Theorem 3.4. Denote V 43 = M − V 42 . For problem (3.1) with any nonnegative nontrivial initial value η, we have the following stability results: (i) If R1 ≤ 1 and R2 ≤ 1, then limm→∞ v(y, t+mT ;η) = (0,M); 10 J. ZHU, J. ZHOU, Z. LIN EJDE-2020/86 (ii) If R1 > 1 and R2 ≤ 1, then limm→∞ v(y, t+mT ;η) = (V 41 ,M); (iii) If R1 ≤ 1 and R2 > 1, then limm→∞ v(y, t+mT ;η) = (0, V 43 ); (iv) when R1 > 1, R2 > 1, (a1b1 )m(1− 1 R1 ) > M2 and (a2b2 )m(1− 1 R2 ) > M1, we have lim m→+∞ v(y, t+mT ;η) = (V 1, V 3)(y, t), if (0, 0) ≤ η ≤ (V 1, V 3) in Ω(0); and lim m→+∞ v(y, t+mT ;η) = (V 1, V 3)(y, t), if (V 1, V 3) ≤ η ≤ (M1,M) in Ω(0). Proof. (i) It follows from Theorem 2.3 that problem (1.7) admits the unique trivial solution (0, 0) when R1 ≤ 1 and R2 ≤ 1 which implies that V 1 = V 1 = V1 = 0, V 2 = V 2 = V2 = 0. Noticing that V3 = M − V2, we have V 3 = M − V 2 = M = M − V 1 = V 3. Recalling (3.5), we have (0,M) = lim inf m→+∞ v(y, t+mT ;η) ≤ lim sup m→+∞ v(y, t+mT ;η) = (0,M). Thus, limm→+∞ v(y, t+mT ;η) exists and equals (0,M). (ii) It is easy to verify that (M1,M) and (0,M−ce−λ 4 2 tφ2(y, t)) is a pair of order upper and lower solutions of (3.2) for some positive constant c satisfying M − cφ2(y, 0) ≤ v3,0(y) ≤M. Then, from Proposition 3.1 (i) it follows that for any ε > 0, there is a positive constant T ∗ such that M − ε ≤ V 3 ≤ V 3 ≤M, for any t ≥ T ∗. Letting t→ +∞, we have V 1t − d1 ρ2(t) ∆V 1 = V 1(a1 − nρ̇ ρ − c1V 1 − b1(M − V 3)) ≥ V 1(a1 − nρ̇ ρ − c1V 1 − ε), y ∈ Ω(0), t > 0, V 1 = 0, y ∈ ∂Ω(0), t > 0, V 1(y, 0) = V 1(y, T ), y ∈ Ω(0); and V 1t − d1 ρ2(t) ∆V 1 = V 1(a1 − nρ̇ ρ − c1V 1 − b1(M − V 3)) ≤ V 1(a1 − nρ̇ ρ − c1V 1), y ∈ Ω(0), t > 0, V 1 = 0, y ∈ ∂Ω(0), t > 0, V 1(y, 0) = V 1(y, T ), y ∈ Ω(0). EJDE-2020/?? DYNAMICS OF A DIFFUSIVE COMPETITIVE MODEL 11 Letting ε→ 0 we have V 1t − d1 ρ2(t) ∆V 1 = V 1(a1 − nρ̇ ρ − c1V 1), y ∈ Ω(0), t > 0, V 1 = 0, y ∈ ∂Ω(0), t > 0, V 1(y, 0) = V 1(y, T ) y ∈ Ω(0). (3.6) Similarly, we have V 1t − d1 ρ2(t) ∆V 1 = V 1(a1 − nρ̇ ρ − c1V 1), y ∈ Ω(0), t > 0, V 1 = 0, y ∈ ∂Ω(0), t > 0, V 1(y, 0) = V 1(y, T ) y ∈ Ω(0). (3.7) According to [10, Theorem 27.1], both (3.6) and (3.7) admit a unique periodic solution. Thus, V 1 = V 1 (:= V 41 ). Recalling back to (3.5), we have (V 41 ,M) = lim inf m→+∞ v(y, t+mT ;η) ≤ lim sup m→+∞ v(y, t+mT ;η) = (V 41 ,M). Thus, limm→+∞ v(y, t+mT ;η) exists and equals (V 41 ,M). (iii) The proof is similar to (ii), so we omit it. (iv) According to Theorem 2.3, Proposition 3.1 and the transformation v1 = M − v3, we deduce that problem (3.2) admits a minimal positive periodic solution (V 1, V 3) and a maximal positive periodic solution (V 1, V 3). Thus, (M1,M) and (V 1, V 3) can be viewed as a pair of ordered upper and lower solution of (3.2). Take V (0) 1 = V1, V (0) 3 = V 3, V (0) 1 = M1, V (0) 3 = M − δϕ2 as initial iterations in (3.3). Then we have another maximal positive periodic solu- tion of problem (3.2) denoted by (V ′ 1, V ′ 3), and another minimal positive periodic solution of problem (3.2) denoted by (V ′1, V ′ 3). Obviously, V ′1 = V ′ 1 = V 1, V ′3 = V 3 = V ′ 3. According to Proposition 3.1, problem (3.2) admits the unique periodic solution (V 1, V 3). And from the Lemma 3.3, we have lim m→∞ v(y, t+mT ;η) = (V 1, V 3), if (V 1, V 3) ≤ η ≤ (M1,M). Similarly, we have lim m→∞ v(y, t+mT ;η) = (V 1, V 3), if (0, 0) ≤ η ≤ (V 1, V 3). � Coming back to problem (1.6), we have the following results directly achieved from the Theorem 3.4 and the transformation v3 = M − v2. Theorem 3.5. Denote ζ = (v1,0(y), v2,0(y)) and (v1, v2)(y, t; ζ) the solution of (1.6) with any nonnegative nontrivial initial value η. (i) If R1 ≤ 1 and R2 ≤ 1, then limm→∞(v1, v2)(y, t+mT ; ζ) = (0, 0); (ii) If R1 > 1 and R2 ≤ 1, then limm→∞(v1, v2)(y, t+mT ; ζ) = (V 41 , 0); (iii) If R1 ≤ 1 and R2 > 1, then limm→∞(v1, v2)(y, t+mT ; ζ) = (0, V 42 ); 12 J. ZHU, J. ZHOU, Z. LIN EJDE-2020/86 (iv) If R1 > 1, R2 > 1, (a1b1 )m(1− 1 R1 ) > M2 and (a2b2 )m(1− 1 R2 ) > M1, we have lim m→+∞ (v1, v2)(y, t+mT ; ζ) = { (v1, v2)(y, t), if (v1, 0) ≤ ζ ≤ (M1, v2) in Ω(0), (v1, v2)(y, t), if (0, v2) ≤ ζ ≤ (v1,M2) in Ω(0). 4. Impact of evolution To study the impact of periodic evolution of domain on the competitive model, here we first present the result of (1.6) on a fixed domain, that is (1.6) with ρ ≡ 1: v1t − d1∆v1 = v1(a1 − c1v1 − b1v2), y ∈ Ω(0), t > 0, v2t − d2∆v2 = v2(a2 − b2v1 − c2v2), y ∈ Ω(0), t > 0, v1(y, t) = v2(y, t) = 0, y ∈ ∂Ω(0), t > 0, v1(y, 0) = v1,0(y), v2(y, 0) = v2,0(y), y ∈ Ω(0). (4.1) According to [27, Lemma 13.1.1], the principal eigenvalue of (4.1) is Ri ∣∣ ρ=1 = ∫ T 0 ai(t)dt diλ0 ∫ T 0 1 ρ2(t)dt ∣∣∣ ρ=1 = ∫ T 0 ai(t)dt Tdiλ0 (i = 1, 2), (4.2) and is denoted by R∗i . The periodic problem corresponding to (4.1) is V1t − d1∆V1 = V1(a1 − c1V1 − b1V2), y ∈ Ω(0), t > 0, V2t − d2∆V2 = V2(a2 − b2V1 − c2V2), y ∈ Ω(0), t > 0, V1(y, t) = V2(y, t) = 0, y ∈ ∂Ω(0), t > 0, V1(y, 0) = V1(y, T ), V2(y, 0) = V2(y, T ), y ∈ Ω(0). (4.3) Theorem 4.1. Let āi = 1 T ∫ T 0 aidt (i = 1, 2). There is a positive constant D∗i = āi λ0 such that (i) if d1 ∈ (D∗1 ,+∞) and d2 ∈ (D∗2 ,+∞), then (4.3) admits a trivial solution which is globally asymptotically stable for problem (4.1); (ii) if d1 ∈ (0, D∗1) and d2 ∈ (D∗2 ,+∞), (4.3) admits a semi-trivial solution, which is a global attractor for problem (4.1); (iii) if d1 ∈ (D∗1 ,+∞) and d2 ∈ (0, D∗2), then (4.3) admits a semi-trivial solu- tion, which is global attractor for problem (4.1); (iv) if d1 ∈ (0, D∗1) and d2 ∈ (0, D∗2), then (4.3) admits the maximal and mini- mal periodic solutions, which are local attractors of problem (4.1). The assertion of the above theorem is easy to verified by letting R∗i = 1 and recalling Theorem 3.5. So omit its proof. Next, we consider the impact of the evolving rate on the long time behavior of the solution to problem (1.6). There are corresponding results in the evolving domain. Theorem 4.2. Let ρ−2 = 1 T ∫ T 0 1 ρ2 dt. Then there is a positive constant Di = āi λ0 1 ρ−2 such that (i) if d1 ∈ [D1,+∞) and d2 ∈ [D2,+∞), then (1.7) admits a trivial solution which is globally asymptotically stable; (ii) if d1 ∈ (0, D1) and d2 ∈ (D2,+∞), then (1.7) admits a semi-trivial solution (V 41 , 0), which is a global attractor of problem (1.6); EJDE-2020/?? DYNAMICS OF A DIFFUSIVE COMPETITIVE MODEL 13 (iii) if d1 ∈ (D1,+∞) and d2 ∈ (0, D2), then (1.7) admits a semi-trivial solution (0, V 42 ), which is a global attractor of problem (1.6); (iv) if d1 ∈ (0, D1) and d2 ∈ (0, D2), then (1.7) admits the maximal and minimal periodic solutions, which are local attractors of problem (1.6). It can be found that Di are thresholds in terms of diffusion, and D∗i are that in a fixed domain, and from the expressions of D∗i and Di, we have the following assertions. Proposition 4.3. Recalling that D∗i = āi λ0 and Di = āi λ0 1 ρ−2 , we have (i) D∗i = Di if ρ−2 = 1; (ii) D∗i > Di if ρ−2 > 1; (iii) D∗i < Di if ρ−2 < 1. The above proposition implies that the evolution with a larger rate allows indi- viduals to move with more freedom so that benefits the survival of both species, which competes each other, while the evolution with a smaller rate goes against. 5. Numerical experiments In this section, we use Matlab to do some numerical simulations in terms of problem (1.6) to support the theoretical results obtained in section 4. To emphasis the impact of the evolution, we assume that the diffusion rates d1 = 0.2 and d2 = 0.1, intrinsic population growth rates a1 = a2 = 1.2, interspecific competition factors b1 = b2 = 0.013, intraspecific competition factors c1 = c2 = 0.012 and Ω(0) = (0, 1) followed with λ0 = π2. Set the evolution rate ρ(t) = 1−m| sinπt|, −1 < m < 1, and hence ρ−2  = 1, if m = 0, > 1, if 0 < m < 1, < 1, if − 1 < m < 0. Next, we select m for different evolution ratios of the domain and then observe the develop trends of v1 and v2. The situation of m = 0 will be presented at first for comparison. Example 5.1. Set ρ(t) = 1. Correspondingly, one has ρ−2 = 1. Meanwhile, from (4.2) it follows that R∗1 = ∫ T 0 a1(t)dt Td1λ0 = 1.2 0.2π2 ≈ 0.6079 < 1, R∗2 = ∫ T 0 a2(t)dt Td2λ0 = 1.2 0.1π2 ≈ 1.2159 > 1. According to Theorem 3.4 (iii), we know that v1 in such fixed domain will vanish, while v2 will survive. As what we have concluded, Figure 2 (a) shows that the variable v2 tends to a positive steady state while v1 tends to zero, which means that the species denoted by v2 will persist and v1 is vanishing as time goes on. 14 J. ZHU, J. ZHOU, Z. LIN EJDE-2020/86 (a) (b) (c) Figure 2. ρ(t) = 1. It is taken in a fixed domain. Graph (a) shows that the variable v2 stabilizes to an equilibrium while v1 van- ishes. Graphs (b) and (c), respectively, are the cross-sectional view and contour view of graph (a). Example 5.2. Set ρ(t) = 1 + 0.5| sin t|. Correspondingly, one has ρ−2 = 1 2 ∫ 2 0 1 (1 + 0.5| sin t|)2 dt ≈ 0.6020. Meanwhile, from (2.3) it follows that R1 = ∫ T 0 a1(t)dt d1λ0 ∫ T 0 1 ρ2(t)dt = 1.2 0.2π2 1 1 2 ∫ 2 0 1 (1+0.5| sin t|)2 dt ≈ 0.6079 0.6020 > 1, R2 = ∫ T 0 a2(t)dt d2λ0 ∫ T 0 1 ρ2(t)dt = 1.2 0.1π2 1 1 2 ∫ 2 0 1 (1+0.5| sin t|)2 dt ≈ 1.2159 0.6020 > 1. EJDE-2020/?? DYNAMICS OF A DIFFUSIVE COMPETITIVE MODEL 15 It follows from Theorem 3.4 (iv) that both v1 and v2 in such evolving domain will persist. As what we have concluded, Figure 3 (a) shows that the variables v1 and v2 tend to positive steady states. Figure 3 (b) and (c) are the corresponding cross- sectional view and contour one for v1 and v2, respectively, and they also clearly indicate not only that the variables v1 and v2 keep positive, but also that the domain, to which v1 and v2 belong to, is periodically evolving. (a) (b) (c) Figure 3. ρ(t) = 1 + 0.5| sin t|. For the bigger evolution ratio ρ(t), we acquire Ri > 1 (i = 1, 2), which results in the persistence of the competitive species for both sides. Graph (a) shows that both v1 and v2 stabilize to an equilibrium, and graphs (b) and (c) are the cross-sectional view and contour one, respectively. Also, we can clearly observe the periodic evolution of domain from (b) and (c). Example 5.3 Setting ρ(t) = 1− 0.3| sin t|, we have ρ−2 = 1 2 ∫ 2 0 1 (1− 0.3| sin t|)2 dt ≈ 1.5853. 16 J. ZHU, J. ZHOU, Z. LIN EJDE-2020/86 (a) (b) (c) Figure 4. ρ(t) = 1− 0.3| sin 5t|. For the smaller evolution ratio ρ(t), we acquire Ri < 1(i = 1, 2), which results in the vanishing of the two competitive species. Graph (a) shows that both v1 and v2 decay to the zero. The graphs (b) and (c) are the cross-sectional view and contour one of Graph (a), respectively, which shows the periodic evolution of the habitat. Meanwhile, from (2.3) it follows that R1 = ∫ T 0 a1(t)dt d1λ0 ∫ T 0 1 ρ2(t)dt = 1.2 0.2π2 1 1 2 ∫ 2 0 1 (1−0.2| sin t|)2 dt ≈ 0.6079 1.5853 < 1, R2 = ∫ T 0 a2(t)dt d2λ0 ∫ T 0 1 ρ2(t)dt = 1.2 0.1π2 1 1 2 ∫ 2 0 1 (1−0.2| sin t|)2 dt ≈ 1.2159 1.5853 < 1. 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Differential Equations, 2020 (36) (2020), 1-14. [27] X. Q. Zhao; Dynamical Systems in Population Biology, Second Edition, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, Cham, 2017. Jiazhen Zhu School of Mathematical Science, Yangzhou University, Yangzhou 225002, China Email address: luckyjiazhenzhu@foxmail.com Jiazheng Zhou Departamento de Matemática, Universidade de Braśılia, BR 70910-900, Braśılia-DF, Brazil Email address: zhoumat@hotmail.com, zhou@mat.unb.br Zhigui Lin (corresponding author) School of Mathematical Science, Yangzhou University, Yangzhou 225002, China Email address: zglin68@hotmail.com, zglin@yzu.edu.cn 1. Introduction and model formulation 2. Ecological reproduction index 3. Dynamics of periodic solutions 4. Impact of evolution 5. Numerical experiments References